UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
POSGRADO EN CIENCIAS FÍSICAS
FROM QUANTUM RADIAL ANHARMONIC OSCILLATOR TO ZEEMAN EFFECT IN
HYDROGEN ATOM
TESIS
QUE PARA OPTAR POR EL GRADO DE:
DOCTOR EN CIENCIAS (FÍSICA)
PRESENTA:
M. EN C. JUAN CARLOS DEL VALLE ROSALES
TUTOR PRINCIPAL:
DR. ALEXANDER TURBINER ROSENBAUM (ICN-UNAM)
COMITÉ TUTOR:
DR. JUAN CARLOS LÓPEZ VIEYRA (ICN-UNAM)
DR. JORGE ALEJANDRO REYES ESQUEDA (IF-UNAM)
CIUDAD DE MÉXICO, 2020
 
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iii
Abstract
In the present thesis, two fundamental systems of quantum mechanics are studied: (i) the D-
dimensional radial polynomial anharmonic potential, and (ii) a hydrogen atom subjected to a
uniform constant magnetic field. For both systems, the corresponding Schrödinger equation is
non-exactly solvable: closed analytical expressions (not in terms of infinite asymptotic series)
for the energies and wave functions may be found only in approximate form. This thesis presents
the general formalism to construct locally accurate approximations of the wave functions in the
coordinate space, which leads to highly accurate energies and expectations values.
The first part of this thesis is devoted to constructing a locally accurate approximation,
called the Approximant, for the wave functions of the low-lying states of the D-dimensional
radial polynomial anharmonic potential. Perturbation Theory in the weak and strong coupling
regimes, asymptotic expansions at small and large distances, and the true semi-classical approx-
imation are basic ingredients of the formalism to construct the Approximant. The key result of
this study is a compact expression for the exponential phase of the ground state wave function,
which is parameter-dependent. This expression is the building block for constructing the Ap-
proximants of the excited states. To fix the value of some parameters, the exact reproduction
of all growing terms of the exponential phase at large distances is imposed as constraint. This
procedure effectively reduces the number of parameters. Taking the Approximant as trial func-
tion, the remaining free parameters are optimized in the framework of the Variational Method.
In this manner, the energy is calculated as an expectation value of the Hamiltonian by means
of a locally accurate Approximant.
Explicit calculations are carried out for the first four low-lying states of the cubic, quartic, and
sextic anharmonic potentials of the form V (r) = r2+gm−2rm, withm = 3, 4, 6, respectively. For
the cubic case, a one-parametric Approximant leads to unprecedented accurate energies with a
relative accuracy of order . 10−5 with respect to the exact result. In turn, the relative deviation
of the Approximant from the exact wave function is . 10−2 at any point of the r-space. For
the case of the quartic/sextic anharmonicity, a 2/4-parametric Approximant leads to energies
with a relative accuracy of order . 10−8, while the relative accuracy of the Approximant is
of order 10−6 for any point of the r-space. Consistency of those results is checked inside the
Non-Linearization Procedure and the Lagrange-Mesh method.
In the second part, the formalism used for the radial anharmonic potential is employed to
construct a locally accurate Approximant for the ground state (1s0) of the hydrogen atom
in a uniform constant magnetic field. A non-relativistic consideration with a static proton
is assumed. The natural extension of the true semi-classical expansion was obtained for this
system. The main result of this part is a compact 8-parametric expression for the Approximant
of the ground state. When it is used as a trial function in the Variational Method, it leads to
highly accurate energy (relative accuracy . 10−6), and an accurate quadrupole moment in the
whole domain of accessible magnetic fields in Nature.
iv
Resumen
En la presente tesis se estudian dos sistemas fundamentales de mecánica cuántica: (i) el po-
tencial anarmónico radial D-dimensional, y (ii) un átomo de hidrógeno en presencia de un
campo magnético constante y uniforme. Para ambos sistemas, la ecuación de Schrödinger cor-
respondiente no admite soluciones exactas: las enerǵıas y las funciones de onda solamente se
pueden encontrar de forma aproximada. Esta tesis presenta el formalismo general para construir
aproximaciones localmente precisas de las funciones de onda en el espacio de coordenadas, esto
conduce a enerǵıas y valores de expectaćıon altamente precisos.
La primera parte de la tesis está dedicada a construir una aproximación localmente pre-
cisa (llamada Aproximante) para las funciones de onda de los primeros estados ligados del
potencial anarmónico radial D-dimensional. La teoŕıa de perturbaciones en los reǵımenes de
acoplamiento débil y fuerte, las expansiones asintóticas a pequeñas y grandes distancias, y la
verdadera aproximación semiclásica son los ingredientes básicos del formalismo para construir
el Aproximante. El resultado principal de este estudio es una expresión compacta para la fase
exponencial de la función de onda del estado base, la cual depende de ciertos parámetros.
Esta expresión es la pieza clave para construir Aproximantes de estados excitados. Para fijar
el valor de algunos parámetros, se impone como constricción la reproducción exacta de todos
los términos crecientes de la fase exponencial a grandes distancias. Este procedimiento reduce
efectivamente el número de parámetros. Tomando el Aproximante como función de prueba, los
parámetros restantes se optimizan usando el Método Variacional. De esta manera, la enerǵıa
se calcula como el valor esperado del Hamiltoniano evaluado por medio de un Aproximante
localmente preciso.
Se realizan cálculos expĺıcitos para los primeros cuatro estados ligados de los potenciales
anarmónicos cúbico, cuártico, y séxtico de la forma V (r) = r2 + gm−2rm, con m = 3, 4, 6,
respectivamente. Para el caso cúbico, un Aproximante con un solo parámetro conduce a enerǵıas
sin precedentes de una precisión relativa de orden 10−5 en cualquier punto del espacio r. A su
vez, la desviación relativa del Aproximante con respecto a la función de onda exacta es de orden
10−2. Para los casos de anarmonicidad cuártica/sextica, un Aproximante de 2/4 parámetros
conduce a enerǵıas de una precisión relativa de orden 10−8, mientras que la precisión relativa
del Aproximante es de orden 10−6 para cualquier punto del espacio r. La consistencia de esos
resultados se verifica usando el Procedimiento de No-Linealización y por medio del método de
la malla de Lagrange.
En la segunda parte, el mismo formalismo utilizado para el potencial anarmónico radial se
emplea para construir un Aproximante localmente preciso para el estado base (1s0) del átomo
de hidrógeno en un campo magnético constante y uniforme. Las consideraciones del estudio
son no-relativistas con un protón estático. El resultado principal de esta parte es una expresión
compacta para el Aproximante del estado base. Cuando el Aproximante se usa como una
función de prueba en el Método Variacional, este conduce a una enerǵıa altamente precisa (de
precisión relativa 10−6), y a un momento cuadrupolar preciso en todo el dominio de campos
magnéticos accesibles en la Naturaleza ( . 2.35× 1013G).
Contents
Overview ix
1 The Basic Tool: Variational Method in Quantum Mechanics 1
1.1 Variational Principle and Upper Bounds for the Energy . . . . . . . . . . . . . . 1
1.2 Variational Energy and Perturbation Theory . . . . . . . . . . . . . . . . . . . . 2
I Radial Anharmonic Oscillator 5
2 Introduction 7
2.1 Anharmonic Oscillator and Diatomic Molecules . . . . . . . . . . . . . . . . . . 7
2.2 Renewed Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Generalities 11
3.1 Radial Potential in D-dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Radial Polynomial Anharmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 13
3.3 The Ground State: Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3.1 Riccati-Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3.2 Generalized Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 The Weak Coupling Regime 19
4.1 Riccati-Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1.1 D-dimensional Radial Anharmonic Oscillator . . . . . . . . . . . . . . . . 21
4.2 Generalized Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Connection Between the Expansions for Y and Z . . . . . . . . . . . . . . . . . 25
4.4 Connection with WKB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 The Strong Coupling Regime 29
5.1 Riccati-Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1.1 Physically Relevant Trial Function . . . . . . . . . . . . . . . . . . . . . 31
5.1.2 Interpolation: Locally Accurate Approximation . . . . . . . . . . . . . . 32
v
vi Contents
5.2 Generalized Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6 The Approximant 37
6.1 The Approximant for the Ground State . . . . . . . . . . . . . . . . . . . . . . 37
6.2 Approximants for Excited States . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7 Results for Two-Term Potentials 41
7.1 Numerical Realization of the Variational Method . . . . . . . . . . . . . . . . . 41
7.2 Cubic Anharmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7.2.1 Perturbation Theory in the Weak Coupling Regime . . . . . . . . . . . . 43
7.2.2 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
7.2.3 The Approximant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7.2.4 Numerical Results: Variational Calculations . . . . . . . . . . . . . . . . 47
7.2.5 First Terms in the Strong Coupling Expansion . . . . . . . . . . . . . . . 52
7.3 Quartic Anharmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.3.1 Perturbation Theory in the Weak Coupling Regime . . . . . . . . . . . . 54
7.3.2 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.3.3 The Approximant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.3.4 Numerical Results: Variational Calculations . . . . . . . . . . . . . . . . 59
7.3.5 First Terms in the Strong Coupling Expansion . . . . . . . . . . . . . . . 62
7.4 Sextic Anharmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.4.1 Perturbation Theory in the Weak Coupling Regime . . . . . . . . . . . . 64
7.4.2 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.4.3 The Approximant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.4.4 Numerical Results: Variational Calculations . . . . . . . . . . . . . . . . 68
7.4.5 First Terms in the Strong Coupling Expansion . . . . . . . . . . . . . . . 72
8 Conclusions 75
II Hydrogen Atom in Constant Magnetic Field 81
9 Introduction 83
9.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.2 Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10 Generalities 87
10.1 The Ground State: Riccati Equation: . . . . . . . . . . . . . . . . . . . . . . . 89
10.2 Riccati-Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
10.3 Generalized Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Contents vii
11 Weak Coupling Regime 93
11.1 Riccati Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
11.1.1 Behavior of the Phase Φ at Small Distances . . . . . . . . . . . . . . . . 95
11.2 Generalized Bloch Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
11.2.1 Asymptotic Behavior of Φ at Large Distances . . . . . . . . . . . . . . . 97
11.3 Connection between RB and GB Equations . . . . . . . . . . . . . . . . . . . . 98
12 The Approximant 101
12.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
12.1.1 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
12.1.2 Quadrupole Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
13 Conclusions 109
General Conclusions 115
Appendices 119
A The WKB (Semi-Classical) Approximation 119
B First PT Corrections and Generating Functions 121
B.1 Cubic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
B.2 Quartic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
B.3 Sextic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C Plots of Variational Parameters: Anharmonic Oscillator 125
C.1 Cubic: (7.2.24) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C.2 Quartic: (7.3.26) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
C.3 Sextic: (7.4.27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
D General Aspects of the Two-Term Radial Anharmonic Oscillator 130
E System of Coordinates {ρ, r, ϕ} 131
E.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
E.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
F Perturbation Theory for Hydrogen Atom in Constant Magnetic Field 133
F.1 Computational Code: Non-Linearization Procedure . . . . . . . . . . . . . . . . 133
F.2 Corrections in Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . 134
G Computational Code: Hydrogen Atom in Magnetic Field 137
viii Contents
H Lagrange-Mesh Method for Hydrogen Atom in Magnetic Field: Code 143
I Plots of Variational Parameters: Hydrogen Atom in Magnetic Field 145
Bibliography 153
Overview
It is well-known that the time-independent Schrödinger equation can be solved exactly for a
few physical systems. Therefore, we need to use different approximation methods to handle
the overwhelming majority of quantum-mechanical problems. In fact, much of the success of
quantum mechanics has been achieved thanks to approximation methods. Among them, the
Variational Method is one of the most powerful tools which allow us to deal with many physical
systems of different nature. In all books on quantum mechanics, e.g. [1–3], the Variational
Method plays a central role among the approximation methods presented. In the variational
approach, the trial function is key for leading to approximations of the spectrum of a given
system. A natural question is: how to choose an adequate trial function that yields reliable
approximate energies and wave functions? The Variational Method per se does not provide a
prescription about the choice of a trial function. Thus, the construction of a trial function is
handmade, usually based on physical intuition. Symmetries of the system, perturbation theory,
asymptotic analysis, semi-classical approximations, features of nodal surfaces, etc., are sources
of information of the exact function that may be coded in the trial one. Nevertheless, one could
choose a trial function based on technical arguments. For example, one may choose a trial
function in such a way that all integrals involved in the realization of the Variational Method
are calculated in closed analytical form or reduced to one-dimensional integrals. In fact, most
of the trial functions chosen in atomic physics follow this line.
Any trial function represents an approximate solution of the Schrödinger equation. With
an approximate solution, one can calculate the expectation values of relevant operators. It is
known that a high accuracy reached in the variational energy does not guarantee high accuracy
in such expectation values, see [4]. This reflects one fact: a trial function may be close to
the exact one in the domain where the dominant contribution of the integrals (involved in the
Variational Method) occurs, but it may differ in the domains required by the calculation of the
expectation value. Let us discuss a simple example, see [5], that illustrates these phenomena
described above.
We consider a particular system where the exact solutions are known: a hydrogen atom. The
exact ground state wave function is well-known1
ψexact(r) = e−r ,
1Atomic units used: ~ = me = −e = 1, where me and e are the mass and the electric charge of the electron,
respectively.
ix
x Overview
as well as its energy2
E = −1Ry .
Suppose we are not aware of the exact solution, and we want to use the Variational Method to
estimate the ground state energy. In particular, we can consider a Gaussian trial function
ψI(r) = e−α r2
where α is a positive variational parameter. With this trial function, the realization of the
Variational Method is completely analytical. The optimal parameter α turns out to be α =
8
9π
,
and it leads to the following variational energy
Evar = −0.849Ry .
This variational estimate deviates from the exact value around 15%. This result is surprising
due to the high accuracy of the variational estimate and the simplicity of the trial function.
The deviation can be dramatically reduced to 3% by considering a two-term superposition of
the previous trial function, namely
ψII(r) = e−α r2 + γ e−β r2 , α, β > 0 .
Here α, β, and γ are free parameters that are fixed by means of the Variational Method.3 With
this new trial function, the variational estimate of the energy is
Evar = −0.972Ry .
One could continue to reduce the deviation taking as trial function a larger combination of
Gaussian functions. However, no matter how many Gaussian functions we consider, it is clear
that their sum will never converge to ψexact. This simple example evidences one important fact:
although a trial function can provide an accurate estimate of the energy, the trial function with
optimal parameters may not be very close to the exact one. In our particular example, it can
be roughly seen if we plot the exact solution and compare it with the trial ones, see Fig. 1.
The closeness of the trial functions to the exact one may be qualitatively studied by means
of the relative deviation (Rd), which is defined as
Rd(r) =
∣
∣
∣
∣
ψexact(r)− ψi(r)
ψi(r)
∣
∣
∣
∣
, i = I, II . (1)
For both ψI and ψII , the plot of Rd as a function of r is presented in Fig. 2. In the domain 0 ≤
r . 3.5, the dominant contribution of the integrals (required by the Variational Method) occurs
for both trial functions. Inside this domain, the relative deviation reaches a local maximum:
∼ 0.6 and ∼ 0.3 for ψI and ψII , respectively. Therefore, in percentage terms, ψexact may
deviate from ψI around 60%, and 30% from ψII . Outside this domain, the relative deviation is
a monotonically increasing function for both trial functions. Due to the large relative deviation
xi
0 1 2 3 4
0.5
0.25
0.75
1
Figure 1: Plot of ψexact, ψI , and ψII . As normalization, we set ψexact(0) = ψI(0) = ψII(0) = 1.
0 1 2 3 4 5
1
2
3
4
Figure 2: Relative deviation of the functions ψI and ψII (from ψexact) as a function of r
throughout the domain, we may say that neither ψI nor ψII are locally accurate approximations
of ψexact.
Non-locally accurate approximation may lead to inaccurate estimates of some expectation
values. For instance, let us now compute 〈r2−n〉 using ψexact, ψI , and ψII . The numerical results
are presented in Table 1. One can note that the deviation of the variational results, compared
with the exact one, is significant for n > 5: it exceeds 30%. This deviation contrasts with the
fact that for the trial functions ψI and ψII , the deviation of the variational energy from the
exact one is 15% and 3%, respectively. The calculation of expectation values of more complex
functions than simple monomials could be catastrophic for reaching highly accurate results. For
example, one relevant quantity which involves expectation values is the cusp parameter C,
C ≡
〈
ψ
∣
∣ δ(~r) ∂
∂r
∣
∣ψ
〉
〈ψ | δ(~r) |ψ〉 ,
2Ry stands for the Rydberg constant, 1 Ry ≈ 13.6 eV.
3The optimal choice of (rounded) parameters is α = 0.20, β = 1.33, and γ = 1.38. The integration required by
the Variational Method can be done analytically, while the optimization of parameters is a numerical procedure.
xii Overview
Table 1: Expectation values 〈rn−2〉 for different values of n using ψexact, ψI , and ψII . Interest-
ingly, at n = 3 the trial function ψI provides the exact result. Results rounded to first displayed
s. d. Atomic units used.
n ψexact ψI ψII
0 2 1.13 1.71
1 1 0.85 0.97
3 1.5 1.5 1.48
4 3 2.65 2.80
5 7.5 5.30 6.28
6 22.5 11.71 16.01
7 78.75 28.11 44.90
see [6]. In atomic physics, this parameter is usually employed to measure the local quality of
the approximate wave function near the Coulomb singularities. Using the exact wave function
of the hydrogen atom ψexact, the exact cusp parameter results in
Cexact = −1 .
This is an exact number that the ground state wave function of the hydrogen atom must provide.
On the other hand, if we compute the cusp parameter associated with the approximate functions
ψI and ψII , it is zero for both! This is nothing but a reflection of the fact that the logarithmic
derivative of ψI and ψII do not have the correct asymptotic behavior at small r. In fact, the
difference between ψexact, ψI , and ψII is more pronounced when we plot their corresponding
logarithmic derivatives, see Fig. 3.
0 1 2 3 4
0.5
1
1.5
2
Figure 3: Logarithmic derivative of ψexact, ψI , and ψII .
This simple and illustrative example exhibits some important niceties that should be taken
xiii
into account when designing trial functions. However, it has to be noted that the above dis-
cussion is based on the knowledge of the exact energy and wave function, which is frequently
unknown. So, the question is: how to estimate the accuracy of the approximate energy (wave
function) that the Variational Method provides when the exact energy (wave function) is not
available? The answer to this highly non-trivial question lies in the connection between the vari-
ational energy and perturbation theory discovered in [7]. This connection plays a fundamental
role in this thesis.
The present work aims to construct locally accurate trial functions for one fundamental sys-
tem: the quantum radial polynomial anharmonic oscillator in arbitrary dimension D. This
study is strongly motivated by the work presented in [8]. Due to the separation of the angular
variables, this D-dimensional system is reduced to a one-dimensional problem. We present a
formalism to design locally accurate trial functions of any state for an arbitrary anharmonic
potential. The formalism culminates in a simple expression that produces trial functions that
lead to unprecedented results in the weak and strong coupling regime: highly accurate estimates
of the exact energies, and locally accurate approximate wave functions. It is demonstrated that
an extension of these ideas and the formalism may be applied to a different physical system: the
hydrogen atom in a constant uniform magnetic field in its ground state. The formalism leads
to an adequate trial function that yields highly accurate results for the energy and quadrupole
moment for all magnetic fields available in Nature. For both systems, the obtained approxi-
mate wave functions actually represent approximate solutions of the corresponding Schrödinger
equation. In fact, they can be considered as exact and compact solutions inside their domain
of applicability. This will be discussed in detail for both systems.
The thesis is divided in two parts. Part I is devoted to the D-dimensional quantum radial
anharmonic oscillator. The hydrogen atom in a constant magnetic field is studied in the Part
II. It is worth mentioning that Part I and II can be read independently since they are self-
contained. Each one begins with a short Introduction and ends with Conclusions. Regarding
Part I, some of the results shown in this thesis were presented in [9, 10]. For Part II, they will
appear in [11].

Chapter 1
The Basic Tool: Variational Method in
Quantum Mechanics
In this Chapter, we present a brief and general description of the Variational Method (VM) in
the framework of quantum mechanics. We focus the discussion on the ground state. However,
when additional information about the nodal surface is available, it may be extended to study
excited states.
1.1 Variational Principle and Upper Bounds for the En-
ergy
Let us consider an arbitrary Hamiltonian Ĥ which describes a particular system of interest.
We assume that the wave function of the ground state, as well as the corresponding energy, are
unknown. If the Schrödinger equation of Ĥ is non-solvable, the natural question is: How to
find the energy and wave function approximately? The variational principle gives us an answer
in terms of the variational energy, which is defined as
Evar[ψ] =
∫
ψ∗
t Ĥψt dV
∫
ψ∗
tψt dV
. (1.1)
Here, ψt is an arbitrary normalizable function,
∫
ψ∗ψ dV <∞ , (1.2)
usually called trial function. The importance of the variational energy is established via the
Variational Principle.
Variational Principle. Consider a system described by the Hamiltonian Ĥ with exact ground
state energy E0, then the variational energy satisfies the inequality
Evar[ψt] ≥ E0 (1.3)
1
2 Chapter 1. The Basic Tool: Variational Method in Quantum Mechanics
for any trial function ψt. Besides, the equality is fulfilled if and only if ψt is the exact ground
state wave function of the system.
Proof. The trial function ψt can be expanded in terms of the (unknown) exact basis of or-
thonormal eigenfunctions {ψk} of Ĥ. Thus,
ψt =
∑
k
ckψk , Ĥψk = Ekψk , (1.4)
where ck are some coefficients, and Ek is the eigenvalue associated to ψk. Without loss of
generality, we assume that eigenvalues are sorted, E0 ≤ E1 ≤ E2 ≤ ... . Hence, the variational
energy (1.1) reads
Evar[ψt] =
∑
k |ck|2Ek
∑
k |ck|2
. (1.5)
Since E0 is the lowest energy, the following inequality holds
∑
k
|ck|2Ek ≥ E0
∑
k
|ck|2 . (1.6)
Using it in (1.5), it follows the variational principle
Evar[ψt] ≥ E0 . (1.7)
From another side, it is clear that the equality is fulfilled if and only if ψt is the exact ground
state wave function of Ĥ. In this case, all coefficient ck is zero except c0.
As one could imagine, the variational principle is a useful tool to estimate upper bounds of
the exact ground state energy. The lower upper bound, the closer Evar is to the exact energy.
In practice, the trial function can depend on a finite number of free parameters. In this manner,
the variational energy will also be parameter-dependent. A proper choice of parameters may
lead to the lowest upper bound that a given trial function can provide. This choice of parameters
is known as the optimal configuration. The efficient way to find it is through an optimization
procedure, which is, for most of the adequate trial functions, a numerical procedure. The
Variational Method covers from the construction of the trial function to the calculation of the
lowest upper bound.
1.2 Variational Energy and Perturbation Theory
We now discuss an interesting connection between the variational energy and Perturbation
Theory (PT). For a given trial function ψt, it is straightforward to calculate the Hamiltonian
Ĥt for which ψt is an exact ground state eigenfunction. We call this procedure the Inverse
Problem. In fact, since the kinetic operator T̂ of the Hamiltonian is usually given, we can
1.2. Variational Energy and Perturbation Theory 3
focus only on the construction of the associated potential1 Vt of such Hamiltonian. Thus, let
us assume that Ĥt is of the form
Ĥt = T̂ + Vt . (1.8)
Since ψt is the exact eigenfunction of Ĥt, it is clear that it obeys
Ĥtψt = T̂ψt + Vtψt = E
(t)
0 ψt , (1.9)
where E
(t)
0 is the ground state energy2 of Ĥt. From equation (1.9) we can trivially solve for the
(unknown) Vt,
Vt = E
(t)
0 − T̂ψt
ψt
. (1.10)
Since E
(t)
0 can be absorbed in the definition of Vt, its value is unimportant. It is evident that
when solving the inverse problem, we can generate zillions of potentials for which a single
eigenstate - the lowest - is known exactly. Once Vt is known, the connection between the
variational energy and PT can be established.
Suppose that we are interested in the ground state of the operator
Ĥ = T̂ + V . (1.11)
However, it turns out that the corresponding Schrödinger equation
Ĥψ = Eψ (1.12)
is non-solvable. To estimate E, we use the VM with a trial function ψt as entry. Thus, we
calculate the variational energy associated to ψt
Evar[ψt] =
∫
ψ∗
t (T̂ψt + V ψt) dV
∫
ψ∗
tψt dV
. (1.13)
Using (1.10), we can write (1.13) as follows
Evar[ψt] = E
(t)
0 +
∫
ψ∗
t (V − Vt)ψt dV
∫
ψ∗
tψt dV
. (1.14)
If we define
E
(t)
1 =
∫
ψ∗
t (V − Vt)ψt dV
∫
ψ∗
tψt dV
, (1.15)
the expression shown in (1.14) acquires a more compact form,
Evar[ψt] = E
(t)
0 + E
(t)
1 . (1.16)
This result suggests reinterpreting the variational energy as the first two terms of an expansion
in PT developed for the perturbed Hamiltonian
Ĥ = T̂ + Vt + λ (V − Vt) . (1.17)
1In this work, potentials are always real functions.
2The meaning of the subscript will be clarified, for now, it is just notation.
4 Chapter 1. The Basic Tool: Variational Method in Quantum Mechanics
Here, λ is a formal parameter that ultimately should be placed equal to one, λ = 1. The
meaning of the superscripts placed in (1.16) is now clear: E
(t)
0 plays the role of the zero-order
correction of the energy, while E
(t)
1 - the expectation value of perturbation potential (V − Vt) -
is the first order correction. To summarize,
The computation of the variational energy of a given Hamiltonian Ĥ = T̂ + V , taking as trial
function ψt, is equivalent to calculate the first two terms in a formal PT. The trial function ψt
is the exact ground state of the unperturbed Hamiltonian Ĥt = T̂ + Vt, while V − Vt plays the
role of perturbation.
If higher order corrections, E
(t)
2 , E
(t)
3 , ..., are known, they may be used to estimate the accuracy
of the variational energy. In principle, the exact value of the lowest energy of Ĥ is the sum of
those corrections,
E =
∞∑
n=0
E(t)
n . (1.18)
In general, the question about the convergence of the expansion (1.18) is crucial but difficult
to answer. We restrict ourselves by saying that if ψt is chosen with the appropriate asymptotic
behavior at large distances, a convergent expansion will occur. In this situation, our initial
variational estimate of the energy can be improved by adding next-order corrections. In practice,
it is sufficient to calculate few terms in the expansion (1.18) to estimate rates of convergence as
well as highly accurate estimates of the exact energy via partial sums. For all one dimensional
physical systems, a considerable number of corrections E
(t)
n can be calculated. This situation
changes dramatically when we deal with true multidimensional systems.
The calculation of any energy correction, i.e. E
(t)
2 , E
(t)
3 , ..., requires the knowledge of the
corrections to the wave function in PT. In this manner, using corrections for ψt, the deviation
(between the trial function and exact one) can also be analyzed and, more important, estimated.
Part I
Radial Anharmonic Oscillator
5

Chapter 2
Introduction
2.1 Anharmonic Oscillator and Diatomic Molecules
The first modern study of the one-dimensional quantum anharmonic oscillator dates back to
1925. This study was carried out by Heisenberg and presented to the scientific community
in [12]. In this work, Heisenberg considered the cubic and quartic anharmonic oscillators
described by the classical equations of motion
ẍ + w2
0 x + λx2 = 0 ,
ẍ + w2
0 x + λx3 = 0 ,
respectively. We have denoted the coupling constant as λ, in turn, w0 is the natural frequency
of the harmonic oscillator. In the framework of what would later be called Matrix (quantum)
Mechanics, he presented not only the energy spectrum of the harmonic oscillator (λ = 0) for
the first time, but also gave the first corrections to the energy in PT in powers of λ. For the
cubic case, a first order (vanishing) correction was obtained. For the quartic case, the first two
corrections were presented. Results at λ = 0 were confirmed in the framework of quantum
mechanics exposed by Schrödinger just one year later [13]. Around those years, the main
motivation to study anharmonic perturbations of the harmonic oscillator came from atomic
and molecular physics. A concrete example is the discovery of the spectral band at 1.76µm
of the diatomic molecule HCl (hydrogen chloride). This result, obtained by Brinsmade and
Kemble in 1917 [14], was of importance since it established that the HCl molecule behaves as
an anharmonic oscillator and showed how this could be interpreted on the basis of the early
quantum theory introduced by Planck. Once the Schrödinger equation was presented to the
world in 1926, the diatomic molecule described by a 3-dimensional anharmonic potential was
studied immediately, see e.g. [15]. Due to the exactly solvable nature of the one-dimensional
Morse potential, it was widely used to model the interaction between the two atoms of a
diatomic molecule near their equilibrium position [16,17]. The Morse potential is given by
V (r) = V0(e
−2a(r−r0) − 2 e−a(r−r0)) ,
7
8 Chapter 2. Introduction
where V0 is the energy of dissociation, r the nuclear separation, r0 the equilibrium value of r,
and a is a positive constant. From the Taylor expansion of this potential around r0,
V (r) = V0
(
−1 + a2(r − r0)
2 − a3(r − r0)
3 +
7a4
12
(r − r0)
4 + . . .
)
, (2.1.1)
one can easily check that the Morse potential contains an infinite number of anharmonic terms.
Following the vibrational spectroscopic information, one can determine parameters a, V0, and
r0. For example, the diatomic molecule BeO (beryllium oxide, see [16]) in the state Σ1 is
characterized by
a = 2.12 a−1
0 , V0 = 5.245 eV , r0 = 1.27 a0 ,
where a0 = 5.29×10−9cm is the Bohr radius. As far as vibrational energy levels are concerned,
the Morse potential provides accurate results. However, when the information given by the
rotational energy levels is taken into account, certain discrepancies are found [15]. A natu-
ral direction, to avoid such discrepancies, was to consider a more general radial anharmonic
polynomial potential1 of the form
V (r) = K(r − r0)
2
(
1 + b (r − r0) + c (r − r0)
2 + d (r − r0)
3 + . . .
)
,
with K, b, c, d, ..., being real parameters called anharmonic constants. With this kind of
potential, the corresponding Schrödinger equation is non-exactly solvable: energies and wave
functions can be found only in approximate form. The first approaches to solve it (∼1926) were
based on the Bohr theory [18], and second order PT [19]. Over the next years, several studies
were focused on determining the first anharmonic constants of several diatomic molecules using
those approaches.
2.2 Renewed Interest
As we have seen, immediately after the foundation of quantum mechanics, the study of anhar-
monic oscillators was strongly motivated by molecular physics. From a more general point of
view, it was recognized that the spectral problem of a quantum anharmonic oscillator has very
interesting properties. The publication of seminal papers by Bender and Wu [20] (from 1969)
refreshed the interest in the quantum mechanical properties of anharmonic oscillators. First,
they studied the one-dimensional quartic anharmonic oscillator characterized by the quantum
Hamiltonian
Ĥ = − d2
dx2
+
1
4
x2 +
1
4
λx4 ,
where λ is a positive coupling constant. In particular, this Hamiltonian is interesting because
it allows us to demonstrate that the perturbation series of the energy in powers of λ diverges
for any state. They investigated the reason for this divergence using an analytical continuation
of the energy levels into the complex λ-plane as well as the WKB approximation. It turned out
1Series are truncated up to some order.
2.3. Present Work 9
that the presence of an infinite number of branch points with a limit point at λ = 0 was the
reason for this divergence. A generalization of this study was presented a few months later by
Bender [21]. This time he considered a more general case
Ĥ = − d2
dx2
+
1
4
x2 +
1
2M
λx2M ,
where M = 2, 3, ... . Analytical features of the energy levels were investigated in this work. He
concluded that those properties are qualitatively similar to the case M = 2. Interestingly, it
was shown that the exact solution exists in the limit M → ∞. In a following paper (1973),
Bender and Wu [22] developed two independent mathematical techniques for determining the
large-n behavior of PT for the quartic anharmonicity. In fact, asymptotic expressions at large
n of perturbation coefficients were immediately called Bender-Wu formulas.
After the publication of their results, thousands of physical systems were suddenly connected
in one way or another. For example, the Hamiltonian of the one-dimensional quantum quartic
anharmonic oscillator represents a φ4 scalar quantum field theory in (0 + 1) dimensions. Many
properties that appear in (0 + 1) dimension, like divergence in PT, can also be present in
a realistic field theory. The apparently simple problem of a quantum anharmonic oscillator
revealed an extremely rich internal structure, which looks intrinsic for any non-trivial system
in quantum mechanics. In particular, they formally evidenced the impossibility of finding the
spectrum in PT in powers of λ. These facts encouraged the development of various numerical
and analytical non-perturbative methods over the next years. Thus, the anharmonic oscillator
became a basic system to test new methods. That explains why, from the early ’70s until today,
one can find many papers devoted to the exploration of the quantum spectra of the anharmonic
potential2. It is worth mentioning some of the methods used throughout the years: Variational
Method [23, 24], Rayleigh-Ritz method [25], PT in both the weak and strong coupling regimes
[20,22,26–30], Padé Approximants [31], Hill Determinant method [32–34], WKBmethod [35,36],
Self-Similar Approximations [37], the method of Characteristic Functions [38], among many,
many others.
2.3 Present Work
From previous considerations, it is clear that the anharmonic oscillator has a fundamental
interest in physics. The present study is devoted to studying the energy spectra of the D-
dimensional radial polynomial anharmonic oscillator. One may ask: why to consider it in
D-dimension? Of course, the physical spatial dimensions, D = 1, 2, 3, are evidently the most
popular. However, as it will be shown, in our study the dimension D does not play an essential
role in the formalism that we present below: it appears only as an external parameter. We
may easily deal with an arbitrary D, and ultimately set its value to the dimension of interest.
Furthermore, some non-trivial analytical properties of the energy as a function of D appear at
both physical and not-physical dimensions, [39, 40].
2Specially in one-dimension for quartic and sextic cases.
10 Chapter 2. Introduction
This work aims primarily to study the ground state of the radial anharmonic oscillator in ar-
bitrary D using as tools PT in the weak and strong coupling regimes, asymptotic series at small
and large distances, and a new version of the semi-classical approximation. In general, here we
present the formalism to study an arbitrary polynomial anharmonic oscillator. Nevertheless,
as we will see in Part II, it can be applied to physical systems with different nature.
Our ultimate goal is to construct a locally-accurate approximation of the ground state wave
function in arbitrary (integer) dimension D > 0. We will call this approximation the Approx-
imant, and we will denote it by Ψ
(t)
0,0. Moreover, we will show how the Approximant Ψ
(t)
0,0 can
be modified to construct Approximants for the wave functions of excited states. It will be
demonstrated that when an Approximant is used as a trial function in the Variational Method
(VM), it leads to a highly accurate estimate of the energy with a very small deviation from
the exact result for all coupling constant. In fact, some of our results will serve as benchmarks
for future works. Furthermore, a fast-convergent PT occurs for both the energy and the wave
function if the trial function is used as the zeroth order approximation in a PT scheme3. This
is explicitly checked for three anharmonicities: cubic, quartic, and sextic.
3By means of the connection between PT and the variational energy discussed in the previous Chapter.
Chapter 3
Generalities
In the forthcoming Sections, we present some general results that will be useful for our study
of the D-dimensional anharmonic oscillator.
3.1 Radial Potential in D-dimension
The time-independent Schrödinger equation for a particle subjected to a spherical-symmetric
potential V (r) in a D-dimensional space (D = 1, 2, ...) has the form
[
− ~2
2M
∇2
D + V (r)
]
ψ = E ψ . (3.1.1)
Here ~ is the Planck constant, M is the mass of the particle,
∇2
D =
D∑
k=1
∂2xk
(3.1.2)
is the D-dimensional Laplace operator in Cartesian coordinates {x1, x2, ...xD},
r =
√
x21 + x22 + ...+ x2D , xi ∈ (−∞,∞) , (3.1.3)
i = 1, ..., D, is the radius in D-dimension1, while ψ and E represent wave function and its
corresponding energy, respectively. Due to the spherical symmetry of the potential V (r), in
D > 1 it is convenient to introduce D-dimensional hyperspherical coordinates2 {r,Ω}, with r
as in (3.1.3) and Ω parameterized with (D− 1) Euler angles. In these coordinates, the Laplace
operator ∇2
D takes the form
∇2
D = ∂2r +
D − 1
r
∂r +
∆SD−1
r2
, ∂r ≡
∂
∂r
, (3.1.4)
1In the present work, we assume that at D = 1 the radius becomes r = |x|, see (3.1.3), thus V = V (|x|).
2See [41,42].
11
12 Chapter 3. Generalities
where ∆SD−1 is the Laplacian on the SD−1 sphere. Naturally, if we choose in (3.1.4) a dimension
D = 2, 3, we obtain the familiar expressions of the Laplacian in polar and spherical coordinates,
respectively. For the purposes of the present work, the explicit form of the operator SD−1 is not
needed. It is enough to know that the operator ∆SD−1 is closely related with the D-dimensional
orbital angular momentum L̂,3
∆SD−1 = −L̂2 . (3.1.5)
This relation4 brings two remarkable consequences at D > 1:
(i) For any spherically symmetric potential, the D-dimensional angular momentum L̂ is
conserved.
(ii) In hyperspherical coordinates, we can separate the hyperradial coordinate r from the
angular ones (Ω).
Hence, in D > 1 the wave function ψ and its energy E can be labeled by one radial quantum
number nr, one angular quantum momentum ℓ, and (D − 2) magnetic quantum numbers. As
a consequence, the wave function is represented as the product of two functions
ψnr,ℓ,{mℓ}(r,Ω) = Ψnr,ℓ(r) ξℓ,{mℓ}(Ω) , (3.1.6)
where nr = 0, 1, 2, . . . , ℓ = 0, 1, 2, . . . . The set of (D − 2) magnetic quantum numbers {mℓ},
where each of them takes values from −ℓ to +ℓ, contains N (D, ℓ) different configurations, where
N (D, ℓ) =



(2ℓ+D − 2) (ℓ+D − 3)!
ℓ! (D − 2)!
, ℓ ≥ 1,
1 , ℓ = 0 .
(3.1.7)
For D > 1, the angular part of the wave function ξℓ,{mℓ}(Ω) corresponds to a D-dimensional
spherical harmonic [41, 42] which satisfies the spectral problem
−∆SD−1 ξℓ,{mℓ} = L̂2 ξℓ,{mℓ} = ℓ (ℓ+D − 2) ξℓ,{mℓ} . (3.1.8)
Hence, for given ℓ this equation is satisfied by N (D, ℓ) different linear independent spherical
harmonics, which correspond to the eigenvalue ℓ (ℓ+D−2) with degeneracy5 equals to N (D, ℓ).
3The D(D − 1)/2 independent components of the D-dimensional angular momentum tensor are
Lij = xipj − xjpi , i < j ,
where xi and pi are the ith coordinate and the linear quantum momentum, respectively.
4The operator L̂2 is defined as
L̂
2 =
N∑
j=2
j−1
∑
i=1
L2
ij
in terms of its components, see [43].
5Note that for D = 3, the eigenvalue of L̂2 and its degeneracy are given by the familiar formulas ℓ(ℓ + 1)
and N (3, ℓ) = 2ℓ+ 1, respectively.
3.2. Radial Polynomial Anharmonic Oscillator 13
Not surprisingly, two spherical harmonics with different quantum numbers and same ℓ are
orthogonal,
∫
Ω
ξℓ,{mℓ}ξℓ,{m′
ℓ}
dΩ = δmℓ,m
′
ℓ
(3.1.9)
where δmℓ,m
′
ℓ
is the Kronecker delta, and dΩ is the solid angle in D−1 dimensions. Furthermore,
two spherical harmonics with different angular momentum, say ℓ and ℓ′, are orthogonal too.
Further properties of the spherical harmonics can be found in [41,42]. However, those properties
do not play an important role in the present work. Since the theory of the spherical harmonics
is well established, we have to focus solely on determining the radial part of the wave function
Ψnr,ℓ(r).
After substituting (3.1.4), (3.1.6) and (3.1.8) into (3.1.1), we arrive to a radial Schrödinger
equation6
[
− ~2
2M
(
∂2r +
D − 1
r
∂r −
ℓ (ℓ+D − 2)
r2
)
+ V (r)
]
Ψnr,ℓ(r) = Enr,ℓ Ψnr,ℓ(r) , (3.1.10)
which determines the radial wave function Ψnr,ℓ(r). For any radial potential, the energy Enr,ℓ of
an excited state is always degenerate with respect to the quantum numbers {mℓ}. Specifically,
for a given (nr, ℓ) and D > 1 we have N (D, ℓ) different wave functions with the same energy.
Note that the magnetic quantum numbers do not play an essential role. For this reason, since
now on, we get rid of the label {mℓ} in the energy and wave function.
3.2 Radial Polynomial Anharmonic Oscillator
From now on, we consider the radial polynomial anharmonic potential7
V (r) =
1
g2
m∑
k=2
ak g
k rk , m ≥ 2 , (3.2.1)
where g is a non-negative coupling constant, ak are real parameters. In addition, we consider
only potentials such that V (r) > 0 at r > 0, with a2 and am positive. Hence, the minimum at
r = 0 is global. For m > 3 there may be some other (local) minimums at r > 0. For the sake of
simplicity, we assume that none of them is degenerate. Evidently, under these considerations
the Hamiltonian
Ĥ = − ~2
2M
∇2
D +
1
g2
m∑
k=2
ak g
krk, (3.2.2)
6It is known that by means of a gauge rotation, equation (3.1.10) can be transformed into a one-dimensional
Schrödinger one with an effective potential. However, we do not follow this approach since it does not provide
advantages in the present study.
7Note that any radial polynomial potential, with a leading quadratic term at small r, can be rewritten as in
(3.2.1) employing scalings in coefficients ak.
14 Chapter 3. Generalities
has an infinite and discrete spectrum: the potential (3.2.1) admits bound states. Let us note
that the potential (3.2.1) can be written in a compact form as
V (r) =
1
g2
V̂ (gr) , (3.2.3)
where we have defined
V̂ (gr) =
m∑
k=2
ak (gr)
k . (3.2.4)
As we discussed in the previous Section, separation of variables occurs for the Schrödinger
equation associated to Hamiltonian (3.2.2). Therefore, we only have to determine the radial
part of the wave function using the Schrödinger equation (3.1.10). We are not trying to find the
complete spectra of the Hamiltonian operator (3.2.2). Our study is devoted to the low-lying
states. However, as we will see through these pages, the ground state plays the most important
role between the low-lying states.
3.3 The Ground State: Riccati Equation
The ground state of any spherical symmetricD-dimensional Hamiltonian is non-degenerate (and
nodeless). It has zero angular momentum ℓ = 0 (S-state) and zero radial quantum number
nr = 0. The angular part corresponds to the zeroth spherical harmonic (which is a constant)
with zero eigenvalue in (3.1.8). From (3.1.10), it can be seen that the ground wave function
depends only on the radial coordinate r, we will denote it without labels by Ψ(r). For the D-
dimensional radial anharmonic oscillator (3.2.1), the ground state wave function corresponds
to the lowest energy eigenfunction of the radial Schrödinger operator
ĥr = − ~2
2M
(
∂2r +
D − 1
r
∂r
)
+
1
g2
V̂ (gr) . (3.3.1)
The corresponding spectral problem, with explicit boundary conditions, read
ĥr Ψ(r) = EΨ(r) ,
∫ ∞
0
Ψ2 rD−1dr < ∞ , Ψ(0) = 1, Ψ(∞) = 0 . (3.3.2)
Here E denotes the ground state energy. It is interesting to note that D may be considered as
a continuous parameter in (3.3.2). It will be shown that some non-trivial analytical properties
of the ground state energy as a function of D appear at non-physical dimensions, see e.g. [39]
and [40].
To solve the spectral problem (3.3.2), it is convenient to adopt the so-called exponential
representation,
Ψ(r) = e−
1
~
Φ(r). (3.3.3)
The function Φ(r) has dimension of action, and it is usually called the phase of the wave
function. Thus, instead of looking for Ψ, we have to find the phase Φ. From (3.3.2), it can
3.3. The Ground State: Riccati Equation 15
be shown that Φ satisfies a non-linear ordinary differential equation of first order and second
degree,
~ ∂ry(r) − y(r)
(
y(r)− ~(D − 1)
r
)
= 2M
[
E − 1
g2
V̂ (r)
]
, (3.3.4)
where the function y(r) is nothing but the derivative of Φ(r)
y(r) = ∂rΦ(r) . (3.3.5)
In fact, y(r) coincides with the logarithmic derivative of the wave function Ψ,
y(r) = −~ ∂r (logΨ(r)) . (3.3.6)
Equation (3.3.4) is a particular case of the familiar Riccati (differential) equation. It shows
that the logarithmic derivative y(r) is more fundamental than the wave function Ψ, and even
more than the phase. The Riccati equation is the key equation of this Chapter.
3.3.1 Riccati-Bloch Equation
The Riccati equation (3.3.4) can be transformed into one without the explicit dependence on
the constants ~ and M . This is achieved by introducing the quantum coordinate8
v =
(
2M
~2
) 1
4
r , (3.3.7)
and making the replacements
y = (2M~
2)
1
4 Y (v) , E =
~
(2M)
1
2
ε . (3.3.8)
In this way, the equation (3.3.4) becomes
∂vY(v) − Y(v)
(
Y(v)− D − 1
v
)
= ε (λ) − 1
λ2
V̂ (λv) , ∂v ≡
d
dv
, (3.3.9)
where
λ =
(
~2
2M
) 1
4
g (3.3.10)
plays the role of the effective coupling constant replacing g. Using previous definitions, it is
evident that
λ v = g r . (3.3.11)
The boundary conditions which we impose to the equation (3.3.9) are
∂vY(v)
∣
∣
∣
v=0
=
ε
D
, Y(+∞) = +∞ . (3.3.12)
8We call v the quantum coordinate because it depends explicitly on the Planck constant ~.
16 Chapter 3. Generalities
These two boundary conditions guarantee that the corresponding wave function is normalizable,
leading to a bound state. Evidently, if ~ = 1 and M = 1/2, the equations (3.3.9) and (3.3.4)
coincide with the same coupling constant λ = g. We call the equation (3.3.9) the Riccati-Bloch
(RB) equation. It governs the dynamics of the logarithmic derivative in the v-space.
The RB equation (3.3.9) allows us to construct the asymptotic expansions of Y(v) at small
and large v.9 For fixed λ 6= 0 and small v, we have the Taylor series10
Y(v) =
ε
D
v+
(ε2 − a2D
2)
D2(D + 2)
v3− a3λ
D + 3
v4−a4D
3(D + 2)λ2 − 2 ε (ε2 − a2D
2)
D3(D + 2)(D + 4)
v5 + . . . . (3.3.13)
For large v,
Y(v) = a1/2m λ(m−2)/2vm/2 +
am−1λ
(m−4)/2
2a
1/2
m
v(m−2)/2 +
(4amam−2 − a2m−1)λ
(m−6)/2
8a
3/2
m
v(m−4)/2 + . . . .
(3.3.14)
It is easy to check that for λ = 0, hence, for the harmonic oscillator potential V (r) = a2 r
2, the
equation (3.3.9) has the exact solution
Y(v) =
ε
D
v , ε = a
1/2
2 D . (3.3.15)
Not surprisingly, this solution leads to the well-known ground state Gaussian wave function of
the harmonic oscillator.
Four remarks in a row:
(i) The expansion (3.3.13) at λ = 0 is terminated, and it consists of the first term alone,
leading to Y(v) = (ε/D) v, which is in agreement with (3.3.15).
(ii) In general, the majority of the coefficients in the expansion of Y(v) in powers of v, see
(3.3.13), depends on the energy ε explicitly.
(iii) The fourth and subsequent (unwritten) terms in (3.3.14) can acquire explicit dependence
on ε.
(iv) For M fixed, large v can be obtained in two situations: large r and fixed ~, or fixed r and
small ~, see (3.3.7). Hence, the expansion (3.3.14) can describe the semi-classical limit
~ → 0.
Integrating (3.3.13) and (3.3.14) with respect to the coordinate v, both expansions are con-
verted into series for 1
~
Φ, namely,
1
~
Φ =
ε
2D
v2 +
(ε2 − a2D
2)
4D2(D + 2)
v4 − a3λ
5(D + 3)
v5 − a4D
3(D + 2)λ2 − 2ε(ε2 − a2D
2)
6D3(D + 2)(D + 4)
v6 + . . . ,
(3.3.16)
9Ultimately, it allows us to construct the expansion of y(r) at large distances once variable r is restored.
10This is the expansion around the global minimum.
3.3. The Ground State: Riccati Equation 17
and
1
~
Φ =
2a
1/2
m
m+ 2
λ(m−2)/2v(m+2)/2+
am−1λ
(m−4)/2
ma
1/2
m
vm/2 +
(4amam−2 − a2m−1)λ
(m−6)/2
4(m− 2)a
3/2
m
v(m−2)/2 + . . . ,
(3.3.17)
for small and large v, respectively. These two expansions contain the asymptotic information
about the exact wave function that later will be used.
3.3.2 Generalized Bloch Equation
There exists an alternative way to transform the original Riccati equation (3.3.4) into a new
one in which the constants ~ and M do not appear explicitly. This is achieved by introducing
the classical coordinate11
u = g r , (3.3.18)
and defining a new unknown function
Z (u) =
(
g2
2M
)1/2
y . (3.3.19)
It is straightforward to show that the function Z(u) satisfies a non-linear differential equation
λ2 ∂uZ(u) − Z(u)
(
Z(u)− λ2(D − 1)
u
)
= λ2 ε(λ) − V̂ (u) , ∂u ≡ d
du
, (3.3.20)
c.f. (3.3.9). Here ε and λ are the same as in (3.3.8) and (3.3.10). Therefore, these two quantities
play the role of energy and effective coupling constant in both spaces: the v-space and the u-
space. Boundary conditions, which guarantee the normalizability of the wave function, are
similar to those imposed for the RB equation,
∂uZ(u)
∣
∣
∣
u=0
=
ε
D
, Z(+∞) = +∞ . (3.3.21)
At D = 1, the equation (3.3.20) was called in [44–46] the (one-dimensional) Generalized Bloch
equation. Here, the equation (3.3.20) is a natural extension of the one-dimensional GB equation
to the D-dimensional radial case. For this reason, we continue to call it as the Generalized
(radial) Bloch equation. In contrast with the RB equation, by setting ~ = 1 and M = 1/2 the
GB equation does not take the form of the Riccati one shown in (3.3.4). This is one of the
principal features that differentiate the RB equation from the GB one.
Naturally, (3.3.20) can also be used to construct the asymptotic expansions similar to (3.3.13)
and (3.3.14) for small u,
Z(u) =
ε
D
u+
(ε2 − a2D
2)
D2(D + 2)λ2
u3 − a3
(D + 3)λ2
u4 − a4D
3(D + 2)λ2 − 2 ε (ε2 − a2D
2)
D3(D + 2)(D + 4)λ4
u5 + . . . ,
(3.3.22)
11We called it like this since it is ~-independent. In fact, in the classical counterpart of the anharmonic
oscillator, the classical equations of motion written in variable u are g-independent, see [44, 45].
18 Chapter 3. Generalities
and large u,
Z(u) = a1/2m um/2 +
am−1
2a
1/2
m
u(m−2)/2 +
(4amam−2 − a2m−1)
8a
3/2
m
u(m−4)/2 + . . . , (3.3.23)
respectively. We note the following:
(i) Small u can be obtained in two situations, small r and fixed g, or fixed r and small
coupling constant g.
(ii) Large u occurs when r is large for fixed g, or for fixed r and g large.
Therefore, it is not a surprise that expansions in v-space (3.3.13) and (3.3.14) can be connected
with (3.3.22) and (3.3.23) in u-space, respectively. This is achieved by using the remarkable
relation
u = λ v , (3.3.24)
which connects the two spaces.
Chapter 4
The Weak Coupling Regime
Assuming a small effective coupling constant λ, we describe how to solve the RB and GB
equations using PT in λ. In particular, we will show that the perturbative solution of the
RB equation is an application of the Non-Linearization Procedure [7]. On the other hand, the
perturbative approach developed for the GB equation leads to a purely algebraic procedure
to calculate arbitrary corrections. Finally, we discuss the non-trivial connection between these
two solutions and its relation with the WKB approximation.
4.1 Riccati-Bloch Equation
Let us take the RB equation (3.3.9) with an arbitrary potential V (v;λ) in the right-hand side1,
∂vY(v) − Y(v)
(
Y(v)− D − 1
v
)
= ε (λ) − V (v;λ) , (4.1.1)
We assume that potential V (v;λ) admits a Taylor expansion in powers of the effective coupling
constant λ, thus
V (v;λ) =
∞∑
n=0
Vn(v)λ
n . (4.1.2)
Here the coefficient functions Vn(v), n = 0, 1, ..., are real functions in v. Let us now assume
that the unperturbed equation at λ = 0,
∂vY0(v) − Y0(v)
(
Y0(v)−
D − 1
v
)
= ε0 − V0(v) , (4.1.3)
1We take (3.3.9) and make the replacement
1
λ2
V̂ (λv) → V (v;λ).
19
20 Chapter 4. The Weak Coupling Regime
can be solved explicitly by any means.2 Evidently, once we know Y0(v), the wave function can
be found since
Ψ0 = exp
(
−
∫ v
Y0(s) ds
)
. (4.1.4)
The only constraint when finding Y0(s) is that the function Ψ0 should be normalizable.
Once the solution of the unperturbed problem is known, we can develop PT in powers of λ,
ε(λ) =
∞∑
n=0
εn λ
n , (4.1.5)
and
Y(v) =
∞∑
n=0
Yn(v)λ
n . (4.1.6)
Now the task is to find the corrections εn and Yn(v). To do so, we substitute (4.1.5) and (4.1.6)
into the RB equation (4.1.1); it is easy to see that the nth correction Yn(v) satisfies the first
order linear ordinary differential equation3
∂v
(
vD−1Ψ2
0 Yn(v)
)
= (εn −Qn(v)) v
D−1Ψ2
0 , (4.1.7)
where
Q1(v) = V1(v) , Qn(v) = Vn(v) −
n−1∑
k=1
Yk(v)Yn−k(v) , n = 2, 3, . . . . (4.1.8)
Evidently, the exact solution for Yn(v) is written in integral form
Yn(v) =
1
vD−1 Ψ2
0
(∫ v
0
(εn −Qn)Ψ
2
0 s
D−1 ds
)
. (4.1.9)
Since we are interested in finding bound states, it is necessary to impose the condition of the
absence of current of particles for both v → 0 and v → ∞ as boundary condition [8],
Yn(v) v
D−1 Ψ2
0
∣
∣
∣
v={0,∞}
→ 0 . (4.1.10)
It can be easily checked that if v → 0, the condition (4.1.10) is satisfied automatically, while at
v → ∞ the correction εn has to be chosen accordingly
εn =
∫∞
0
Qn Ψ
2
0 v
D−1 dv
∫∞
0
Ψ2
0 v
D−1 dv
(4.1.11)
to satisfy (4.1.10). In D = 1, PT developed to solve the Riccati equation (associated with a
given Schrödinger equation) was called the Non-Linearization Procedure [8]. We have given its
straightforward extension to radial potentials in arbitrary D. We will continue to call it the
Non-Linearization Procedure. Some remarks:
2It can always be achieved via the Inverse Problem in the framework of the RB equation: we take some
function Y0(v) and then calculate the left-hand side of (4.1.3). Finally, we solve for the potential V0(v) and ε0.
3The product vD−1Ψ2
0 is the integrating factor of the differential equation.
4.1. Riccati-Bloch Equation 21
(i) In contrast with the Rayleigh-Schrödinger PT, see e.g. [1], the knowledge of the entire
spectrum of the unperturbed problem is not required to find perturbative corrections in
(4.1.5) and (4.1.6). It is sufficient to know the unperturbed state wave function of the
unperturbed problem to which we are looking for corrections.
(ii) This approach gives the closed analytic expression for both corrections εn and Yn(v) in
the form of nested integrals. Therefore, this procedure is an efficient method to calculate
several orders in PT, see e.g. [8] and [47].
(iii) In this framework, the convergence of the perturbation series (4.1.5) and (4.1.6) is guar-
anteed for any D if the first correction Y1(v) is bounded
|Y1(v)| ≤ C (4.1.12)
where C is a positive constant, for discussion and details see [8].
(iv) Using the Non-Linearization Procedure we can exploit the connection between the varia-
tional energy and PT to calculate corrections of a given trial function and its variational
estimate of the energy.
We have presented a brief review of the Non-Linearization Procedure applied to the ground
state. This approach can be modified to study excited states by admitting a finite number of
simple poles with residues equal to one in the corrections Yn(v). The position of the poles is
also found in PT in powers in λ. Explicit formulas can be found in [8] for the one-dimensional
case. The generalization of the D-dimensional radial potential case is straightforward.
4.1.1 D-dimensional Radial Anharmonic Oscillator
Let us now consider the D-dimensional radial anharmonic oscillator potential (3.2.1). We study
the weak coupling regime assuming that the effective coupling constant
λ =
(
~2
2M
) 1
4
g
is small. To apply the Non-Linearization Procedure, we choose in (4.1.2)
Vk = ak+2 v
k+2 for k = 0, 1, . . . ,m− 2 ,
Vk = 0 for k > m− 2 . (4.1.13)
Therefore, our potential is a polynomial of finite degree m − 2 and m, in λ and v, respec-
tively. The corresponding unperturbed equation (4.1.3) describes the D−dimensional spherical
harmonic oscillator: an exactly solvable problem in any dimension D > 0, see e.g. [48]. The
unperturbed ground state wave function is given by
Ψ0 = e
−
√
M
2~2
a
1/2
2 r2
= e−
1
2
a
1/2
2 v2 , (4.1.14)
22 Chapter 4. The Weak Coupling Regime
and it does not depend on the dimension D.
In general, the explicit calculation of the perturbative corrections Yn(v) and εn can be carried
out analytically by using (4.1.9). However, the final expressions involve the incomplete gamma
function and the Meijer G-function explicitly. Due to the form of those expressions, they are
irrelevant for our study since the information that they may provide is hidden in cumbersome
formulas. Hence, we take a different but more relevant approach: asymptotic series in particular
domains.
Let us find the asymptotic expansions of the nth correction Yn(v) in two limits: v → 0, and
v → ∞. For small v, the correction Yn(v) is given by the Taylor expansion
Yn(v) = v
∞∑
k=0
b
(n)
k vk , (4.1.15)
where b
(n)
k are some coefficients. It can be easily shown that the first coefficient b
(n)
0 is related
to the energy correction εn,
b
(n)
0 =
εn
D
. (4.1.16)
The next coefficient always vanishes,
b
(n)
1 = 0 , (4.1.17)
due to the absence of the linear term in the radial anharmonic potential (3.2.1). On the other
hand, for large v we have a Laurent-like series,
Yn(v) = v
∞∑
k=0
c
(n)
k vn−k , n > 0 , (4.1.18)
where c
(n)
k are some coefficients. Interestingly, the leading coefficient c
(n)
0 does not depend on
D while the next-to-leading one always vanishes,
c
(n)
1 = 0 . (4.1.19)
Using expansions (4.1.15) and (4.1.18), it can be shown that in the non-physical dimension
D = 0 all corrections εn vanish,
εn = 0 . (4.1.20)
From the integral form of correction εn shown in (4.1.11), one can show that if D → 0, the
numerator of (4.1.11) is bounded,
∫ ∞
0
Qn Ψ
2
0 v
D−1 dv < ∞ , (4.1.21)
while the denominator behaves like
∫ ∞
0
Ψ2
0 v
D−1 dv =
1
D
a
−D
4
2 + O(D0) (4.1.22)
4.2. Generalized Bloch Equation 23
for D → 0. Consequently, the correction εn vanishes linearly in D when D → 0. Note that the
coefficient b
(n)
0 remains finite in this limit, see (4.1.16). If any correction εn vanishes for arbitrary
n, then their formal sum (4.1.5) results in ε = 0, and the energy vanishes, E = 0. In general,
for D 6= 0 it can be shown that expansion for ε is asymptotic: εn grows at least factorially as
n → ∞, see e.g. [20, 22, 28, 49]. Therefore, series (4.1.5) and (4.1.6) are divergent. This is not
surprising since a Dyson [50] instability is present for most of the radial anharmonic oscillators
of the form (3.2.1). The divergent nature of the series could discourage to calculate corrections
in PT. However, for some particular cases of (3.2.1) it has been proved that perturbation
series for the energy can be summed to the exact one by using different summation techniques :
calculating the Borel sum [27], taking Padé approximants [31] and via renormalization [51].
These regularization procedures can lead to highly accurate estimates of the energy for some
systems. In particular, it is successful for some cases of the anharmonic oscillator.
An interesting situation occurs when all odd monomial terms in r in potential (3.2.1) are
absent, i.e. the potential is (formally) an even function in r, V (r) = V (−r). In this case, all
odd corrections Y2n+1(v) and ε2n+1 vanish. Likewise, any even correction Y2n(v) has the form
of a polynomial of finite degree
Y2n(v) = v
n∑
k=0
c
(2n)
2k v2(n−k) , Y2n(−v) = −Y2n(v) , (4.1.23)
where
c
(2n)
2n =
ε2n
D
. (4.1.24)
This implies that ε2n and Y2n(v) can be calculated by linear algebra means4. In turn, the energy
correction ε2n is a finite-degree polynomial in D,
ε2n =
n+1∑
k=1
d
(2n)
k Dk , (4.1.25)
where d
(2n)
k are real coefficients. However, it is enough to have a single odd monomial term
∼ v2q+1 in the potential (3.2.1) (with q positive and integer) to break the features (4.1.23) and
(4.1.25). Hence, all coefficients in front of the singular terms in the expansion (4.1.18), as well
as all higher order terms in the expansion (4.1.15) at k > n, are proportional to a2q+1.
4.2 Generalized Bloch Equation
We can develop PT in powers of λ for the GB equation (3.3.20). Of course, the expansion of ε
in powers of λ is the same as the one used in the RB equation, see (4.1.5),
ε(λ) =
∞∑
n=0
εn λ
n .
4In this situation, a large number of corrections can be calculated by using a computational code, see e.g. [47].
24 Chapter 4. The Weak Coupling Regime
In turn, the expansion for the function Z(u) reads
Z(u) =
∞∑
n=0
Zn(u)λ
n . (4.2.1)
We assume that the energy corrections εn are already known: they can be found using the Non-
Linearization Procedure, the standard Rayleigh-Schrödinger PT, or any other suitable method.
It is immediate to see that the correction Zn(u) is calculated by algebraic means. Its calculation
depends on corrections of a smaller order, just like it happens in any perturbative approach.
The first 3 corrections are
Z0(u) = ±
√
V̂ (u) ,
Z1(u) = 0 , (4.2.2)
Z2(u) =
u ∂uZ0(u) + (D − 1)Z0(u) − u ε0
2 uZ0(u)
.
In general, the nth correction is given by
Zn(u) =
u ∂uZn−2(u) + (D − 1)Zn−2(u) − u2
∑n−2
i=2 Zi(u)Zn−i(u) − u εn−2
2 uZ0(u)
, n > 2 .
(4.2.3)
Note that the boundary condition (3.3.21), Z(∞) = +∞, implies that the positive sign of
the square root must be chosen in the expression for Z0(u).
Analyzing the Zn(u) correction (4.2.3), one can see that the boundary condition at u = 0,
see (3.3.21), can not be fulfilled in the case of an arbitrary potential (3.2.1) without defined
parity with respect to the change r → −r. In this situation, any Zn(u) (and its derivative) at
n > 1 diverges at small u. For instance, for the cubic potential where a3 6= 0 the nth correction
Zn(u) with n > 1 behaves like
Zn(u) ∼ u−n+2 (4.2.4)
when u → 0. In turn, for the quintic potential with a3 = 0 but a5 > 0 the nth correction at
n > 3 behaves like
Zn(u) ∼ u−n+4 (4.2.5)
when u tends to zero. For the polynomial potential of degree (2k + 1) in u, if all other odd
terms are absent a3 = a5 = . . . = a2k−1 = 0 but a2k+1 > 0, the function Zn(u) behaves like
Zn(u) ∼ u−n+2k (4.2.6)
for small u as long as n > 2k − 1. It might be an indication that the radius of convergence of
the expansion of Z(u) (see (4.2.1)) in 1/u is finite. However, if the anharmonic potential V (r)
is formally even, V (r) = V (−r) the boundary condition at u = 0 can always be satisfied. This
allows us to determine the correction εn by imposing
∂uZn(u)
∣
∣
∣
u=0
=
εn
D
. (4.2.7)
4.3. Connection Between the Expansions for Y and Z 25
At this point, it is worth mentioning that the expansion in powers of λ (4.2.1) is divergent
for any anharmonic oscillator due to the Dyson instability mentioned below. This fact can be
seen in the partial sums of (4.2.1), see Figs. 4.1 and 4.2.
Let us emphasize that the presence of a single odd degree monomial term in u in the potential
V (u) breaks the feature (4.2.7): the expansion for Z(u) in powers of λ only satisfies the single
boundary condition at u = ∞. It is necessary to discuss the connection between the expansions
(4.1.6) and (4.2.1) to explain why this happens.
Figure 4.1: Partial sums of the expansion for Z(u) as functions of u for a general anharmonic
potential with monomials of even degrees. The function Z(u) (solid blue line), represents the
exact solution, see text. Near u = 0, the deviation of the partial sums from the exact solution
is a consequence of the divergent nature of series (4.1.6).
4.3 Connection Between the Expansions for Y and Z
Thus far, we have constructed two different representations for the logarithmic derivative y(r),
see (3.3.6). Due to the uniqueness of the solutions of the Schrödinger equation, there must be
a connection between those representations. According to the RB equation, we have the
y(r) = (2M~
2)
1
4 Y(v(r)) , Y(v) =
∞∑
n=0
Yn(v)λ
n , v(r) =
(
2M
~2
) 1
4
r . (4.3.1)
26 Chapter 4. The Weak Coupling Regime
Figure 4.2: Partial sums of the expansion for Z(u) as functions of u for a generic anharmonic
potential which includes odd monomials. The function Z(u) (solid blue line) represents the
exact solution, see text. The divergence in the vicinity of u = 0 of the partial sums is the result
of the impossibility of satisfying the boundary condition at u = 0.
From the other side, the GB equation leads to
y(r) =
(
2M
g2
) 1
2
Z(u(r)) , Z(u) =
∞∑
n=0
Zn(u)λ
n , u(r) = g r . (4.3.2)
The coupling constant λ is defined in (3.3.10). The connection can be established if we use
explicitly the expansion of Yn(v) at large v, see (4.1.18). In this case, y(r) is given formally by
y = (2M ~
2)
1
4 v
∞∑
n=0
λn
∞∑
k=0
c
(n)
k vn−k . (4.3.3)
By using the relation between the v-space and u-space (3.3.24),
u = λ v ,
and changing the order of summation, it follows that (4.3.3) is equivalent to
y =
(
2M
g2
) 1
2
∞∑
n=0
λn(gr)
∞∑
k=1
c(k)n (gr)k−n . (4.3.4)
Now, the connection is clear. If we compare the expansions (4.3.3) and (4.3.4), we can conclude
that
Zn(u) = u
∞∑
k=1
c(k)n uk−n (4.3.5)
4.4. Connection with WKB 27
due to the uniqueness of the Taylor series. Therefore, Zn(u) is a generating function of the
coefficients c
(n)
k , k = 0, 1, ... , see Fig. 4.3 for a graphical illustration. Equation (4.3.5) exhibits
y(r) = (2M)
1
2
g
( c
(0)
0 (gr)1
+ c(1)0 (gr)2 + λ2 c(1)2 (gr)0 + λ3 c(1)3 (gr)−1 + λ4 c(1)4 (gr)−2 + . . .
+ c(2)0 (gr)3 + λ2 c(2)2 (gr)1 + λ3 c(2)3 (gr)0 + λ4 c(2)4 (gr)−1 + . . .
+ c(3)0 (gr)4 + λ2 c(3)2 (gr)2 + λ3 c(3)3 (gr)1 + λ4 c(3)4 (gr)0 + . . .
+ c(4)0 (gr)5 + λ2 c(4)2 (gr)3 + λ3 c(4)3 (gr)2 + λ4c(4)4 (gr)1 + . . .
+
... +
... +
... +
... + . . . )
︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸ ︸ ︷︷ ︸
≡ (2M)
1
2
g
( Z0(gr) + λ2 Z2(gr) + λ3 Z3(gr) + λ4 Z4(gr) + . . . )
Figure 4.3: Representation of the generating functions Zn constructed from the perturbative
series in powers of λ for the function y(r).
the reason why we cannot satisfy, in general, the boundary condition at u = 0: Zn(u) is
constructed from the expansion (4.1.18), which is valid in the limit r → ∞ (u → ∞ for fixed
g). It also clarifies the reason why this boundary condition can be satisfied only for even
potentials: when formally V (r) = V (−r), the (finite) series of any Yn(v) at small distances is
the same than for large ones.
Some interesting properties of the corrections Zn(u) occur when considering r → ∞. It will
be discussed in detail for the cubic, quartic, and sextic anharmonic oscillators, see below.
4.4 Connection with WKB
Let us conclude this Chapter by showing the relation between the perturbative approach im-
plemented for the GB equation and the semi-classical WKB approximation. By integrating in
r, the expansion (4.3.2) is converted into an expansion in terms of generating functions for the
phase
Φ(r) =
∞∑
n=0
λnGn(r) , G1(r) = 0 , (4.4.1)
where Gn(r) is equal to
Gn(r) =
(
2M
g2
)1/2 ∫ r
Zn(gr) dr . (4.4.2)
28 Chapter 4. The Weak Coupling Regime
Note that keeping g and M fixed, (4.4.1) can be regarded as a semi-classical expansion of the
phase in powers of ~
1
2 , see (3.3.10). The leading order of this expansion is given by
G0(r) =
∫ r√
2M V (r) dr . (4.4.3)
This leading order term is the classical action for vanishing energy5 E = 0, and it is related to
the zeroth order correction of the standard D-dimensional (radial) WKB method6 (developed
in the classically forbidden region), see below. This is the only term in the expansion (4.4.1)
which contains no dependence on D. The second order term in (4.4.1) is
λ2G2(r) = ~
(
D − 1
2
log[r] +
1
4
log [2M V (r)] − ε0
2
∫ r dr
√
V (r)
)
. (4.4.4)
Except for the integral - the third term - it looks like the first order correction to WKB at
E = 0 and arbitrary D, see Appendix A. Thus, it defines the functional determinant of a path
integral formalism [45]. The appearance of the extra term in the form of an integral can be
explained as follows. Let us take the zeroth order term in the standard WKB method (in the
forbidden region) and expand it in powers of ~1/2 using the fact that the energy is ~-dependent,
see (3.3.8) and (3.3.10), thus
∫ r√
2M (V (r)− E) dr =
∫ r√
2M V (r) dr − ~ ε0
2
∫ r dr
√
V (r)
+ O(~3/2) . (4.4.5)
Note that the first term is nothing but G0(r) while the second term is exactly the integral
which appears in (4.4.4). Consequently, one can see that (4.4.1) is the standard semi-classical
WKB expansion in the classically-forbidden region, re-expanded in powers of ~1/2. Higher
order generating functions G3(r), G4(r), . . ., are related with higher order corrections of the
WKB expansion in a similar way.
This is an alternative consideration to the one-dimensional case presented in [44–46]. In these
papers, it was described how to obtain expansion (4.4.1) using a path integral formulation (in
Euclidean time) considering a special classical solution of zero energy (E = 0) called flucton.
In a quite straightforward way, this consideration can be extended to the D-dimensional radial
case, where the flucton path appears in the radial direction. Hence, the perturbative approach
developed for the GB Equation can be called the true semi-classical approximation: both ε(λ)
and Φ(r;λ) are expanded in powers ~1/2.
5A vanishing energy corresponds to the classical energy of a particle located at the global minimum.
6We present a brief review of the WKB method in Appendix A.
Chapter 5
The Strong Coupling Regime
In Chapter 4, assuming a small effective constant
λ =
(
~2
2M
) 1
4
g,
we studied the ground state energy E and the phase in the form of a Taylor expansion around
λ = 0. As a result, the harmonic oscillator was taken as the unperturbed problem. We now
consider the opposite limit: large λ in both RB and GB equations. In this case, we take as
unperturbed potential the leading monomial term amg
m−2rm in (3.2.1). It implies that all
other terms in the potential (3.2.1), except for the leading one, are taken as perturbations.
This situation can be achieved if we consider the expansion of the energy in fractional powers
1/λ.
5.1 Riccati-Bloch Equation
To develop the strong coupling expansion in the framework of the RB equation, let us first
introduce the new quantum coordinate1
w = λ
m−2
m+2 v , (5.1.1)
and define the new function
Ỹ(w) = λ
2−m
m+2Y(v) . (5.1.2)
When we introduce these two expressions in the RB equation (3.3.9), we have
∂wỸ(w) − Ỹ(w)
(
Ỹ(w)− D − 1
w
)
= ε̃(λ̃) − 1
λ̃m
V̂
(
λ̃ w
)
, ∂w ≡ d
dw
, (5.1.3)
1The insertion of this coordinate in the RB equation is equivalent to perform a Symanzik scaling transfor-
mation to the Schrödinger equation, see [9].
29
30 Chapter 5. The Strong Coupling Regime
where
ε̃ = λ
4−2m
m+2 ε , λ̃ = λ
4
m+2 , (5.1.4)
cf. (3.3.9). This equation does not contain explicit dependence on ~ and M , while ε̃ plays the
role of energy. Furthermore, (5.1.3) is the RB equation that is appropriate to construct the
strong coupling expansion. This can be easily seen if we note that the potential is now given
by
1
λ̃m
V̂
(
λ̃ w
)
=
1
λ̃m
m∑
k=2
ak (λ̃ w)
k = amw
m +
am−1
λ̃
wm−1 + . . .+
a2
λ̃m−2
w2 . (5.1.5)
Thus, the parameter λ̃−1 is now the coupling constant. Therefore, when large λ̃ is considered,
the potential becomes a single monomial in the variable w in power m. This monomial now
represents the unperturbed potential. The latter implies that Ỹ(w) and ε̃(λ̃) should eventually
be expanded in series in the following form
Ỹ(w) =
∞∑
n=0
Ỹn(w) λ̃
−n . (5.1.6)
and
ε̃(λ̃) =
∞∑
n=0
ε̃n λ̃
−n , (5.1.7)
respectively. Here ε̃n, n = 0, 1, ... , are the coefficients of the strong coupling expansion. A simi-
lar expansion to (5.1.7) occurs for the excited states. Contrary to the weak coupling expansion,
it has been demonstrated that, for some particular cases of the anharmonic potential (3.2.1), PT
developed in the strong coupling regime (5.1.7) has a finite radius of convergence, see e.g. [52]
and references therein. It is related to the fact that the dominant term in the potential (3.2.1)
is rm at r → ∞ while other monomials are subdominant. Therefore, the phenomenon of the
Dyson instability does not occur.
It has to be emphasized that the equation (5.1.3) has a similar form than the RB equation
with arbitrary potential (4.1.1). This implies that all the formulas, derived in Section 4.1 in
the framework of the Non-Linearization Procedure, are suitable to calculate Ỹn(w) and ε̃n: we
only have to make the replacement v → w in (4.1.1). Thus, let us choose the potential that
appears in (4.1.2) in the following form
Vk = am−k w
m−k for k = 0, 1, ..., m− 2 ,
Vk = 0 for k > m− 2 . (5.1.8)
In this case, the unperturbed equation (4.1.3) is
∂vỸ0(w) − Ỹ0(w)
(
Ỹ0(w)−
D − 1
w
)
= ε̃0 − amw
m . (5.1.9)
If the solution of this equation were known, the corresponding square-integrable unperturbed
wave function Ψ0 would be obtained using (4.1.4). Then, we would use the formulas (4.1.9) and
5.1. Riccati-Bloch Equation 31
(4.1.11) to construct PT. As a result, we would obtain the coefficients ε̃k of the strong coupling
expansion. However, since the exact solution of (5.1.9) for m > 2 is unknown, we need to find
Ỹ0(w) in approximate form. In the spirit of the present work, there are two possible ways to
find Ỹ0(w):
(i) One can choose the simplest physically relevant trial function for the potential
W0(w) = a2pw
2p , m = 2p (5.1.10)
which defines the unperturbed dimensionless radial Schrödinger equation
− 1
2
(
∂2w +
D − 1
w
∂w
)
Ψ0,0 + W0(w)Ψ0,0 = ε̃0,0Ψ0,0 . (5.1.11)
Then, we use the Non-Linearization Procedure employing the connection between PT and
the variational energy, see (1.16).
(ii) We can construct a more sophisticated locally-accurate approximation of Ỹ0(w) for w ≥ 0.
Let us describe both options.
5.1.1 Physically Relevant Trial Function
Probably, the simplest physically relevant trial function, see [8] for discussion, was proposed
for the first time in [7, 53] handling the one-dimensional case. Undoubtedly, the similar trial
function is also appropriate for the D-dimensional case of the radial potentials,
Ψ0,0(w) = e−a
1/2
2p
wp+1
p+1 , Ỹ0,0(w) = a
1/2
2p wp , (5.1.12)
where the second subscript marks the order of the correction to Ψ0. By solving the inverse
problem, one can find the potential W0,0(w) for which Ψ0,0 is the exact solution, namely
W0,0(w) = a2pw
2p − a
1/2
2p (p+D − 1)wp−1 , ε̃0,0 = 0 . (5.1.13)
Taking the difference between the potentials (5.1.10) and (5.1.13),
W0,1(w) = W0(w) − W0,0(w) = a
1/2
2p (p+D − 1)wp−1 , (5.1.14)
one can expand PT as in (4.1.5) and (4.1.6) with W0,1(w) as the perturbative potential for
W0,0(w), see Section 1.2 for details. In this manner, it allows us to calculate the zeroth order
coefficient ε̃0 in the strong coupling expansion (5.1.7) iteratively, which appears in the form of
an expansion
ε̃0 = ε̃0,0 + ε̃0,1 + ε̃0,2 + . . . . (5.1.15)
32 Chapter 5. The Strong Coupling Regime
The first two coefficients can be found analytically,
ε̃0,0 = 0 , (5.1.16)
ε̃0,1 = a
1
p+1
2p (p+D − 1)
(
p+ 1
2
) p−1
p+1 Γ(D+p−1
p+1
)
Γ( D
p+1
)
. (5.1.17)
The next ones can be obtained numerically2. The function Ỹ0(w) is expanded in a similar way
as in (5.1.15),
Ỹ0(w) = Ỹ0,0(w) + Ỹ0,1(w) + Ỹ0,2(w) + . . . , (5.1.18)
where Ỹ0,0(w) is given by (5.1.12). Evidently, this expansion can be converted into the expansion
of the phase,
Φ0 = Φ0,0 + Φ0,1 + Φ0,2 + . . . (5.1.19)
where Φ0,0 = a
1/2
2p ~
wp+1
p+ 1
. Using Ψ0,0 as entry one can also calculate explicitly the first ap-
proximation to ε̃1, the subdominant coefficient in the strong coupling expansion (5.1.7), which
results in
ε̃1,0 = am−1
(
p+ 1
2a
1/2
2p
) 2p−1
p+1 Γ(D+2p−1
p+1
)
Γ( D
p+1
)
. (5.1.20)
The next terms, e.g. ε̃1,1, ε̃1,2, ..., can be only written in the form of nested integrals, similar
to those as in expression (4.1.11). Their computation is numerical. Of course, this procedure
can be extended to calculate higher coefficients in the expansion (5.1.7).
In general terms, it can be shown that in this approach series (5.1.15) and (5.1.19) are
convergent. However, the rate of convergence is very slow. It implies that several corrections
are needed to obtain highly accurate results from their partial sums. Since corrections are
calculated numerically, the accumulation of error in the computations is a drawback. This
inconveniences can be avoided if we accelerate the convergence using a more appropriate zeroth
order approximation than (5.1.12).
5.1.2 Interpolation: Locally Accurate Approximation
One can construct another physically relevant trial function for the RB Equation (5.1.9) that
leads to a fast convergent series (5.1.15) and (5.1.19). To do so, we employ the asymptotic
behavior of Ỹ0(w) at small w,
Ỹ0(w) =
(
ε̃0
D
w +
ε̃20
D2 (D + 2)
w3 − a2p δp,3/2
D + 3
w4 +
2ε̃30 − a2pD
3(D + 2)δp,2
D3(D + 2)(D + 4)
w5 + . . .
)
,
(5.1.21)
2 In Section 7.2, which is devoted to the cubic anharmonic oscillator, the first coefficients will be presented
explicitly.
5.2. Generalized Bloch Equation 33
and at large w
Ỹ0(w) =
(
a
1/2
2p wp +
1
2
(D + p− 1)w−1 − 1
2
ε̃0w
−p + . . .
)
. (5.1.22)
The dominant term in (5.1.22) corresponds to (5.1.12). A more sophisticated approximation
of Ỹ0,0(w) is the result of the interpolation between expansions (5.1.21) and (5.1.22). The con-
struction of such interpolation will be presented in Chapter 6. We show how this interpolation
leads to a locally accurate approximation for the cubic, quartic, and sextic anharmonic oscil-
lators. Furthermore, it leads to highly accurate estimates of the first two coefficients of the
strong coupling expansion.
5.2 Generalized Bloch Equation
There is an interesting connection between the strong coupling expansion developed for the
GB equation (3.3.20) and the expansion of y(r) at small r, see (3.3.13). To construct the GB
equation for the strong coupling regime, it is necessary to note that u and w are related in a
very simple way,
u = λ̃ w . (5.2.1)
This can be checked straightforwardly just by noting that
λ̃ w = λ
4
m+2
(
λ
m−2
m+2 v
)
= λ v
= u , (5.2.2)
see (5.1.1) and (5.1.4). Interestingly, the equation (5.2.2) looks very similar to (3.3.24). This
remark suggests introducing into the RB equation (5.1.3) the classical coordinate
u = λ̃ w , (5.2.3)
and a new function
Z̃(u) = λ̃ Ỹ (w) . (5.2.4)
In this way, we obtain
λ̃2 ∂uZ̃(u) − Z̃(u)
(
Z̃(u) − λ̃2
(D − 1)
u
)
= λ̃2 ε̃ − λ̃−m+2 V̂ (u) . (5.2.5)
This equation is the GB Equation for the strong coupling regime: the left-hand side is the
same as in (3.3.20) - the GB equation for weak coupling regime - while the right-hand side is
different3. Now we can develop PT in inverse powers of λ̃−1 in the form
ε̃(λ̃) =
∞∑
n=0
ε̃n λ̃
−n.
3Except when m = 2. In this irrelevant case - the harmonic oscillator - there is no coupling constant, see
(3.2.1).
34 Chapter 5. The Strong Coupling Regime
see (5.1.7), and
Z̃(u) =
∞∑
n=0
Z̃n(u) λ̃
−n , (5.2.6)
This is nothing but the expansions at λ̃ = ∞. We assume the coefficients ε̃n are known a
priori. In particular, they may be computed from PT developed for the RB equation in the
strong coupling regime. Inserting the series (5.1.7) and (5.2.6) in (5.2.5), one can determine
corrections Z̃n(u) iteratively by solving the same first order differential equation with a different
right-hand side
∂uZ̃i(u) +
(D − 1)
u
Z̃i(u) = ε̃i , i = 0, 1 , (5.2.7)
and
∂uZ̃n(u) +
(D − 1)
u
Z̃n(u) = ε̃n +
n−2∑
i=0
Z̃i(u) Z̃n−i−2(u) − δm,n V̂ (u) , n = 2, 3, . . . . (5.2.8)
Imposing the boundary condition Z̃(0) = ε̃/D to the original equation (5.2.5), we can find the
boundary condition for the nth correction,
∂uZ̃(u)
∣
∣
∣
u=0
=
ε̃n
D
. (5.2.9)
The equations (5.2.7) and (5.2.8) are solved in an elementary way. In particular,
Z̃0(u) =
ε̃0
D
u ,
Z̃1(u) =
ε̃1
D
u ,
Z̃2(u) =
ε̃2
D
u +
ε̃20
D2(D + 2)
u3 ,
Z̃3(u) =
ε̃3
D
u +
2 ε̃0 ε̃1
D2(D + 2)
u3 − δm,3
m∑
k=2
ak
D + k
uk+1 ,
Z̃4(u) =
ε̃4
D
u +
2 ε̃0 ε̃2 + ε̃21
D2(D + 2)
u3 +
2 ε̃30
D3(D + 2)(D + 4)
u5 − δm,4
m∑
k=2
ak
D + k
uk+1 .
The general nth correction is given by
Z̃n(u) =
ε̃n
D
u + u1−D
n−2∑
k=0
∫ u
0
Z̃k(s) Z̃n−k−2(s) s
D−1 ds − δm,n
m∑
k=2
ak u
k+1
D + k
(5.2.10)
at n ≥ 2. Note that it is not guaranteed that the boundary condition Z̃(∞) = +∞ is fulfilled
for all these corrections. It implies that PT in inverse powers of λ̃ for Z̃(u) has a finite radius of
convergence in u. Thus, the corrections Z̃n(u) make sense in a bounded domain in u only, see
5.2. Generalized Bloch Equation 35
discussion below. From (5.2.10), one can see that any correction is a finite-degree polynomial
in u,
Z̃n(u) =



[n
2
]
∑
k=0
α
(n)
2k u
2k+1 − δm,n
m∑
k=2
ak u
k+1
D + k
for n ≤ m+ 1 ,
n∑
k=0
α
(n)
k uk+1 for n > m+ 1 ,
(5.2.11)
where α
(n)
k are real coefficients, and the symbol [ ] denotes the integer part function. It is easy
to see that
α
(n)
0 =
ε̃n
D
(5.2.12)
following (5.2.9). The next coefficient always vanishes,
α
(n)
1 = 0 , (5.2.13)
for any n. The coefficient α
(n)
2 can be found in the form of finite sum
α
(n)
2 =
1
D2(D + 2)
n−2∑
k=0
ε̃kε̃n−k−2 , (5.2.14)
while the next coefficient again vanishes,
α
(n)
3 = 0 , (5.2.15)
for any n, c.f. (5.2.13). From (5.2.11), one can see that Z̃n(u) for n < m and n = m + 1 is
always an odd polynomial in u:
Z̃n(−u) = −Z̃n(u) ,
while for n = m and n > m + 1 this property may be lost. In the case of an even anharmonic
potential, V (r) = V (−r), all odd corrections Z̃2n+1(u) and ε̃2n+1 vanish. In turn, all even
corrections Z̃2n(u) are odd polynomials in u of degree (2n+ 1) if 2n > m+ 1.
The perturbative approach, which was used to calculate the strong coupling expansion of
Z̃(u), is related to the asymptotic behavior of Z̃(u) at small u. Once we established the poly-
nomial structure of the corrections Z̃n(u), one can sum up some subseries of the perturbative
expansion (5.2.6), keeping the degree of u fixed. For example, the sum
(
∞∑
n=0
α
(n)
0 λ̃−n
)
u =
ε̃
D
u , (5.2.16)
corresponds to summing up all terms O(u) in (5.2.6). The next sum is carried out with all
terms O(u3). Using (5.2.14) one can show that it corresponds to
∞∑
n=2
α
(n)
2 λ̃−nu3 − a2 λ̃
−m
D + 2
u3 =
λ̃−2 ε̃2 − a2 λ̃
−mD2
D2(D + 2)
u3 . (5.2.17)
36 Chapter 5. The Strong Coupling Regime
The next sum contains coefficients in front of u4 terms. Following (5.2.15), this sum has a single
term,
− a3 λ̃
−m
D + 3
u4 . (5.2.18)
This summation procedure can be extended to higher order terms un, n > 4. After performing
such summations, we obtain the coefficients in front of all terms in the Taylor expansion in
powers of u for the function Z̃(u), and, eventually, for y(r). From another side, one can
construct the expansion in powers of u directly from (5.2.5),
Z̃(u) =
ε̃
D
u +
λ̃−2 ε̃2 − a2 λ̃
−mD2
D2(D + 2)
u3 − a3 λ̃
−m
D + 3
u4 −
a4 λ̃
−mD3(D + 2)− 2 ε̃ λ̃−2(ε̃2λ̃−2 − a2 λ̃
−mD2)
D3(D + 2)(D + 4)
u5 + . . . . (5.2.19)
One can note the exact correspondence between the first three terms of the expansion (5.2.19)
and the subseries previously defined in (5.2.16), (5.2.17) and (5.2.18). In this way, we can see the
connection between the strong coupling expansion, developed on the basis of the GB equation,
and the small u one: both expansions lead to the same representation - the Taylor series - of
Z̃(u) and, eventually, of y(r). Finally, this connection clarifies why the boundary condition
Z̃(∞) = +∞ can not always be fulfilled.
Chapter 6
The Approximant
Now we formulate a prescription for the construction of a locally-accurate, uniform approxima-
tion of the wave function of a given state. Such approximate function is called The Approximant.
6.1 The Approximant for the Ground State
We denote this approximation by Ψ
(t)
0,0 and call it the Approximant. Here, the subscripts indicate
explicitly the quantum numbers of the ground state (nr = 0, l = 0) at any D.1 It is convenient
to assume the exponential representation of the wave function,
Ψ
(t)
0,0(r) = e−
1
~
Φt(r) , (6.1.1)
since the phase Φ of the exponential representation is not that sharp as the wave function Ψ.2
Hence, we focus on the construction of the approximate phase Φt. We follow the prescription
that the phase Φt has to interpolate the expansions for small and large distances, (3.3.16) and
(3.3.17), respectively. Besides, it should interpolate the same expansions constructed in the
strong coupling regime (5.1.21) and (5.1.22). One of the simplest interpolations with these
characteristics is of the following form
1
~
Φt(r) =
ã0 + ã1 g r +
1
g2
V (r ; ã2, . . . , ãm)
√
1
g2 r2
V (r ; b̃2, . . . , b̃m)
+ Logarithmic Terms (r ; {c̃}) .
(6.1.2)
1 At D = 1, the angular quantum number is not defined. However, we may fix it (formally) to zero, ℓ = 0.
2 This was explicitly seen in a particular example described in the Overview.
37
38 Chapter 6. The Approximant
Without loss of generality, we put b̃2 = 1 as normalization. Here, V (r; {ã}) and V (r; {b̃})
are modified potentials of the form of the original one (3.2.1): instead of the external given
parameters {a} some free parameters {ã} and {b̃} are taken, respectively. The logarithmic
terms can depend on free parameters {c̃} as well. These terms will eventually insert a prefactor
in (6.1.1). Parameters {c̃} are chosen in such a way that this prefactor has no real nodes.
All those remaining free parameters will be fixed by making the variational calculation with
(6.1.2) taken as the phase of the trial function.
At g = 0, the phase (6.1.2) becomes ã0 + ã2r
2, which corresponds to the radial harmonic
oscillator phase. Formula (6.1.2) is the key result of this Chapter.
The phase Φt has the outstanding property that, with an appropriate choice of parameters
(ã0, ã1; ã2, b̃2, . . . ãk, b̃k; {c̃}), the generating function G0(r) is partially (or exactly) reproduced.
In particular, to reproduce exactly the dominant term in the expansion (3.3.17), the condition
ãm =
2 a
1/2
m
m+ 2
b̃m−2 (6.1.3)
is imposed3. Further constraints to parameters can be imposed in order to reproduce all growing
terms in the expansion of the exact function Y(v), see (3.3.14). If such constraints are imposed,
all the corrections in PT to Y0 = ∂rΦt will be bounded, and it will guarantee the convergence
of the perturbation series (1.18).
To mimic the possible appearance of logarithmic terms in the expansion (3.3.17), in particular,
those which may occur in G0(r) and G2(r), some logarithmic terms to phase Φt should be added,
see (4.4.4). For example, for the two-term anharmonic oscillator potential
V (r) = r2 + gm−2 rm , (6.1.4)
the generating function G2(r) always contains logarithmic terms,
g2G2(r)
(2M)1/2
=
1
4
log
[
1 + (gr)m−2
]
+
D
m+ 2
log
[
1 +
√
1 + (gr)m−2
]
, (6.1.5)
see (4.4.2). In this case, the logarithmic terms added in (6.1.2) should be a certain minimal
modification of the logarithmic terms that appear in G0(r) and G2(r). For instance, a minimal
modification of g2G2(r)/(2M)1/2 is of the form
1
4
log
[
1 + c̃ (gr)m−2
]
+
D
m+ 2
log
[
1 +
√
1 + c̃ (gr)m−2
]
, (6.1.6)
where c̃ is a free parameter. If c̃ = 1, it coincides with (6.1.5).
In general, some constraints on parameters have to be imposed to guarantee that Φt has the
structure of (3.3.16) and that Ψ
(t)
0,0 is a normalizable function. Other constraints come from
3Without loss of generality, this constraint is written for the case ~ = 1 and M = 1/2.
6.2. Approximants for Excited States 39
the symmetry of the phase. For example, if the potential is formally even, V (r) = V (−r), the
function Φt has to be even too.
After imposing all the constraints described above, the remaining free parameters can be
fixed by giving them the values of optimal parameters obtained in variational calculations. To
realize it, we use the VM for the radial Hamiltonian (3.3.2) taking the Approximant4
Ψ
(t)
0,0(r) = Prefactor(r ; {c̃})× exp

−
ã0 + ã1 g r +
1
g2
V (r ; ã2, . . . , ãm)
√
1
g2 r2
V (r ; b̃2, . . . , b̃m)

 ,
(6.1.7)
as a trial function, see (6.1.1) and (6.1.2). The accuracy of variational calculations can be
estimated using the connection between the variational energy and PT5. This kind of estimates
will be presented in the forthcoming Chapter for particular anharmonicities.
6.2 Approximants for Excited States
Case D = 1
In the one-dimensional situation r = |x|, where x ∈ (−∞,∞), see (3.1.3). Therefore, the
potential is an even function V = V (|x|). Due to the symmetry of the potential under the
reflection x → −x the quantum number which indicates the parity ν = (−1)p, p = 0, 1, is well
defined. It implies that we can use it to label any wave function6, namely,
Ψ(t)
nr,p(x) = xpP (p)
nr
(x2)e−
1
~
Φt(x) , nr = 0, 1, ..., p = 0, 1 , (6.2.1)
where P
(p)
nr (x
2) is a polynomial of degree nr with real coefficients and all real roots, and with
P
(p)
nr (0) = 1 chosen for normalization. The polynomial in (6.2.1) defines the nodal surface of
the approximate wave function. With vanishing coupling constant (g = 0) the polynomial
xpP
(p)
nr (x
2) is well-known, it is written in terms of the Hermite polynomials
xpP (p)
nr
(x2) = H2nr+p(x) . (6.2.2)
The phase Φt has exactly the same form as in (6.1.2) but with a different set of free parameters
{ã}, {b̃}, {c̃}. The same constraints imposed on the parameters of the ground state should be
4It is beyond our purpose to study the analytical properties of the Approximant in the complex r-plane.
However, except for the real axis, it is clear that the Approximant develops nodes in the complex plane.
5See Section 1.2
6The energies (which now are labeled as Enr,p) obey the following inequality Enr,0 < Enr,1 < Enr+1,0 for
any coupling constant g.
40 Chapter 6. The Approximant
imposed for the excited ones. To fix the value of the coefficients of P
(p)
nr (x
2), some orthogonality
constraints are imposed. For fixed (nr, p), the (nr − 1) free parameters of P
(p)
nr (x
2) are found by
demanding7
〈Ψ(t)
nr,p|Ψ
(t)
kr,p
〉 = 0 , kr = 0, . . . , (nr − 1) . (6.2.3)
These constraints will ensure that the function Ψ
(t)
nr,p(r) has the correct number of nodes (n =
2nr + p).8 The remaining free parameters are fixed through the VM.
Case D > 1
Due to the separation of the Schrödinger equation (3.1.1) in spherical coordinates, it is sufficient
to approximate the radial part of the wave function. One can label the eigenstates by the
pair of quantum numbers (nr, ℓ), omitting the magnetic quantum number. In this case, the
Approximant is of the form
Ψ
(t)
nr,ℓ
(r) = rℓ P (ℓ)
nr
(r2) e−Φt , nr = 0, 1, ..., ℓ = 0, 1, ..., (6.2.4)
where P
(ℓ)
nr (r
2) is a polynomial of degree nr with real roots. We set P
(ℓ)
nr (0) = 1 as normalization.
Note that at g = 0, the polynomial P
(ℓ)
nr (r
2) is given in terms of the Laguerre associated
polynomials [48]
P (ℓ)
nr
(r2) = Lℓ−D/2−1
nr
(r2) . (6.2.5)
Analogously to the one-dimensional case, in (6.2.4) Φt has the same form of the phase con-
structed for the ground state (6.1.2), but with a different set of parameters {ã}, {b̃}, {c̃}. The
same set of constraints imposed on the parameters of the ground state should be imposed for
the excited ones. For a given (nr, ℓ), the coefficients of the polynomial P
(ℓ)
nr (r) are fixed imposing
the orthogonality constraints
〈Ψ(t)
nr,ℓ
|Ψ(t)
kr,ℓ
〉 = 0 , kr = 0, . . . , (nr − 1) . (6.2.6)
These constraints will make sure that the function Ψ
(t)
nr,p(r) has the correct number of nodes
(n = 2nr + l). Note that two states with different angular momentum, say ℓ and ℓ′ are already
orthogonal, independently of the value of their radial quantum numbers, see Section 3.1. The
remaining free parameters are fixed by means of the VM.
7 We used the familiar bra-ket notation to represent the scalar product.
8 In practice, n can be used as the only quantum number to label the states.
Chapter 7
Results for Two-Term Potentials
The formalism, described in the previous Chapters, is now applied to 3 particular cases of
two-term radial anharmonic potentials: cubic, quartic, and sextic. Hence, the forthcoming
discussion is focused on potentials of the form
V (r) = r2 + gm−2rm , (7.0.1)
with m = 3, 4, 6. For each anharmonicity, we present results of analytical and numerical nature
in the following order. We begin by presenting some general features of the PT coming from
the RB and GB equations developed in the weak coupling regime.1 Then, we construct the
Approximants of the first 4 low-lying states for which we estimate their corresponding energy
by means of the VM at gm−2 = 0.1, 1, 10 in D = 1, 2, 3, 6. For some of those states, we use
the connection between PT and the variational energy to estimate its accuracy, as well as the
accuracy of the Approximant. Finally, employing the Approximant, we compute in VM the
first two coefficients of the strong coupling expansion of the ground state energy. As we describe
below, the realization of the VM and the estimation of its accuracy is a numerical procedure. In
the next Section, we briefly describe the computational codes used in numerical computations.
Full details can be found in [54], [55], and [56]. Finally, let us mention that the Section for each
anharmonicity is self-contained, written in such a way that they can be read independently.
7.1 Numerical Realization of the Variational Method
It is not a surprise that the calculation of the variational energy Evar[ψt] with trial function ψt =
Ψ
(t)
0,0, shown in (6.1.7), requires to perform a one-dimensional numerical integration, see (1.13).
In addition, a numerical minimization is needed in order to obtain the optimal configuration of
parameters.
A computational code was written in FORTRAN 77. It uses the integration routine D01FCF
1 Some general properties of the two-term potential are summarized in Appendix D.
41
42 Chapter 7. Results for Two-Term Potentials
from the NAG-LIB, which was built using the algorithm described in [57]. The routine of
optimization that we used to find the optimal variational parameters was MINUIT of the
CERN-LIB. Needless to say, the use of Approximants for excited states as trial functions in
VM of the form (7.2.25) and (7.2.26) also require numerical integration and minimization.
Without loss of generality in (numerical) variational calculations we set ~ = 1 and M = 1/2,
thus v = r, ε = E, λ = g, and Y = y see (3.3.7), (3.3.8) and (3.3.10). With this setting, it is
clear that Evar corresponds to a variational estimate of the exact energy ε.
For each anharmonicity in D = 1, we performed variational calculations for the first three
low-lying states which are characterized by the quantum numbers (0, 0), (0, 1), and (1, 0), see
(6.2.1). In turn, at D = 2, 3, 6 we consider the first four states with quantum numbers (0,0),
(1,0), (0,1), and (0,2), see (6.2.4). We calculate the variational energy
Evar = E
(t)
0 + E
(t)
1 (7.1.1)
for those states. Using the connection between the variational energy and PT2, we take y0 =
∂rΦt as zeroth order correction. Then, the accuracy of the variational estimate Evar is found
by calculating the second correction3
E
(t)
2 (7.1.2)
in the expansion (1.18). Furthermore, we can define the corrected variational energy
Evar + E
(t)
2 (7.1.3)
which, in principle, should provide a more accurate estimate of the exact energy. Correction
E
(t)
2 is found numerically using the formalism of the Non-Linearization Procedure presented
in Section 4.1. The calculation of E
(t)
2 requires the knowledge of y1, which is already known
in integral form, see (4.1.9). A computational code in Mathematica package was designed; it
includes all relevant formulas. Note that once y1 is known, the accuracy of the Approximant
itself can be estimated via the relative deviation from the exact wave function
∣
∣
∣
∣
∣
Ψ0,0(r)−Ψ
(t)
0,0(r)
Ψ
(t)
0,0(r)
∣
∣
∣
∣
∣
. 10−δ . (7.1.4)
Here δ is a positive constant that is obtained numerically. In (7.1.4), the function Ψ0,0 denotes
the (unknown) exact one. If δ is sufficiently large, we will say that Ψ
(t)
0,0 is a locally accurate
approximation of the exact wave function Ψ0,0.
We can compare the corrected variational energies Evar + E
(t)
2 with those coming from an
independent numerical method. We choose the Lagrange-Mesh method [58]: an approximate
variational method simplified by a Gauss quadrature associated with the mesh. It is known
that this method, in the form proposed and developed by D. Baye and co-authors, provides
easily 12 - 13 s. d. in energies for various problems of atomic and molecular physics. A detailed
and complete description can be found in [58]. At D = 1, we use the Hermite mesh, while for
D > 1, the Laguerre one.
2Presented in Section 1.2
3 It is the second correction in formal PT, but the first one to the variational energy.
7.2. Cubic Anharmonic Oscillator 43
7.2 Cubic Anharmonic Oscillator
The simplest radial anharmonic oscillator is characterized by a cubic anharmonicity,
V (r) = r2 + g r3 , (7.2.1)
see (7.0.1) at m = 3. It is worth mentioning that many features that the cubic anharmonic
potential exhibits are also present in the general potential without symmetry of reflection
r → −r, i.e. V (r) 6= V (−r). Thus, the cubic anharmonic oscillator covers the basis of
understanding of more complex potentials. Let us note that at D = 1, r → |x|. Therefore, the
one-dimensional version of the potential (7.2.1) is
V (x) = x2 + g|x|3 , −∞ < x <∞ , (7.2.2)
which clearly holds bound states.
7.2.1 Perturbation Theory in the Weak Coupling Regime
For the cubic anharmonic oscillator, the perturbative expansions of ε and Y(v) can be derived
for the RB equation (3.3.9),
∂vY(v) − Y(v)
(
Y(v)− D − 1
v
)
= ε (λ) − v2 − λ v3 , ∂v ≡
d
dv
, (7.2.3)
with v and λ defined in (3.3.7) and (3.3.10), respectively. Those expansions remain functionally
the same as for the general oscillator,
ε = ε0 + ε1 λ + ε2 λ
2 + . . . , ε0 = D ,
and
Y(v) = Y0(v) + Y1(v)λ + Y2(v)λ
2 + . . . , Y0(v) = v .
Since potential (7.2.1) contains an odd degree monomial r3, the correction Yn(v) is characterized
by the infinite series (4.1.15) and (4.1.18) at small and large v, respectively. The first three
corrections Yn(v), at both v → 0 and v → ∞, are presented in Appendix B. The first corrections
ε1 and Y1(v) can be calculated in closed analytic form using (4.1.9) and (4.1.11),
ε1 =
Γ(D+3
2
)
Γ(D
2
)
, Y1(v) =
ev
2
2 vD−1
{
ε1 γ
(
D
2
, v2
)
− γ
(
D + 3
2
, v2
)}
, (7.2.4)
where Γ(a) and γ(a, b) denote the complete and incomplete gamma functions, respectively,
see [59]. Higher energy corrections εn can be computed only numerically.4 For example, the
first six corrections5 at D = 1 are
ε1 = 0.564 190 , ε2 = −0.373 405 , ε3 = 0.512 464 ,
ε4 = −0.941 502 , ε5 = 2.051 176 , ε6 = −5.022 987 . (7.2.5)
4Interestingly, the behavior of εn at large D can be obtained analytically in the 1/D-expansion, see [29].
5Rounded to the first displayed s. d.
44 Chapter 7. Results for Two-Term Potentials
7.2.2 Generating Functions
We can determine the coefficients of the expansion of Yn(v) at large v, c
(n)
k in (4.1.18), by
algebraic means. In general, they are written in terms of the corrections ε0, ε1, . . ., εn, see
Appendix B for explicit formulas for n = 1, 2, 3.
It was pointed out in [8,39,60,61] that the general expression for the coefficients c
(n)
0 , c
(n)
2 , . . .
can be obtained by solving recurrence relations. For example, c
(n)
0 satisfies non-linear recurrence
relation
c
(n)
0 = −1
2
n−1∑
k=1
c
(k)
0 c
(n−k)
0 , c
(1)
0 =
1
2
. (7.2.6)
In turn, the recurrence relation for c
(n)
2 is linear
c
(n)
2 =
1
2
(2n+D) c
(n)
0 −
n−1∑
k=1
c
(k)
0 c
(n−k)
2 , c
(1)
2 =
1
4
(D + 2) . (7.2.7)
The solution of (7.2.7) requires the knowledge of the coefficients c
(n)
0 . It is evident that in order
to find c
(n)
3 , the coefficients c
(n)
0 and c
(n)
2 are needed, and so on. The simplest way to solve these
recurrence relations is by means of generating functions. Surprisingly, we can calculate these
generating functions straightforwardly via the algebraic and iterative procedure derived from
the GB equation in the weak coupling regime. For the cubic anharmonic oscillator, the GB
equation reads
λ2 ∂uZ(u) − Z(u)
(
Z(u)− λ2(D − 1)
u
)
= λ2 ε(λ) − u2 − u3 , ∂u ≡ d
du
. (7.2.8)
For the particular cubic anharmonicity, the function Z(u) admits an expansion of the form
Z(u) = Z0(u) + Z2(u)λ
2 + Z3(u)λ
3 + . . . , (7.2.9)
with
Zk(u) = u
∞∑
n=1
c
(n)
k un−k . (7.2.10)
Evidently, the generating function of the coefficient c
(n)
k is nothing but Zk(u). Generating
functions can be easily calculated using formulas presented in (4.2.2) and (4.2.3). For example,
the first two are
Z0(u) = u
√
1 + u , (7.2.11)
Z2(u) =
u+ 2D
(
1 + u−
√
1 + u
)
4u(1 + u)
. (7.2.12)
From these expressions, the explicit solutions of the equations (7.2.6) and (7.2.7) may be easily
derived,
c
(n)
0 =
(−1)n+1 Γ(2n+ 1)
22n−1Γ(n) Γ(n+ 1)
, c
(n)
2 =
(−1)n+1
2
(
1 +D
Γ(2n+ 1)
22n Γ(n+ 1)2
)
. (7.2.13)
7.2. Cubic Anharmonic Oscillator 45
Note that the coefficients c
(n)
0 are D-independent, while c
(n)
2 depends on D linearly. As men-
tioned before, an important property of the generating functions Z0(u), Z2(u), Z3(u), ... is
related to the asymptotic expansion of the function y(r) at large r, keeping fixed the effective
coupling constant λ. This expansion is conveniently written in variable v,
y = (2M~
2)
1
4
(
λ1/2v3/2 +
1
2λ1/2
v1/2 − 1
8λ3/2
v−1/2 +
2D + 1
4
v−1 − (8λ2ε− 1)
16λ5/2
v−3/2 + . . .
)
(7.2.14)
see (3.3.7). Note that the first four terms of this expansion are ε-independent, but only first
three are D-independent. From another side, the expansion of (2M
g2
)1/2Z0(u) at large u is
(
2M
g2
)1/2
Z0(u) =
(
2M
g2
)1/2(
u3/2 +
1
2
u1/2 − 1
8
u−1/2 +
1
16
u−3/2 + . . .
)
. (7.2.15)
To compare the expansions (7.2.14) and (7.2.15), let us replace the classical coordinate u by
the quantum one v, taking into account that u = λv, see (3.3.24). Then, the expansion (7.2.15)
becomes
(
2M
g2
)1/2
Z0(λv) = (2M~
2)
1
4
(
λ1/2v3/2 +
1
2λ1/2
v1/2 − 1
8λ3/2
v−1/2 +
1
16λ5/2
v−3/2 + . . .
)
.
(7.2.16)
One can intermediately see that it reproduces exactly the expansion (7.2.14) up to O(v−1/2).
With a similar approach, we can demonstrate that at large r the function
(
2M
g2
)1/2
λ2Z2(λv)
contributes to the expansion (7.2.14) from O(v−1), since
(
2M
g2
)1/2
λ2Z2(λv) = (2M~
2)
1
4
(
2D + 1
4
v−1 − D
2λ1/2
v−3/2 − 1
4λ
v−2 + . . .
)
. (7.2.17)
Note that the expansion of (2M
g2
)1/2(Z0(gr) + λ2Z2(gr)) at large v reproduces exactly the first
four terms in (7.2.14). The expansion of the next generating functions (2M
g2
)1/2λ3Zn(λv), with
n > 2, starts from O(v−3/2). Explicitly,
(
2M
g2
)1/2
λnZn(λv) = (2M~
2)
1
4
(
−εn−2 λ
n−5/2
2
v−3/2 + . . .
)
, v → ∞ , n > 2 , (7.2.18)
where εn−2 is the energy PT correction of the order (n− 2). As a consequence, no matter how
many generating functions we consider in the expansion (2M
g2
)1/2(Z0(λv) + λ2Z2(λv) + ...), the
term of order O(v−3/2) in (7.3.18) can not be reproduced exactly.
46 Chapter 7. Results for Two-Term Potentials
7.2.3 The Approximant
By means of (4.4.2), we can calculate the first two6 generating functions, G0(r) and G2(r) in
the expansion (4.4.1). They are given by
G0(r) =
(2M)1/2
g2
(
2(−2 + 3gr)(1 + gr)3/2
15
)
, (7.2.19)
G2(r) =
(2M)1/2
g2
(
1
4
log[1 + gr] +D log
[
1 +
√
1 + gr
])
. (7.2.20)
In contrast with G2(r), the function G0(r) contains no logarithmic terms. Theses generating
functions serve as the building blocks for the construction of the Approximant. Following the
general prescription (6.1.2), we write the Approximant of the ground state in the exponential
representation Ψ
(t)
0,0 = e−
1
~
Φt with
1
~
Φt(r) =
ã0 + ã1 gr + ã2 r
2 + ã3 g r
3
√
1 + b̃1 g r
+
1
4
log[1 + b̃1 g r] + D log
[
1 +
√
1 + b̃1 g r
]
.
(7.2.21)
We set
ã3 =
2
5
b̃
1/2
1 , (7.2.22)
in order to reproduce exactly the leading term in the asymptotic behavior of the phase at
r → ∞. Here, the logarithmic terms in Φt represent a minimal modification of those coming
G2(r), cf. (7.2.20). Additionally, we impose the constraint
ã1 =
b̃1
4
(2 ã0 −D − 1) (7.2.23)
in order to guarantee a vanishing Φ′
t at r = 0. Thus, there is no linear term in the expansion
of (7.2.21) at small distances, just as it is indicated in (3.3.16). Eventually, the Approximant
in its final form depends on three free parameters {ã0, ã2, b̃1} only, and it reads
Ψ
(t)
0,0(r) =
1
(
1 + b̃1 g r
)1/4
(
1 +
√
1 + b̃1 g r
)D
exp

− ã0 + ã1 g r + ã2 r
2 + ã3 g r
3
√
1 + b̃1 g r

 ,
(7.2.24)
6The next two functions G3(r) and G4(r) are presented explicitly in Appendix B.
7.2. Cubic Anharmonic Oscillator 47
where ã3,1 are defined by constraints (7.2.22), (7.2.23). This is the eventual expression of the
3-parametric approximate ground state wave function.
To construct 3-parametric Approximants for excited states, we follow the prescription pre-
sented in Section 6.2. Hence, at D = 1 we have
Ψ(t)
nr,p(x) =
xpP
(p)
nr (x
2)
(
1 + b̃1g|x|
)1/4
(
1 +
√
1 + b̃1g|x|
)D
exp

− ã0 + ã1g|x|+ ã2x
2 + ã3g|x|3
√
1 + b̃1g|x|

 ,
(7.2.25)
while at D > 1
Ψ
(t)
nr,ℓ
(r) =
rℓ P
(ℓ)
nr (r
2)
(
1 + b̃1 g r
)1/4
(
1 +
√
1 + b̃1 g r
)D
exp

− ã0 + ã1 g r + ã2 r
2 + ã3 g r
3
√
1 + b̃1 g r

 .
(7.2.26)
Definitions of P
(p)
nr (x
2) and P
(ℓ)
nr (r
2) are presented in (6.2.1) and (6.2.4), explicitly.
7.2.4 Numerical Results: Variational Calculations
The calculations of the variational energy for the first low-lying states with quantum numbers
(0, 0), (0, 1), (0, 2), (1, 0) are presented in Tables 7.1 - 7.4 for different values of D and g. For
some states, the variational energy Evar = E
(t)
0 + E
(t)
1 , the first correction E
(t)
2 to it as well as
its corrected value Evar + E
(t)
2 are shown. All states are studied in dimension D = 1, 2, 3, 6
and coupling constant g = 0.1, 1, 10. For all cases, the variational energy Evar is obtained with
absolute deviation from the exact result of the order 10−7−10−8 (7−8 s.d.7). This deviation is
found by calculating the second correction E
(t)
2 (the first correction to the variational energy).
To check the energies obtained in Non-Linearization Procedure, we calculated them numeri-
cally in the Lagrange-Mesh method. These numerical calculations show that all digits presented
for Evar +E
(t)
2 - the variational energy with the first (second order in PT) correction E
(t)
2 taken
into account - are exact. We obtain not less than 9 d.d.8 correctly in Non-Linearization Proce-
dure. This accuracy is confirmed independently by calculating the second correction E
(t)
3 to the
7From now on, s.d. stands for significant digits.
8Decimal digits is abbreviated as d.d.
48 Chapter 7. Results for Two-Term Potentials
variational energy: this correction is . 10−9 for any D and g that we have studied. It indicates
a very fast rate of convergence in Non-Linearization Procedure with trial functions (7.2.24),
(7.2.25), and (7.2.26) taken as zeroth order approximations of the wave function. In general,
the energies grow with the increase of D and/or g. At D > 1, we can classify the states by
radial quantum number nr and angular momentum ℓ as (nr, ℓ); we mention the hierarchy of
eigenstates which holds for any fixed integer D > 1 and g > 0: (0,0), (0,1), (0,2), (1,0).
It has to be noted that to the best of our knowledge, these results represent the first accu-
rate calculations of the energy of the low-lying states for the D-dimensional cubic anharmonic
oscillator. Hence, our results will serve as the benchmark for future studies. The Devi-
ation of the Approximant from the exact (unknown) eigenfunction can be estimated via the
Non-Linearization Procedure. It can be shown that for the ground state function the relative
deviation is extremely small and bounded.
∣
∣
∣
∣
∣
Ψ0,0(r)−Ψ
(t)
0,0(r)
Ψ
(t)
0,0
∣
∣
∣
∣
∣
. 10−4 , (7.2.27)
This small deviation occurs in the whole range of r ∈ [0,∞) at any dimension D that we
explored and at any coupling constant g > 0 considered.
Therefore, the Approximant leads to a locally accurate approximation of the exact wave
function Ψ0,0(r) once the optimal parameters for Ψ
(t)
0,0 are chosen. In general, optimal variational
parameters are smooth and slow changing functions of g without a strong D dependence. As
an example, for the ground state (0,0) the plots of the parameters ã0,2,3 vs g for D = 2, 3, 6
are shown in Appendix C, see Fig. C.1. A similar situation appears for the excited states for
different D and g. Analyzing those plots, one can see the (accidental) appearance of another
constraint,
ã0 =
125 ã23 + 12
150 ã3
. (7.2.28)
It is fulfilled by the optimal parameters with high accuracy. This constraint guarantees an
approximate reproduction of the subdominant term in the expansion (7.2.14). One can demand
the Approximant to fulfill exactly this constraint. When doing so, no deterioration in the
accuracy of the wave function and energy occurs.
In all cases studied, the first correction y1 to the logarithmic derivative y0 = ∂rΦt is un-
bounded. The reason is clear: we have only imposed the constraint (7.2.22) to Φt, the exact
reproduction of the dominant term in the expansion (7.2.14).9 If in addition to constraints
(7.2.22), (7.2.23), and (7.2.28), we impose
− g2ã1 =
9375 a43 − 1000 a23 + 48
15000 a33
, (7.2.29)
Φt reproduces (exactly) all growing terms of the exact phase at r → ∞: r5/2, r3/2, r1/2, and log r
see (7.2.14). Thus, y1 will be bounded. Surprisingly, the accuracy of the variational estimate
9The approximate reproduction of the subdominant term was imposed by the VM.
7.2. Cubic Anharmonic Oscillator 49
Table 7.1: Ground state energy for the cubic potential V (r) = r2 + g r3 for D = 1, 2, 3, 6 and
g = 0.1, 1, 10, labeled by quantum numbers (0,0) at any D. Variational energy Evar = Evar, the
first correction E
(t)
2 (rounded to three s.d.) found with use of Ψ
(t)
0,0, see text, and the corrected
energy Evar +E
(t)
2 = Evar +E
(t)
2 shown. Evar +E
(t)
2 coincides with Lagrange Mesh results (see
text) in 9 displayed d. d., hence all printed digits are exact.
g
D = 1 D = 2
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 1.053 120 300 5.39× 10−7 1.053 119 761 2.124 027 648 4.40× 10−7 2.124 027 208
1.0 1.387 428 891 4.00× 10−8 1.387 428 851 2.877 490 906 3.76× 10−8 2.877 490 868
10.0 2.729 533 139 6.56× 10−7 2.729 532 483 5.794 213 459 5.58× 10−7 5.794 212 901
g
D = 3 D = 6
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 3.208 922 743 4.00× 10−7 3.208 922 343 6.528 432 540 2.02× 10−7 6.528432338
1.0 4.442 965 260 3.15× 10−8 4.442 965 229 9.465 319 951 1.85× 10−8 9.465319933
10.0 9.094 985 589 4.23× 10−7 9.094 985 166 19.981 458 504 1.96× 10−7 19.981 458 308
Table 7.2: The first excited state energy for the cubic potential V (r) = r2 + g r3 for different
D and g. Variational energy Evar, the first correction E
(t)
2 found with use of Ψ
(t)
0,1 for D = 1
and Ψ
(t)
0,1 for D > 1, see (7.2.25) and (7.2.26). The corrected energy Evar + E
(t)
2 = Evar + E
(t)
2
shown, the correction E
(t)
2 rounded to 3 s.d. All printed digits for Evar + E
(t)
2 are exact.
g
D = 1 D = 2
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 3.208 922 765 4.21× 10−7 3.208 922 343 4.305 557 665 3.55× 10−7 4.305 557 309
1.0 4.442 965 265 3.59× 10−8 4.442 965 229 6.068 723 537 2.92× 10−8 6.068 723 507
10.0 9.094 985 630 4.64× 10−7 9.094 985 166 12.579 594 377 3.48× 10−7 12.579 594 029
g
D = 3 D = 6
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 5.412 425 220 2.86× 10−7 5.412 424 933 8.784 695 351 1.21× 10−7 8.784 695 230
1.0 7.745 092 165 2.41× 10−8 7.745 092 141 13.018 486 318 1.49× 10−8 13.018 486 303
10.0 16.215 748 127 2.66× 10−7 16.215 747 861 27.841 430 199 1.37× 10−7 27.841 430 061
50 Chapter 7. Results for Two-Term Potentials
Table 7.3: The second excited state energy for the cubic potential V (r) = r2 + g r3 for different
D and g. Variational energy Evar found with use of Ψ
(t)
1,0 for D = 1 and Ψ
(t)
0,2 for D > 1, see
(7.2.25) and (7.2.26). The first correction E
(t)
2 and the corrected energy Evar+E
(t)
2 = Evar+E
(t)
2
shown. Correction E
(t)
2 rounded to 3 s.d. All 9 printed d. d. in Evar + E
(t)
2 are exact.
g
D = 1 D = 2
Evar Evar −E(t)
2 Evar + E
(t)
2
0.1 5.436 849 553 6.528 432 582 2.43× 10−7 6.528 432 338 34
1.0 7.879 141 644 9.465 319 955 2.21× 10−8 9.465 319 932 56
10.0 16.641 305 904 19.981 458 531 2.23× 10−7 19.981 458 308 14
g
D = 3 D = 6
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 7.652 743 974 1.87× 10−7 7.652 743 787 11.069 434 802 6.73× 10−8 11.069 434 735
1.0 11.224 406 591 1.87× 10−8 11.224 406 573 16.699 837 135 1.22× 10−8 16.699 837 123
10.0 23.860 743 313 1.78× 10−7 23.860 743 135 36.070 426 676 1.01× 10−7 36.070 426 576
Table 7.4: The third excited state energy for the cubic potential V = r2 + g r3 for different D
and g. The radial node r0 calculated in Lagrange-Mesh method with 9 correct d.d. Variational
energy Evar and its radial node r
(0)
0 given by underlined digits, both found with Ψ
(t)
1,0, see (7.2.26).
g
D = 2 D = 3 D = 6
Evar r
(0)
0 Evar r
(0)
0 Evar r
(0)
0
0.1 6.570 941 476 0.953 355 772 7.709 696 057 1.162 462 764 11.158 149 471 1.626 259 519
1.0 9.690 376 045 0.780 270 156 11.517 370 455 0.942 382 083 17.128 462 904 1.289 488 891
10.0 20.681 622 942 0.532 038 126 24.758 598 047 0.638 712 790 37.346 045 150 0.865 039 246
will not be dramatically deteriorated for the ground state: it can provide not less than 5 s.d
for the energy in the whole domain studied. Furthermore, the Approximant results in a locally
accurate approximation of the exact ground state wave function. The relative deviation does
not exceed ∼ 0.02. This is surprising, taking into account that the Approximant will be one
parametric. Interestingly, the dependence of the single parameter as function of D and g is
very weak, see Fig. 7.1.
Results of similar accuracy occur for excited states.
In our present calculations10, the ratio |y1/y0| is bounded and small. Thus, y1 is a small
function in comparison with y0. In Fig. 7.2, it is presented y0 and y1 vs r for g = 1 for
dimension D = 1. Similar plots appear for D = 2, 3, 6. Making an analysis of these plots, one
can see that in the domain 0 ≤ r . 1 , which gives the dominant contribution to the integrals
10In which we have only imposed (7.3.27) and (7.2.23) as constraints.
7.2. Cubic Anharmonic Oscillator 51
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
g
a 3
Figure 7.1: Variational parameter ã3 for the one-parametric ground state (0, 0) Approximant
as function of g for D = 1, 2, 3, 6. From g ≈ 2.3 the single parameter ã3 becomes approximately
D-independent.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
1
2
3
4
5
6
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.000
0.002
0.004
0.006
0.008
0.010
0.012
Figure 7.2: Function y0 = ∂rΦt (on left) and its first order correction y1 (on right) at g = 1 as
a function of r at D = 1.
which define the energy corrections, |y1| is extremely small compared to |y0|, being close to
zero. It explains why the energy correction E
(t)
2 is small being of order ∼ 10−7, or ∼ 10−8.
In a similar way, one can show numerically that higher corrections y2, y3, . . . drop down to
zero in the domain 0 ≤ r . 1 even faster. This is a clear indication of the convergence of the
expansion (1.18) as n→ ∞.
At D > 1, we checked that the Approximant Ψ
(t)
1,0 also provides an accurate estimate of
the position of the radial node of the exact wave function, see (7.2.26). The orthogonality
condition described in Section 6.2 provides a simple analytic expression for the zeroth order
approximation r
(0)
0 to r0. Making a comparison of r
(0)
0 with numerical estimates which come
from the Lagrange-Mesh method with ∼ 50 mesh points, we can see the coincidence of r0 and
r
(0)
0 for at least 5 d.d. at integer dimension D at any coupling constant g ≥ 0. The results are
presented in Table 7.4.
52 Chapter 7. Results for Two-Term Potentials
7.2.5 First Terms in the Strong Coupling Expansion
We present the estimates of the first two terms in the strong coupling expansion (5.1.7) for
the ground state energy of the cubic oscillator. Assuming for simplicity 2M = ~ = 1, this
expansion has the form
E = g2/5
(
ε̃0 + ε̃1g
−4/5 + ε̃2g
−8/5 + . . .
)
. (7.2.30)
It is worth mentioning that similar expansion holds for any excited state. This expansion
corresponds to PT in powers of λ̂ for the potential
V (w) = w3 + λ̂ w2 , λ̂ = g−4/5 , (7.2.31)
defined in w ∈ [0,∞). Evidently, it has a finite radius of convergence.
As the first step, we focus on the calculation of ε̃0 in (7.2.30), which is the ground state
energy in the pure cubic potential V = w3 (the ultra-strong coupling regime). To do it, we use
the simplest physically relevant trial function (5.1.12),
Ψ
(t)
0,0 = e−
2
5
w5
. (7.2.32)
Taking Ψ
(t)
0,0 as zero approximation, we calculate variational energy analytically and then develop
the PT procedure described in Section 5.1. In this manner we obtain different PT corrections11
ε̃0,k , k = 0, 1, 2, 3, . . . to the exact value of ε̃0 and then form the partial sums (5.1.15). The first
non-trivial partial sum (ε̃0,k + ε̃0,1) is equal to variational energy. Explicit results, including
partial sums up to sixth order, are presented in Table 7.5 for the one-dimensional case. We have
to note a slow convergence of the partial sums to the exact result with respect to the increase
of the number of perturbative terms in the expansion (5.1.15). For example, the partial sum
∑6
k=0 ε̃0,k with seven terms included allows us to reproduce 7 s.d. of the exact result12 It is
natural to accelerate convergence taking more advanced trial functions as entry.
Table 7.5: Different partial sums (5.1.15) for the leading term ε̃0 of the strong coupling ex-
pansion (7.2.30) for the one-dimensional cubic anharmonic oscillator in PT with trial function
(7.2.32) as zero approximation.
ε̃0,0
∑1
k=0 ε̃0,k
∑2
k=0 ε̃0,k
∑3
k=0 ε̃0,k
∑4
k=0 ε̃0,k
∑5
k=0 ε̃0,k
∑6
k=0 ε̃0,k Exact ε̃0
0 1.053 006 976 1.021 174 929 1.022 989 568 1.022 956 899 1.022 946 414 1.022 947 763 1.022 947 875
As an alternative to the trial function (7.2.32), let us use the Approximant13 (7.2.21) in order
to calculate the first two terms in the strong coupling expansion (7.2.30), see also (5.1.7). In
Table 7.6, we present for different D the leading coefficient of the strong coupling expansion
11The calculation of corrections is a numerical process.
12By saying exact, we mean that the (truncated) displayed d.d. in ε̃0 are exact digits.
13We fix g = 1 as normalization of the free parameters.
7.2. Cubic Anharmonic Oscillator 53
ε̃0 found in VM, denoted by ε̃
(1)
0 , and the second PT correction to it ε̂2 calculated via the
Non-Linearization Procedure. We introduce the partial sum ε̃
(2)
0 = ε̃
(1)
0 + ε̂2. The final results
are verified using the Lagrange-Mesh method with 200 mesh points. One can see that system-
atically, the correction ε̂2 is of order ∼ 10−7. Hence, the first six d. d. in variational energy are
correct. It defines the accuracy of variational calculations of ε̃0 with the Approximant (7.2.21)
as the trial function. One can estimate the order of the third PT correction: for all D studied
ε̂3 ∼ 10−2 ε̂2. It indicates an extremely high rate of convergence ∼ 10−2 in PT. In Table 7.7 we
present the first two approximations of the coefficient ε̃1 - the expectation value of w2 - , see
(7.2.30).
Table 7.6: Ground state energy ε̃0, see (7.2.30), for the potential W (w) = w3 for D = 1, 2, 3, 6
found in PT with the Approximant Ψ
(t)
0,0 as the entry: ε̃
(1)
0 is the variational energy, ε̂2 is the
second PT correction and ε̃
(2)
0 = ε̃
(1)
0 + ε̂2 is corrected variational energy. Eight d.d. in ε̃
(2)
0
confirmed in Lagrange mesh calculation.
D = 1 D = 2
ε̃
(1)
0 −ε̂2 ε̃
(2)
0 ε̃
(1)
0 −ε̂2 ε̃
(2)
0
1.022948250 3.75× 10−7 1.022 947 875 2.187 461 809 3.09× 10−7 2.187 461 499
D = 3 D = 6
ε̃
(1)
0 −ε̂2 ε̃
(2)
0 ε̃
(1)
0 −ε̂2 ε̃
(2)
0
3.450 562 918 2.29× 10−7 3.450 562 689 7.647 118 254 1.01× 10−7 7.647 118 153
Table 7.7: Subdominant (next-to-leading) term ε̃1 in the strong coupling expansion (7.2.30) of
the ground state energy for the cubic anharmonic radial potential for D = 1, 2, 3, 6. The first
correction to it ε̃1,1 as well as the corrected value ε̃
(2)
1 = ε̃
(1)
1 + ε̃1,1 displayed.
D = 1 D = 2
ε̃
(1)
1 ε̃1,1 ε̃
(2)
1 ε̃
(1)
1 ε̃1,1 ε̃
(2)
1
0.410 598 524 6.78× 10−7 0.410 599 202 0.766 573 847 5.24× 10−7 0.766 574 371
D = 3 D = 6
ε̃
(1)
1 ε̃1,1 ε̃
(2)
1 ε̃
(1)
1 ε̃1,1 ε̃
(2)
1
1.092 125 224 2.67× 10−7 1.092 125 491 1.967 599 668 1.42× 10−7 1.967 599 810
54 Chapter 7. Results for Two-Term Potentials
7.3 Quartic Anharmonic Oscillator
The simplest and most popular formally even radial anharmonic oscillator potential is charac-
terized by a quartic anharmonicity
V (r) = r2 + g2 r4 , (7.3.1)
cf. (7.0.1) at m = 4. Many properties that the quartic anharmonic radial oscillator exhibits
are typical for any even anharmonic potential, V (r) = V (−r). In particular, the polynomial
nature of corrections εn and Yn(v), see (4.1.11) and (4.1.9) is one of such common features.
7.3.1 Perturbation Theory in the Weak Coupling Regime
For the quartic anharmonic oscillator, the perturbative expansions of ε and Y(v) may be derived
from the RB equation (3.3.9),
∂vY(v) − Y(v)
(
Y(v)− D − 1
v
)
= ε (λ) − v2 − λ2 v4 , ∂v ≡
d
dv
, (7.3.2)
where v and λ are defined in (3.3.7) and (3.3.10), respectively. These expansions are of the
form
ε = ε0 + ε2 λ
2 + ε4 λ
4 + . . . , ε0 = D , (7.3.3)
and
Y(v) = Y0(v) + Y2(v)λ
2 + Y4(v)λ
4 + . . . , Y0(v) = v , (7.3.4)
respectively. All odd terms in λ vanish in both expansions, see (4.1.5) and (4.1.6). In general,
a large number of corrections can be calculated by linear algebra means. In particular, the first
non-vanishing corrections are
ε2 =
1
4
D (D + 2) , Y2(v) =
1
2
v3 +
1
4
(D + 2) v . (7.3.5)
The next two corrections ε4,6 and Y4,6(v) are presented in Appendix B. The algebraic procedure
of finding PT corrections holds for all even anharmonic potential, V (r) = V (−r). However, it
is enough to have one single odd monomial term in the potential to break this property down.
In this situation, the calculation of the correction εn becomes a numerical procedure, just like
it happened for the cubic case. In general, all corrections Y2n(v) are odd-degree polynomials in
v of the form,
Y2n(v) = v
n∑
k=0
c
(2n)
2k v2(n−k) , (7.3.6)
where any coefficient c
(2n)
2k is a polynomial in D of degree k,
c
(2n)
2k = P
(2n)
k (D) , c
(2n)
2n =
ε2n
D
. (7.3.7)
7.3. Quartic Anharmonic Oscillator 55
The energy corrections ε2n are of the form [60]
ε2n(D) = D (D + 2)Rn−1(D) , (7.3.8)
where Rn−1(D) is a polynomial of degree (n− 1) in D. In particular, R0(D) = 1
4
.
From (7.3.8), one can see that any energy correction ε2n vanishes when D = 0. In this
situation, the formal sum of corrections results in ε = 0, and ultimately in E = 0. Thus, the
radial Schrödinger equation is reduced to
− ~2
2M
(
d2Ψ(r)
dr2
− 1
r
dΨ(r)
dr
)
+ (r2 + g2 r4)Ψ(r) = 0 (7.3.9)
at D = 0. This equation defines the zero mode of the radial Schrödinger operator at D = 0. It
can be solved exactly in terms of Airy functions [39],
Ψ = C1 Ai
(
1 + (λv)2
λ4/3
)
+ C2 Bi
(
1 + (λv)2
λ4/3
)
, (7.3.10)
see (3.3.7) and (3.3.10) for the definition of λ and v, respectively. However, this linear com-
bination can not be made normalizable14 by any choice of constants C1 and C2. Therefore, it
is an indication of the absence of a normalizable zero mode at D = 0. Interestingly, at the
non-physical dimension D = −2, ε2n with n > 1 also vanish. Thus, the formal sum of the
corrections results is ε = −2, see (7.3.8). In this case, no exact solution for the corresponding
radial Schrödinger equation was found.
7.3.2 Generating Functions
For the quartic anharmonic oscillator, the function Z(u) satisfies the GB equation
λ2 ∂uZ(u) − Z(u)
(
Z(u)− λ2(D − 1)
u
)
= λ2 ε(λ) − u2 − u4 , ∂u ≡ d
du
, (7.3.11)
cf. (3.3.20). The solution of this equation can be written as an expansion in terms of generating
functions, namely,
Z(u) = Z0(u) + Z2(u)λ
2 + Z4(u)λ
4 + . . . , (7.3.12)
where each coefficient is of the form
Z2k(u) = u
∞∑
n=k
c
(2n)
2k u2(n−k) , k = 0, 1, . . . . (7.3.13)
Here, the expansion of ε in powers of λ is given by (7.3.3). Note that all generating functions
Z2k+1(u), k = 1, 2, ..., of odd order λ2k+1 are absent in expansion (4.2.1). Interestingly, from the
14At D = 0, the normalizability of the wave function is guaranteed if
∫
∞
0
ψ2 r−1dr <∞.
56 Chapter 7. Results for Two-Term Potentials
polynomial form of the coefficient c
(2n)
2k in D, see (7.3.8), one can find the structure of generating
function in the variable D, namely
Z2k(u) = u
k∑
n=0
f (k)
n (u2)Dn , (7.3.14)
where f
(k)
n (u2), n = 0, 1, ..., k, are real functions. Hence, Z2k(u) is a polynomial in D of degree
k. For instance, this feature can be explicitly noted in the first two terms of the expansion
(7.3.12),
Z0(u) = u
√
1 + u2 , (7.3.15)
Z2(u) =
u2 +D
(
1 + u2 −
√
1 + u2
)
2u(1 + u2)
. (7.3.16)
Thus,
f
(0)
0 (u2) =
√
1 + u2 , f
(2)
0 (u2) =
1
2(1 + u2)
, f
(2)
1 (u2) =
1 + u2 −
√
1 + u2
2u2(1 + u2)
. (7.3.17)
The asymptotic behavior of the generating functions Z2k(u), k = 0, 1, 2, . . ., in the expansion
(7.3.12) at large u is related to the asymptotic behavior of the function y(r) at large r in a
quite interesting manner. It can be easily found that for fixed effective coupling constant λ,
the asymptotic expansion of y at large r, written in variable v, see (3.3.7), has the form
y = (2M~
2)
1
4
(
λv2 +
1
2λ
+
D + 1
2
v−1 − 4λ2ε+ 1
8λ3
v−2 + . . .
)
, v → ∞ . (7.3.18)
Note that the first three terms of the expansion are ε-independent, but only the first two are
D-independent. On the other hand, the first three terms in the expansion of lowest generating
function (2M
g2
)1/2Z0(u) at large u are
(
2M
g2
)1/2
Z0(u) =
(
2M
g2
)1/2(
u2 +
1
2
− 1
8
u−2 + . . .
)
, u→ ∞ , (7.3.19)
see (7.3.15). To compare the expansions (7.3.18) and (7.3.19), let us replace the classical
coordinate u by the quantum one v via u = λ v, see (3.3.24). Evidently, large v implies large
u and vice versa (as long as λ is fixed). Then the expansion (7.3.19) becomes
(
2M
g2
)1/2
Z0(λv) = (2M~
2)
1
4
(
λv2 +
1
2λ
− 1
8λ3
v−2 + . . .
)
, v → ∞ . (7.3.20)
It reproduces exactly the first two terms in (7.3.18) but fails to reproduce the term of order
O(v−1), which is absent. However, the next generating function (2M
g2
)1/2λ2Z2(λv) at large v-
expansion reproduces the term O(v−1) exactly in the original expansion (7.3.18),
(
2M
g2
)1/2
λ2Z2(λv) = (2M~
2)
1
4
(
D + 1
2
v−1 − D
2λ
v−2 + . . .
)
, v → ∞ . (7.3.21)
7.3. Quartic Anharmonic Oscillator 57
In turn, it fails to reproduce the term O(v−2) correctly. Thus, the expansion of the sum
(2M
g2
)1/2(Z0(λv) + λ2Z2(λv)) at large v reproduces exactly the first three ε-independent terms
in the expansion (7.3.18). These three terms are responsible of the normalizability of the wave
function at large v.
All higher generating functions Z4(λv), Z6(λv) . . . contribute at large v to the same term
O(v−2) as follows
(
2M
g2
)1/2
λ2nZ2n(λv) = (2M~
2)
1
4
(
−ε2n−2 λ
2n−3
2
v−2 + . . .
)
, v → ∞ , n > 2 , (7.3.22)
where ε2n−2 is the energy PT correction of the order (2n − 2). As a consequence, no matter
how many generating functions we consider in the expansion (2M
g2
)1/2(Z0(λv) + λ2Z2(λv) + ...),
the term of order O(v−2) of (7.3.18) can not be reproduced exactly.
7.3.3 The Approximant
The first two15 generating functions of the phase in the expansion (4.4.1), G0(r) and G2(r), are
given by
G0(r) =
(2M)1/2
g2
(
(1 + g2r2)
3/2
3
)
, (7.3.23)
G2(r) =
(2M)1/2
g2
(
1
4
log[1 + g2r2] +
D
2
log
[
1 +
√
1 + g2r2
])
. (7.3.24)
These two functions serve as building blocks for the construction of the Approximant. Following
the general prescription (6.1.2), we write the Approximant of the ground state in the exponential
representation Ψ
(t)
0,0 = e−
1
~
Φt where
1
~
Φt(r) =
ã0 + ã2 r
2 + ã4 g
2 r4
√
1 + b̃4 g2 r2
+
1
4
log
[
1 + b̃4 g
2 r2
]
+
D
2
log
[
1 +
√
1 + b̃4 g2 r2
]
,
(7.3.25)
here ã0,2,4, and b̃4 are free parameters. The logarithmic terms, added in (7.3.25), generate a
prefactor to the exponential function. These terms are just a certain minimal modification of
those which occur in the second generating function (7.3.24). As a result, the Approximant of
the ground state for arbitrary D = 1, 2, 3, . . . is given by
15The generating functions G4(r) and G6(r) are presented explicitly in Appendix B.
58 Chapter 7. Results for Two-Term Potentials
Ψ
(t)
0,0(r) =
1
(
1 + b̃4 g2 r2
)1/4
(
1 +
√
1 + b̃4 g2 r2
)D/2
exp

− ã0 + ã2 r
2 + ã4 g
2 r4
√
1 + b̃4 g2 r2

 .
(7.3.26)
This is the central formula of this Section devoted to the quartic anharmonicity. We will
show that it provides a highly accurate uniform approximation of the exact ground state wave
function. We impose the constraint
b̃4 = 9 ã24 , (7.3.27)
which allows us to reproduce the dominant term in the expansion (7.3.18) exactly16. Hence,
it reproduces the exact asymptotic behavior of the phase at large distances. Therefore, the
final form of the Approximant contains only 3 free parameters, say {ã0, ã2, ã4}. Note that by
choosing
ã0 =
(2M)1/2
3g2
, ã2 =
(8M)1/2
3
, ã4 =
(2M)1/2
3
, (7.3.28)
the phase Φt reproduces exactly the first two terms in the expansion in generating functions
(4.4.1). However, parameters ã0, ã2, ã4 in (7.3.28) are not optimal from the point of view of
the VM. To construct 3-parametric Approximants for excited states, we follow the prescription
presented in the previous Section 6.2. Hence, at D = 1 we have
Ψ(t)
nr,p(x) =
xpP
(p)
nx (x
2)
(
1 + b̃4 g2 x2
)1/4
(
1 +
√
1 + b̃4 g2 x2
)D/2
exp

− ã0 + ã2 x
2 + ã4 g
2 x4
√
1 + b̃4 g2 x2

 ,
(7.3.29)
while at D > 1
Ψ
(t)
nr,ℓ
(r) =
rℓ P
(ℓ)
nr (r
2)
(
1 + b̃4 g2 r2
)1/4
(
1 +
√
1 + b̃4 g2 r2
)D/2
exp

− ã0 + ã2 r
2 + ã4 g
2 r4
√
1 + b̃4 g2 r2

 .
(7.3.30)
Definitions of P
(p)
nr (x
2) and P
(ℓ)
nr (r
2) are presented in (6.2.1) and (6.2.4), explicitly.
16In numerical computations we set ~ = 1 and M = 1/2.
7.3. Quartic Anharmonic Oscillator 59
7.3.4 Numerical Results: Variational Calculations
The calculations of the variational energy for the first low-lying states with quantum numbers
(0,0), (0,1), (0,2), (1,0) are presented in Tables 7.8 - 7.11 for different values of D and g2. For
some of those states, the variational energy Evar = E
(t)
0 +E
(t)
1 , the first correction E
(t)
2 to it, as
well as its corrected value Evar+E
(t)
2 are shown. All states are studied in dimensionD = 1, 2, 3, 6
and coupling constant g2 = 0.1, 1, 10. The variational energy Evar is obtained with absolute
deviation (from the exact result) in the order 10−8−10−14 (8−14 s.d.). This deviation is found
by calculating the second correction E
(t)
2 (the first correction to the variational energy).
To check the energies obtained in Non-Linearization Procedure, we calculated them numeri-
cally in the Lagrange-Mesh method. These numerical calculations show that all digits presented
for Evar + E
(t)
2 are exact: we obtain not less than 12 exact d.d. in the Non-Linearization Pro-
cedure. This accuracy is confirmed independently by calculating the second correction E
(t)
3 to
variational energy: this correction is . 10−13 for any D and g2 that we have studied. It indi-
cates a very fast rate of convergence in Non-Linearization Procedure with trial function either
(7.3.29) or (7.3.30) taken as zero approximation. In general, the energies grow with an increase
of D and/or g2. At D > 1 we can classify the states by radial quantum number nr and angular
momentum ℓ as (nr, ℓ), we mention the hierarchy of eigenstates which holds for any fixed integer
D > 1 and g2: (0,0), (0,1), (0,2), (1,0). There is a considerable number of calculations devoted
to estimating the energy of the first low-lying states. Our results, see Tables 7.8 - 7.11, are in
complete agreement with [23] and [24] for D = 1, and superior considerably of those obtained
for different D > 1 and g2, see e.g. [25], [26] and [32]. At D = 6 the calculations are carried
out for the first time.
The Deviation of the Approximant from the exact (unknown) eigenfunction can be estimated
via the Non-Linearization Procedure. For the ground state function, this deviation is extremely
small and bounded, ∣
∣
∣
∣
∣
Ψ0,0(r)−Ψ
(t)
0,0(r)
Ψ
(t)
0,0(r)
∣
∣
∣
∣
∣
. 10−6 (7.3.31)
in the whole range of r ∈ [0,∞) at any D and g2. Therefore, the Approximant of the ground
state is a locally accurate approximation of the exact wave function Ψ0,0(r) once the optimal pa-
rameters are chosen. In general, the optimal variational parameters are smooth, slow changing
as functions of g2 without a strong D dependence.
For the ground state (0,0), the plots of the parameters ã0,2,3 vs g for D = 1, 2, 3, 6 are shown
in Appendix C, Fig. C.2. In turn, Fig. C.3 shows the plots of the parameters as functions of
D for fixed g2. Similar plots appear for the excited states for different D and g2. Making an
analysis of the parameters ã0,2,4 vs g2 for different D, one can see the (accidental) appearance
of another D-independent constraint on the parameters, namely
ã2 =
1 + 27 ã24
18ã4
. (7.3.32)
It corresponds to the fact that the coefficient in front of r - another growing term at r → ∞ in
60 Chapter 7. Results for Two-Term Potentials
Table 7.8: Ground state energy for the quartic potential V (r) = r2 + g2 r4 for D = 1, 2, 3, 6 and
g2 = 0.1, 1, 10, labeled by quantum numbers (0,0) at any D. Variational energy Evar = Evar,
the first correction E
(t)
2 (rounded to 3 s.d.) found with use of Ψ
(t)
0,0, see text, and the corrected
energy Evar +E
(t)
2 = Evar +E
(t)
2 shown. Evar +E
(t)
2 coincides with Lagrange Mesh results (see
text) in 12 displayed d. d., hence all printed digits are exact.
g2
D = 1 D = 2
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 1.065 285 509 544 3.00× 10−14 1.065 285 509 544 2.168 597 211 269 5.28× 10−14 2.168 597 211 269
1.0 1.392 351 641 563 3.37× 10−11 1.392 351 641 530 2.952 050 091 995 3.17× 10−11 2.952 050 091 962
10.0 2.449 174 072 588 4.69× 10−10 2.449 174 072 118 5.349 352 819 751 3.44× 10−10 5.349 352 819 462
g2
D = 3 D = 6
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 3.306 872 013 152 2.20× 10−13 3.306 872 013 152 6.908 332 111 232 9.80× 10−14 6.908 332 111 232
1.0 4.648 812 704 237 2.69× 10−11 4.648 812 704 210 10.390 627 295 514 9.68× 10−12 10.390 627 295 504
10.0 8.599 003 455 030 2.22× 10−10 8.599 003 454 807 19.936 900 374 076 6.48× 10−11 19.936 900 374 011
Table 7.9: The first excited state energy for the quartic potential V (r) = r2 + g2 r4 for different
D and g2. Variational energy Evar, the first correction E
(t)
2 found with use of Ψ
(t)
0,1 for D = 1
and Ψ
(t)
0,1 for D > 1, see (7.3.29) and (7.3.30). The corrected energy Evar + E
(t)
2 = Evar + E
(t)
2
shown, the correction E
(t)
2 rounded to 3 s.d. All printed digits for Evar + E
(t)
2 are exact.
g2
D = 1 D = 2
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 3.306 872 013 236 8.33× 10−11 3.306 872 013 153 4.477 600 360 878 1.10× 10−10 4.477 600 360 768
1.0 4.648 812 707 206 2.99× 10−9 4.648 812 704 212 6.462 906 003 251 3.39× 10−9 6.462 905 999 864
10.0 8.599 003 467 556 1.27× 10−8 8.599 003 454 810 12.138 224 752 729 1.38× 10−8 12.138 224 738 901
g2
D = 3 D = 6
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 5.678 682 663 377 1.33× 10−10 5.678 682 663 243 9.447 358 518 278 1.80× 10−10 9.447 358 518 099
1.0 8.380 342 533 658 3.56× 10−9 8.380 342 530 101 14.658 513 816 952 3.39× 10−9 14.658 513 813 563
10.0 15.927 096 988 667 1.40× 10−8 15.927 096 974 709 28.536 810 849 436 1.21× 10−8 28.536 810 837 360
7.3. Quartic Anharmonic Oscillator 61
Table 7.10: The second excited state energy for the quartic potential V (r) = r2 + g2 r4 for
different D and g2. Variational energy Evar found with use of Ψ
(t)
1,0 for D = 1 and Ψ
(t)
0,2 for
D > 1, see (7.3.29) and (7.3.30). The first correction E
(t)
2 and the corrected energy Evar+E
(t)
2 =
Evar + E
(t)
2 shown. Correction E
(t)
2 rounded to 3 s.d. All 12 printed d. d. in Evar + E
(t)
2 are
exact.
g2
D = 1 D = 2
Evar Evar −E(t)
2 Evar + E
(t)
2
0.1 5.747 959 269 942 6.908 332 112 167 9.35× 10−10 6.908 332 111 232
1.0 8.655 049 995 062 10.390 627 321 799 2.63× 10−8 10.390 627 295 506
10 16.635 921 650 401 19.936 900 479 247 1.05× 10−7 19.936 900 374 040
g2
D = 3 D = 6
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 8.165 006 438 494 1.00× 10−9 8.165 006 437 493 12.084 471 853 886 1.11× 10−9 12.084 471 852 776
1.0 12.485 556 075 670 2.47× 10−8 12.485 556 051 000 19.217 523 515 555 1.97× 10−8 19.217 523 495 879
10.0 24.145 857 689 623 9.48× 10−8 24.145 857 594 824 37.811 402 320 699 6.90× 10−8 37.811 402 251 702
Table 7.11: The third excited state energy for the quartic potential V = r2+g2 r4 for different D
and g2. The radial node r0 calculated in Lagrange-Mesh method with 9 correct d.d. Variational
energy Evar and its radial node r
(0)
0 given by underlined digits, both found with Ψ
(t)
1,0, see (7.3.30).
g2
D = 2 D = 3 D = 6
E
(1)
0 r
(0)
0 E
(1)
0 r
(0)
0 E
(1)
0 r
(0)
0
0.1 7.039 707 584 0.918 783 458 8.352 677 825 1.111 521 078 12.415 256 177 1.522 966 591
1.0 10.882 435 576 0.733 724 778 13.156 803 922 0.875 567 486 20.293 829 707 1.166 753 149
10.0 21.175 135 370 0.524 083 057 25.806 276 215 0.621 795 290 40.388 142 970 0.820 068 428
the trial phase (7.3.25) - is reproduced almost exactly in accordance to (7.3.18). Not surprisingly,
if we demand to fulfill exactly constraint (7.3.32), no deterioration in the accuracy of the energy
and wave function occurs. Thus, it can be concluded that the approximate phase (7.3.25), at
large r, reproduces (almost) exactly all three growing with r terms: r3, r, and log r. Eventually,
if we require to reproduce all those terms exactly by the Approximant in its final form, it will
contain two free parameters {ã0, ã4} only. In this case, the parameters {ã2, b̃4} obey constraints
(7.3.32), (7.3.27), respectively.
The first order correction y1 to the logarithmic derivative of the ground state y0 = ∂rΦt is a
bounded function at different D and g2. For example, for g2 = 1 the first correction y1 has the
bound
|y1|max ∼



0.0101 , D = 1
0.0087 , D = 2
0.0077 , D = 3
0.0060 , D = 6
(7.3.33)
It is the consequence of the fact that the derivative of Φt reproduces exactly the growing terms
at large r in the expansion (7.3.18). “Boundness” of y1, together with the small value of the
62 Chapter 7. Results for Two-Term Potentials
maximum, implies that we deal with smartly designed zeroth order-approximation Ψ
(t)
0,0; it leads,
in the framework of the Non-Linearization Procedure, to a fast convergent series of the energy
and wave function. In Fig. 7.3 y0 and y1 vs r are presented for g2 = 1 in D = 1. It is not
a surprise that a similar plot should appear for D = 2, 3, 6 (not shown). An analysis of these
plots indicates that |y1| is an extremely small function in comparison with |y0| in the domain
0 ≤ r . 1 where the dominant contribution to variational integrals occurs, see (1.1). It explains
why the energy correction E
(t)
2 is small being the order of ∼ 10−8, or ∼ 10−11. In a similar
way one can show numerically that the higher corrections y2, y3, . . ., drop down to zero in the
domain 0 ≤ r . 1 even faster, indicating the convergence of the expansion (1.18) as n→ ∞.
0.5 1 1.5 2 2.5 3
0
2
4
6
8
10
5 10 15 20 25
0
0.002
0.004
0.006
0.008
0.01
0.012
Figure 7.3: Quartic oscillator at D = 1: function y0 = (Φt)
′ (on left) and its first correction y1
(on right) vs r for g2 = 1.
At D > 1 we checked that the Approximant Ψ
(t)
1,0 also provides an accurate estimate of
the position of the radial node of the exact wave function, see (7.2.26). The orthogonality
condition described in Section 6.2 provides a simple analytic expression for the zeroth order
approximation r
(0)
0 to r0. Making a comparison of r
(0)
0 with numerical estimates which come
from the Lagrange-Mesh method with 200 mesh points, we can see the coincidence of r0 and
r
(0)
0 for at least 6 d.d. at integer dimension D at any coupling constant g2 ≥ 0. The results are
presented in Table 7.11.
7.3.5 First Terms in the Strong Coupling Expansion
In this Section, we will explicitly find the first two terms of the strong coupling expansion
(5.1.7) of the ground state energy for the quartic anharmonic oscillator (7.3.1),
E = g2/3(ε̃0 + ε̃2g
−4/3 + ε̃4g
−8/3 + . . .) , (7.3.34)
assuming 2M = ~ = 1. In contrast with the weak coupling expansion, the expansion (7.3.34)
has a finite radius of convergence. This expansion corresponds to PT in powers of λ̂ for the
potential
V (w) = w4 + λ̂ w2 , λ̂ = g−2/3 , (7.3.35)
7.4. Sextic Anharmonic Oscillator 63
in the radial Schrödinger equation defined in w ∈ [0,∞). In order to calculate the first two
terms ε̃0 and ε̃2 of the strong coupling expansion (7.3.34), we use the Approximant (7.3.26).
In Table 7.12, the leading coefficient ε̃0 and the second perturbative correction to it ε̂2, as
well as the corrected value ε̃
(2)
0 = ε̃
(1)
0 + ε̂2, calculated via the Non-Linearization Procedure,
are presented for different D. Numerical results for ε̃0, based on the Lagrange-Mesh method
and obtained with 12 d.d., indicate that ε̃
(2)
0 = ε̃
(1)
0 + ε̂2 reproduce not less than 10 d.d. This
accuracy is verified independently by calculating the next correction ε̂3, which results in the
order of ε̂3 ∼ 10−2ε̂2. In turn, Table 7.13 contains the results of the first two approximations for
the coefficient ε̃2 in (7.3.34). It should be mentioned that our final results for the coefficient ε̃0
reproduce and sometimes exceed the best results available in the literature, see [62], [63], [64].
Hence, we have established some benchmarks. At D = 6 the calculations are carried out for
the first time.
Table 7.12: Ground state energy ε̃0 for the potential W = w4 (see (7.3.35)) for D = 1, 2, 3, 6
found in PT based on the Approximant Ψ
(t)
0,0: ε̃
(1)
0 corresponds to the variational energy, ε̂2 is
the second PT correction, ε̃
(2)
0 = ε̃
(1)
0 + ε̂2 is the corrected variational energy. 10 d.d. in ε̃
(2)
0
confirmed independently in Lagrange-Mesh method.
D = 1 D = 2
ε̃
(1)
0 −ε̂2 ε̃
(2)
0 ε̃
(1)
0 −ε̂2 ε̃
(2)
0
1.060 362 090 491 7.02× 10−12 1.060 362 090 484 2.344 829 072 753 9.27× 10−12 2.344 829 072 744
D = 3 D = 6
ε̃
(1)
0 −ε̂2 ε̃
(2)
0 ε̃
(1)
0 −ε̂2 ε̃
(2)
0
3.799 673 029 810 9.27× 10−12 3.799 673 029 801 8.928 082 199 890 4.07× 10−11 8.928 082 199 850
Table 7.13: Subdominant coefficient ε̃2 in the strong coupling expansion (7.3.34) for the ground
state energy for the quartic radial anharmonic potential (7.3.35) for different D = 1, 2, 3, 6. The
first correction to it ε̃2,1 as well as the corrected value ε̃
(2)
2 = ε̃
(1)
2 + ε̃2,1 displayed
D = 1 D = 2
ε̃
(1)
2 ε̃2,1 ε̃
(2)
2 ε̃
(1)
2 ε̃2,1 ε̃
(2)
2
0.362 022 648 388 3.96× 10−10 0.362 022 648 784 0.651 477 773 845 4.38× 10−10 0.651 477 774 283
D = 3 D = 6
ε̃
(1)
2 ε̃2,1 ε̃
(2)
2 ε̃
(1)
2 ε̃2,1 ε̃
(2)
2
0.901 605 894 682 2.03× 10−9 0.901 605 896 709 1.526 804 282 772 −3.06× 10−8 1.526 804 252 175
7.4 Sextic Anharmonic Oscillator
In this Section, it will be considered the two-term sextic anharmonic radial oscillator potential
V (r) = r2 + g4 r6 , (7.4.1)
see (7.0.1) at m = 6.
64 Chapter 7. Results for Two-Term Potentials
7.4.1 Perturbation Theory in the Weak Coupling Regime
In the Weak Coupling Regime, the perturbative expansion for ε and Y(v) can be developed
from the RB equation (3.3.9),
∂vY(v) − Y(v)
(
Y(v)− D − 1
v
)
= ε (λ) − v2 − λ4 v6 , ∂v ≡
d
dv
. (7.4.2)
The definition of v and λ is shown in (3.3.7) and (3.3.10), respectively. The perturbation series
for ε and Y(v) are of the form
ε = ε0 + ε4 λ
4 + ε8 λ
8 + . . . , ε0 = D , (7.4.3)
and
Y(v) = Y0(v) + Y4(v)λ
4 + Y8(v)λ
8 + . . . , Y0(v) = v , (7.4.4)
respectively. The terms of order λ4n+1 and λ4n+2 are absent in both expansions. Since the
potential (7.4.1) is even, PT can be constructed by algebraic means due to the polynomial
nature of correction Yn(v). The first non-vanishing correction is
ε4 =
1
8
D (D+2) (D+4) , Y4(v) =
1
2
v5 +
1
4
(D+4) v3 +
1
8
(D+2) (D+4) v . (7.4.5)
The next two corrections, ε8,12 and Y8,12(v), are presented in Appendix B. It can be shown that
the correction Y4n(v) has the form of an odd polynomial in v, namely
Y4n(v) = v
2n∑
k=0
c
(4n)
2k v2(2n−k) , (7.4.6)
with coefficients c
(4n)
2k being polynomials in D of degree k,
c
(4n)
2k = P
(4n)
k (D) , c
(4n)
4n =
ε4n
D
, (7.4.7)
cf. (7.3.6), (7.3.7). The correction ε4n has the factorization property
ε4n(D) = D (D + 2) (D + 4)R2n−2(D) , (7.4.8)
where R2n−2(D) is a polynomial in D of degree (2n − 2), cf. (7.3.8). In particular, R0 = 1
8
.
From (7.4.7), one can see that all corrections ε4n vanish at D = 0,−2,−4. Hence, their formal
sum results in ε = 0,−2,−4, respectively. In the case D = 0, the radial Schrödinger equation
takes the form
− ~2
2M
(
d2Ψ(r)
dr2
− 1
r
dΨ(r)
dr
)
+ (r2 + g4 r6)Ψ(r) = 0 , (7.4.9)
cf.(7.3.9). Its formal solution is given in terms of the parabolic cylinder functions [65] (also
known as Weber functions),
Ψ = C1Dν−(λv
2) + C2Dν−
(
i λv2
)
, ν± = −1
2
± 1
4λ2
, (7.4.10)
7.4. Sextic Anharmonic Oscillator 65
where v and λ are defined in (3.3.7) and (3.3.10), respectively. This formal solution has the
meaning of the zero mode of the Schrödinger operator at D = 0. The function shown in (7.4.10)
cannot be made normalizable by any choice of constants C1 and C2. Hence, the Schrödinger
operator at D = 0 for the sextic potential (7.4.1) has no zero mode in the Hilbert space. It
complements a similar statement made for quartic potential (7.3.1). One can guess that the zero
mode in the Hilbert space at D = 0 is absent for the Schrödinger operator with anharmonicity
r2m. At D = −2,−4, all corrections ε4n vanish. As a result, we formally have ε = −2 and
ε = −4, respectively. For both cases, no exact solution has been found for their corresponding
radial Schrödinger equations.
7.4.2 Generating Functions
For the sextic anharmonic oscillator, the GB equation is given by
λ2 ∂uZ − Z
(
Z − λ2(D − 1)
u
)
= λ2 ε(λ) − u2 − u6 , ∂u ≡ d
du
, (7.4.11)
see (3.3.20). The expansion of Z(u) in terms of generating functions has the form
Z(u) = Z0(u) + Z2(u)λ
2 + Z4(u)λ
4 + . . . . (7.4.12)
In turn, the expansion of the energy reads
ε = ε0 + ε4 λ
4 + ε8 λ
8 + . . . , ε0 = D , (7.4.13)
see (7.4.3). Interestingly, (7.4.12) has the same structure as the expansion for the quartic
anharmonic case: all generating functions Z2k+1(u), k = 1, 2, ... of odd orders λ2k+1 are absent
in expansion, cf. (7.3.12). It contrasts with the expansion of Y(v) and ε(λ) in which the
powers λ4n are present only. In fact, for any even radial anharmonic potential, V (r) = V (−r),
the function Z(u) is written in terms of generating functions as an expansion in powers of
λ2. Further properties of the generating function for the sextic anharmonic oscillator can be
established. For example, there are two different families of generating functions,
Z4k(u) = u
∞∑
n=k
c
(4n)
4k u4(n−k) , (7.4.14)
and
Z4k+2(u) = u
∞∑
n=k+1
c
(4n)
4k+2u
4(n−k)−2 . (7.4.15)
Those families occur in correspondence to ε4k+2 = 0 and ε4k 6= 0, respectively. Following (7.4.6)
and (7.4.7), it is easy to see that both families of generating functions are of the form Z2k(u),
Z2k(u) = u
k∑
n=0
f (k)
n (u2)Dn , (7.4.16)
66 Chapter 7. Results for Two-Term Potentials
where f
(p)
n (u2) are some real functions. It implies that Z2k(u) is a polynomial in D of degree
k. For example, the first two terms in the expansion (7.4.12) are
Z0(u) = u
√
1 + u4 , (7.4.17)
Z2(u) =
2u4 +D
(
1 + u4 −
√
1 + u4
)
2u (1 + u4)
. (7.4.18)
From these expressions, we can conclude immediately that
f
(0)
0 (u2) =
√
1 + u4 , f
(2)
0 (u2) =
u2
(1 + u4)
, f
(2)
1 (u2) =
1 + u4 −
√
1 + u4
2u2(1 + u4)
. (7.4.19)
The asymptotic behavior at large u of Z2p(u) is related to the expansion of y at large r. For
the sextic anharmonic oscillator, the expansion of y at large r and fixed effective coupling λ is
rewritten conveniently in variable v, see (3.3.7),
y = (2M~
2)
1
4
(
λ2v3 +
(D + 2)λ2 + 1
2λ2
v−1 − ε
2λ2
v−3 + . . .
)
, v → ∞ . (7.4.20)
The first two terms of this expansion are ε-independent, but only the first one is D-independent.
Following an analogous procedure to that used for the cubic and quartic oscillator, we can
transform the expansion of Z2k(u) at large u into an expansion at large v via the connection
between the classical and quantum coordinate shown in (3.3.24). The first two terms in (7.4.12)
expanded at large v read
(
2M
g2
)1/2
Z0(λv) = (2M~
2)
1
4
(
λ2v3 +
1
2λ2
v−1 − 1
8λ6
v−5 + . . .
)
, v → ∞ , (7.4.21)
and
(
2M
g2
)1/2
λ2Z2(λv) = (2M~
2)
1
4
(
D + 2
2
v−1 − D
2λ2
v−3 − 1
λ4
v−5 + . . .
)
, v → ∞ ,
(7.4.22)
see (7.4.17) and(7.4.18). The sum (2M
g2
)1/2(Z0(λv) + λ2Z2(λv)) at large v reproduces exactly
the first two terms in the expansion (7.4.20). However, higher generating functions, Z4(λv),
Z6(λv), ... contribute to the same order O(v−3) at large v,
(
2M
g2
)1/2
λ2kZ2k(λv) = (2M~
2)
1
4
(
−ε4k−4λ
4k−6
2
v−3 + . . .
)
, v → ∞ . (7.4.23)
Here ε4k−4 denotes the energy correction of order λ4k−4. Therefore, it does not matter how
many generating functions are considered in the expansion (2M
g2
)1/2(Z0(λv) + λ2Z2(λv) + ...)
re-expanded at large v, the ε-dependent coefficient in front of the term of order O(v−3) will
never be reproduced exactly. A similar situation occurred for the cubic and quartic cases.
7.4. Sextic Anharmonic Oscillator 67
7.4.3 The Approximant
The first two17 generating functions of the phase, G0(r) and G2(r), are given by
G0(r) =
(2M)1/2
g2
(
r2
4
√
1 + g4r4 + log
[
g2r2 +
√
1 + g4r4
])
, (7.4.24)
and
G2(r) =
(2M)1/2
g2
(
1
4
log
[
1 + g4r4
]
+
D
4
log
[
1 +
√
1 + g4r4
])
. (7.4.25)
These two functions serve as building blocks for the construction of the Approximant. Following
the general prescription (6.1.2), we write the Approximant of the ground state in the exponential
representation Ψ
(t)
0,0 = e−
1
~
Φt where
1
~
Φt(r) =
ã0 + ã2 r
2 + ã4 g
2 r4 + ã6 g
4 r6
√
1 + b̃4 g2 r2 + b̃6 g4 r4
+
1
4g2
log
[
c̃2 g
2 r2 +
√
1 + b̃4 g2 r2 + b̃6 g4 r4
]
+
1
4
log
[
1 + b̃4 g
2 r2 + b̃6 g
4r4
]
+
D
4
log
[
1 +
√
1 + b̃4 g2 r2 + b̃6g4r4
]
. (7.4.26)
At this stage ã0,2,4,6, b̃4,6, and c̃2 are 7 free parameters. The logarithmic terms added in (7.4.26)
generate prefactor to the exponential function, Those terms are a certain minimal modification
of those that occur in the first two generating functions (7.3.24) and (7.4.25). As a result, the
Approximant of the ground state for arbitrary D = 1, 2, 3, . . . is given by
Ψ
(t)
0,0(r) =
1
(
1 + b̃4 g2 r2 + b̃6 g4r4
)1/4
(
1 +
√
1 + b̃4 g2 r2 + b̃6g4 r4
)D/4
×
1
(
c̃2g2r2 +
√
1 + b̃4 g2 r2 + b̃6 g4 r4
)1/4g2
exp

− ã0 + ã2 r
2 + ã4 g
2 r4 + ã6 g
4 r6
√
1 + b̃4 g2 r2 + b̃6 g4 r4

 .
(7.4.27)
This is the central formula of this Section devoted to the sextic anharmonicity. Once the optimal
parameters coming from the VM are chosen, we will show that it provides a highly accurate
and uniform approximation for the exact ground state eigenfunction. We impose the constraint
b̃6 = 16 ã26 (7.4.28)
17The generating functions G4(r) and G6(r) are presented explicitly in Appendix B.
68 Chapter 7. Results for Two-Term Potentials
which allows us to reproduce the dominant term in the expansion (7.4.20) exactly18. Hence,
it reproduces the exact dominant asymptotic behavior of the phase at large distances. To
construct 5-parametric Approximants for excited states, we follow the prescription presented
in Section 6.2. Hence, at D = 1 we have
Ψ(t)
nr,p(x) =
xpP
(p)
nx (x
2)
(
1 + b̃4 g2 x2 + b̃6 g4x4
)1/4
(
1 +
√
1 + b̃4 g2 x2 + b̃6g4 x4
)D/4
×
1
(
c̃2g2x2 +
√
1 + b̃4 g2 x2 + b̃6 g4 x4
)1/4g2
exp

− ã0 + ã2 x
2 + ã4 g
2 x4 + ã6 g
4 x6
√
1 + b̃4 g2 x2 + b̃6 g4 x4

 .
(7.4.29)
while at D > 1
Ψ
(t)
nr,ℓ
(r) =
rℓ P
(ℓ)
nr (r
2)
(
1 + b̃4 g2 r2 + b̃6 g4r4
)1/4
(
1 +
√
1 + b̃4 g2 r2 + b̃6g4 r4
)D/4
×
1
(
c̃2g2r2 +
√
1 + b̃4 g2 r2 + b̃6 g4 r4
)1/4g2
exp

− ã0 + ã2 r
2 + ã4 g
2 r4 + ã6 g
4 r6
√
1 + b̃4 g2 r2 + b̃6 g4 r4

 .
(7.4.30)
Definitions of P
(p)
nr (x
2) and P
(ℓ)
nr (r
2) are presented in (6.2.1) and (6.2.4), explicitly.
7.4.4 Numerical Results: Variational Calculations
The calculations of the variational energy for the first low-lying states with quantum numbers
(0,0), (0,1), (0,2), (1,0) are presented in Tables 7.14 - 7.17 for different values of D and g4. For
some states, the variational energy Evar = E
(t)
0 +E
(t)
1 , the first correction E
(t)
2 , and the corrected
value Evar + E
(t)
2 are shown. All states are studied in dimension D = 1, 2, 3, 6 and coupling
constant g4 = 0.1, 1, 10. For all cases, the variational energy Evar is obtained with absolute
18In numerical computations we set ~ = 1 and M = 1/2.
7.4. Sextic Anharmonic Oscillator 69
deviation (from the exact result) in the order 10−9−10−12 (9−12 s.d.). This deviation is found
by calculating the second correction E
(t)
2 (the first correction to the variational energy).
To check the energies obtained in Non-Linearization Procedure, we calculated them numeri-
cally in the Lagrange-Mesh method. These numerical calculations show that all digits presented
for Evar + E
(t)
2 are exact: we obtain not less than 13 d.d. correctly in the Non-Linearization
Procedure. In general, the energies grow with an increase of D and/or g4. At D > 1, we classify
the states by radial quantum number nr and angular momentum ℓ as (nr, ℓ); we mention the
hierarchy of eigenstates which holds for any fixed integer D > 1 and g: (0,0), (0,1), (0,2), (1,0).
This hierarchy is the same that we obtained for the cubic and quartic anharmonic oscillators.
In contrast with the quartic oscillator case, the available literature with estimates of the
energy of the low-lying states for the sextic radial anharmonic oscillator is very limited. Most
of the calculations are made for the one-dimensional case. In general, our results reproduce all
known numerical ones that can be found for D = 1, while for D = 2, 3, we sometimes exceeded
their accuracy, see e.g. [25], [26], [66], and [67]. At D = 6, the calculations are carried out for
the first time.
The relative deviation of Ψ
(t)
0,0 from the exact (unknown) ground state eigenfunction Ψ0,0 is
estimated with the Non-Linearization Procedure. The deviation is bounded and very small,
∣
∣
∣
∣
∣
Ψ0,0(r)−Ψ
(t)
0,0(r)
Ψ
(t)
0,0(r)
∣
∣
∣
∣
∣
. 10−6 , (7.4.31)
in the whole range of r ∈ [0,∞) at any integer D and at any coupling constant g4 that we
considered. Thus, our Approximant leads to a locally accurate approximation of the exact
wave function once optimal parameters are chosen. In general, optimal variational parameters
are smooth, slow changing as functions of g4 without strong D dependence. The plots of the
optimal parameters for the ground state (0,0) a0,2,3 vs g for D = 1, 2, 3, 6 are shown in Fig. C.4
in Appendix C. In turn, Fig. C.5 presents the plots of the parameters as functions of D for
fixed g4. Similar plots occur for excited states for different D and g4. Making an analysis of
those plots, one can check that they obey (with high accuracy) another constraint,
b̃4 ≈ 32 ã6 ã4 . (7.4.32)
This constraint guarantees the exact reproduction of the first two terms in the expansion
(7.4.20). If we demand to fulfill exactly constraint (7.4.32), no deterioration in the accuracy
of the energy and wave function occurs. Thus, the Approximants either (7.4.29) or (7.4.30)
contains ultimately 5 free parameters, namely {ã0, ã2, ã4, ã6, c̃2}.
Now we focus on the ground state. For all D and g4, the correction |y1| to y0 = ∂rΦt is a
very small and bounded function in 0 . r . 1.7. In this domain, the dominant contribution of
integrals - required by the variational method - occurs. For example, for g4 = 1 andD = 1, 2, 3, 6
70 Chapter 7. Results for Two-Term Potentials
Table 7.14: Ground state energy for the sextic potential V (r) = r2 + g4 r6 for D = 1, 2, 3, 6 and
g4 = 0.1, 1, 10, labeled by quantum numbers (0,0) at any D. Variational energy Evar = Evar,
the first correction E
(t)
2 (rounded to three s.d.) found with use of Ψ
(t)
0,0, see text, the corrected
energy Evar +E
(t)
2 = Evar +E
(t)
2 shown. Evar +E
(t)
2 coincides with Lagrange Mesh results (see
text) in 12 displayed d. d., hence all printed digits are exact.
g4
D = 1 D = 2
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 1.109 087 078 465 1.20× 10−13 1.109 087 078 465 2.307 218 600 932 7.04× 10−13 2.307 218 600 931
1.0 1.435 624 619 003 3.22× 10−13 1.435 624 619 003 3.121 935 474 246 9.81× 10−13 3.121 935 474 246
10.0 2.205 723 269 598 3.22× 10−12 2.205 723 269 595 4.936 774 524 584 1.72× 10−12 4.936 774 524 582
g4
D = 3 D = 6
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 3.596 036 921 222 1.76× 10−12 3.596 036 921 220 7.987 905 269 800 7.11× 10−13 7.987 905 269 799
1.0 5.033 395 937 721 5.21× 10−12 5.033 395 937 720 11.937 202 695 862 9.62× 10−13 11.937 202 695 862
10.0 8.114 843 118 826 7.60× 10−12 8.114 843 118 819 19.880 256 604 739 3.12× 10−12 19.880 256 604 736
Table 7.15: The first excited state energy for the sextic potential V (r) = r2 + g4 r6 for different
D and g4. Variational energy Evar, the first correction E
(t)
2 found with use of Ψ
(t)
0,1 for D = 1
and Ψ
(t)
0,1 for D > 1, see (7.4.29) and (7.4.30). The corrected energy Evar + E
(t)
2 = Evar + E
(t)
2
shown, the correction E
(t)
2 rounded to 3 s.d. All printed digits for Evar + E
(t)
2 are exact.
g4
D = 1 D = 2
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 3.596 036 921 295 7.50× 10−11 3.596 036 921 220 4.974 197 493 807 9.01× 10−11 4.974 197 493 717
1.0 5.033 395 937 795 7.52× 10−11 5.033 395 937 720 7.149 928 601 496 5.84× 10−11 7.149 928 601 438
10.0 8.114 843 118 966 1.48× 10−10 8.114 843 118 818 11.688 236 0345 77 1.81× 10−10 11.688 236 034 396
g4
D = 3 D = 6
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 6.439 143 322 388 6.64× 10−11 6.439 143 322 321 11.324 899 788 818 8.15× 10−10 11.324 899 788 004
1.0 9.455 535 276 950 1.09× 10−10 9.455 535 276 841 17.387 207 808 723 1.26× 10−9 17.387 207 807 460
10.0 15.619 579 279 334 5.05× 10−10 15.619 579 278 830 29.302 506 554 618 1.22× 10−9 29.302 506 553 402
7.4. Sextic Anharmonic Oscillator 71
Table 7.16: The second excited state energy for the sextic potential V (r) = r2 + g4 r6 for
different D and g4. Variational energy Evar found with use of Ψ
(t)
1,0 for D = 1 and Ψ
(t)
0,2 for
D > 1, see (7.4.29) and (7.4.30). The first correction E
(t)
2 and the corrected energy Evar+E
(t)
2 =
Evar + E
(t)
2 shown. Correction E
(t)
2 rounded to 3 s.d. All 12 printed d. d. in Evar + E
(t)
2 are
exact.
g4
D = 1 D = 2
Evar Evar −E(t)
2 Evar + E
(t)
2
0.1 6.644 391 710 782 7.987 905 270 111 3.12× 10−10 7.987 905 269 799
1.0 9.966 622 004 356 11.937 202 696 127 2.66× 10−10 11.937 202 695 862
10.0 16.641 218 168 076 19.880 256 605 756 1.02× 10−9 19.880 256 604 742
g4
D = 3 D = 6
Evar −E(t)
2 Evar + E
(t)
2 Evar −E(t)
2 Evar + E
(t)
2
0.1 9.617 462 285 440 1.50× 10−10 9.617 462 285 290 14.962 630 328 506 1.60× 10−10 14.962 630 328 346
1.0 14.584 132 948 883 3.15× 10−9 14.584 132 945 729 23.431 551 835 405 2.19× 10−9 23.431 551 833 215
10.0 24.447 468 037 325 5.42× 10−9 24.447 468 031 906 39.815 551 142 800 7.05× 10−9 39.815 551 135 750
Table 7.17: The third excited state energy for the sextic potential V = r2+g4 r6 for different D
and g4. The radial node r0 calculated in Lagrange-Mesh method with 9 correct d.d. Variational
energy Evar and its radial node r
(0)
0 given by underlined digits, both found with Ψ
(t)
1,0, see (7.4.30).
g4
D = 2 D = 3 D = 6
Evar r0 Evar r0 Evar r0
0.1 8.402 580 462 0.837 310 052 10.237 873 721 0.995 787 872 16.154 260 610 1.308 543 484
1.0 12.914 938 793 0.671 821 606 15.989 440 787 0.790 364 964 25.938 441 037 1.019 166 200
10.0 21.792 578 251 0.515 914 526 27.155 085 604 0.604 322 682 44.521 781 513 0.773 860 964
the first correction y1 has the upper bound
|y1|max ∼



0.0078 , D = 1
0.0065 , D = 2
0.0048 , D = 3
0.0031 , D = 6
(7.4.33)
It is the consequence of the fact that, by construction, the derivative y0 = ∂rΦt reproduces the
growing terms in the expansion (7.4.20). It implies that we deal with a smartly designed zeroth
order-approximation Ψ
(t)
0,0 which leads (in the framework of the Non-Linearization Procedure)
to a fast convergent series for the energy and wave function. In Fig. 7.4, y0 and y1 vs r are
presented for g4 = 1 in D = 1. We emphasize that all curves in these figures are slow-changing
vs D. Similar plots appear for D = 2, 3, 6 (not shown) as well as for other values of g4 > 0. An
analysis of these plots indicates that (−y1)2 is an extremely small function in comparison with
y0 in the domain 0 ≤ r . 1.7, thus, in the domain which provides the dominant contribution
72 Chapter 7. Results for Two-Term Potentials
0.5 1 1.5 2
0
2
4
6
8
10
5 10 15
0
0.002
0.004
0.006
0.008
Figure 7.4: Sextic oscillator at D = 1: function y0 = (Φt)
′ (on left) and its first correction y1
(on right) vs r for g4 = 1.
in variational integrals. It is the real reason why the energy correction E
(t)
2 is extremely small
being of order ∼ 10−8, or sometimes even smaller, ∼ 10−10. A similar situation occurs for the
phase (and its derivative) of the Approximants for the excited states.
At D > 1, it was checked that the Approximant provides an estimate of the radial node
position of the wave function Ψ
(t)
1,0 by imposing the orthogonality constraint to the Approximant
of the ground state. For sextic anharmonic potential, the zeroth order approximation r
(0)
0 gives
not less than 7 d.d. of accuracy. The first order correction r
(1)
0 gives a contribution to the 6
d.d. Variational results are presented in Table 7.17. They are compared with ones obtained in
Lagrange-Mesh method with 200 mesh points. It can be noted that the radial node r0 grows
with an increase of D at fixed g4, but decreases with the increase of g4 at fixed D.
7.4.5 First Terms in the Strong Coupling Expansion
In this Section, we calculate the first two terms in the strong coupling expansion (5.1.7). For
the sextic anharmonic oscillator potential (7.4.1) this (convergent) expansion has the form
ε = g(ε̃0 + ε̃6 g
−3 + ε̃12 g
−6 + . . .) . (7.4.34)
We have assumed that 2M = ~ = 1. Evidently, this expansion corresponds to PT in powers of
λ̂ for the potential
V (w) = w6 + λ̂ w2 , λ̂ = g−3 (7.4.35)
in the Schrödinger equation defined in w ∈ [0,∞).
We use the Approximant (7.4.27) and develop the perturbation procedure already described
in Chapter 5 to calculate ε̃0 and ε̃6. In Table 7.18, we present for different D the variational
estimate for ε̃0 (denoted by ε̃
(1)
0 ), the first correction to it ε̂2, and the corrected value ε̃
(2)
0 =
ε̃
(1)
0 + ε̂2 calculated via the Non-Linearization Procedure. Using the Lagrange-Mesh method
it was verified that ε̃
(2)
0 provides not less than 12 exact d.d. at every D considered. Hence,
7.4. Sextic Anharmonic Oscillator 73
the first 10 d.d. in variational energy ε̃
(1)
0 displayed in Table 7.18 are exact. In this case, the
ε̂3 ∼ 10−2ε̂2, it indicates a high rate of convergence in PT. Finally, in Table 7.19, we present
the first two approximations of the coefficient ε̃6 - the expectation value of w2 - , see (7.4.34).
Comparing our results for ε̃0 and ε̃6 with those available in the literature, see e.g. [25] and [66],
one can see that we usually reproduce, although often exceed them.
Table 7.18: Ground state (0, 0) energy ε̃0 for the potential W = r6 at D = 1, 2, 3, 6 found in PT
based on the Approximant Ψ
(t)
(0,0) : ε̃
(1)
0 corresponds to the variational energy, ε̂2 is the second
PT correction, ε̃
(2)
0 = ε̃
(1)
0 + ε̂2 is the corrected variational energy. 10 d.d. in ε̃
(2)
0 confirmed
independently in Lagrange-Mesh method, see text.
D = 1 D = 2
ε̃
(1)
0 −ε̂2 ε̃
(2)
0 ε̃
(1)
0 −ε̂2 ε̃
(2)
0
1.144 802 453 800 3.21× 10−12 1.144 802 453 797 2.609 388 463 259 5.72× 10−12 2.609 388 463 253
D = 3 D = 6
ε̃
(1)
0 −ε̂2 ε̃
(2)
0 ε̃
(1)
0 −ε̂2 ε̃
(2)
0
4.338 598 711 518 4.73× 10−12 4.338 598 711 513 10.821 985 609 895 7.21× 10−12 10.821 985 609 888
Table 7.19: Subdominant coefficient ε̃6 in the strong coupling expansion (7.4.34) of the ground
state (0, 0) energy for the sextic radial anharmonic potential (7.4.1) for D = 1, 2, 3, 6. The first
correction to it ε̃6,1 as well as the corrected value ε̃
(2)
6 = ε̃
(1)
6 + ε̃6,1 displayed
D = 1 D = 2
ε̃
(1)
6 −ε̃6,1 ε̃
(2)
6 ε̃
(1)
6 −ε̃6,1 ε̃
(2)
6
0.307 920 304 114 3.83× 10−10 0.307 920 303 731 0.534 591 069 789 2.85× 10−10 0.534 591 069 504
D = 3 D = 6
ε̃
(1)
6 −ε̃6,1 ε̃
(2)
6 ε̃
(1)
6 -ε̃6,1 ε̃
(2)
6
0.718 220 134 970 1.55× 10−9 0.718 220 133 425 1.137 762 108 070 2.68× 10−10 1.137 762 107 802

Chapter 8
Conclusions
It has been developed a formalism to study the bound states of the quantum radial polynomial
anharmonic potential
V (r) =
1
g2
m∑
k=2
ak g
k rk , m ≥ 2 .
For this potential, our primary goal was to construct a locally accurate approximate solution of
the D-dimensional radial Schrödinger equation. In this context, the key question was: how to
design an appropriate approximation leading to locally accurate solutions and also to accurate
estimates for the energies? Our answer is based on the available analytical information about
the (unknown) exact ground state wave function.
General Overview
There are two parameterizations of the configuration space to study the radial anharmonic
oscillator:
(i) Quantum coordinate v-space, v ∝ r/
√
~. In this case, the anharmonic potential is a
finite-degree perturbation in variable v of the harmonic oscillator through the effective
coupling constant λ ∝
√
~ g. The dynamics of the wave function, taken in the exponential
representation Ψ = e−Φ, is defined by the derivative of the phase y(r) = ∂rΦ. Written in
the quantum variable v, the unknown function y(r) satisfies a non-linear ~-independent
differential equation: the so-called Riccati-Bloch (RB) equation. Relevant analytical
information can be extracted by studying this equation. One can construct the asymptotic
series of y(r) at small and large r. To solve the RB equation, Perturbation Theory (PT)
for the energy and the wave function can be developed in the weak and strong coupling
regimes. For both cases, the construction of PT is nothing but an application of the Non-
Linearization Procedure developed a long time ago. For the general anharmonic potential,
75
76 Chapter 8. Conclusions
the construction of PT is a non-algebraic procedure. However, for even anharmonic
potentials, V (r) = V (−r), the Non-Linearization Procedure in the weak coupling regime
is purely algebraic at any dimension D.
(ii) Classical coordinate u-space, with u = gr. Here, the potential is a finite degree polynomial
in u. The coupling constant appears in front of the potential as the factor 1/g2. In this
space, the dynamics of y(r) = ∂rΦ is governed by the Generalized Bloch (GB) equation:
a new non-linear ~-independent differential equation different from the RB one. Just as
it occurred with the RB equation, we can obtain analytical information from the GB one
as well. In particular, the same asymptotic series for small and large r described in (i)
can be constructed from the GB equation. When PT is developed for the GB equation
in the weak coupling regime, we checked that it generates the true type of semi-classical
expansion (in the non-classical domain, beyond the turning point) for the phase. The
main difference between the standard WKB expansion and the present new one is based
on the fact that in the true expansion the energy does have an explicit dependence on
the Planck constant ~. On the other hand, when PT is developed for the strong coupling
regime, we found an interesting connection between the expansion for large g and that
for small r: both expansions describe the same domain in r.
Both spaces are related through a simple expression: u = λ v. Furthermore, it turned out
that there is a remarkable connection between them in the weak coupling regime. This non-
trivial connection can be summarized as follows: perturbative corrections calculated for the
GB equation are generating functions of those corrections calculated on the basis of the RB
equation.
All the information collected from (i) and (ii), such as asymptotic series and PT in weak
and strong regimes, was used to design a general prescription to construct a locally accurate
approximation of the ground state wave function at any D: the Approximant Ψ
(t)
0,0(r). When
the exponential representation is assumed for the Approximant,
Ψ
(t)
0,0(r) = e−
1
~
Φt(r) ,
we may focus on finding approximately Φt. One key result of this Thesis is a compact expression
for the phase
1
~
Φt(r) =
ã0 + ã1 g r +
1
g2
V (r ; ã2, . . . , ãm)
√
1
g2 r2
V (r ; b̃2, . . . , b̃m)
+ Logarithmic Terms (r ; {c̃}) .
It contains a finite number of parameters, {ã, b̃, c̃}. At the weak and strong coupling regimes,
it interpolates the asymptotic series of the phase at small and large distances. Furthermore,
77
we showed that this formula contains the analytical information obtained from (i) and (ii) if
appropriate constraints are imposed on free parameters. Interestingly, those constraints came
from the asymptotic series at large r. Let us emphasize that the potential V (r) appeared as a
building block of the approximate phase. The recipe to insert logarithmic terms in Φt(r) was
also established: they should mimic the logarithmic terms of the first two generating functions
for the phase: G0(r) and G2(r). Imposing orthogonality constraints, the prescription may be
used to design Approximants of excited states.
Some of these parameters were fixed requiring Φt to reproduce the coefficients of all growing
terms in the expansion of the exact phase at large distances. The remaining parameters are
considered as variational. In this way, the Approximant is taken as a trial function in the VM.
The Approximant is not only suitable to estimate (with high accuracy) the energy of the first
low-lying states for finite values of g, but also, due to its features, it can be used to construct
the strong coupling expansion. Through the connection of PT with the variational energy,
the (relative) deviation between the Approximant and the exact (unknown) wave function
may be estimated. A similar situation occurs with the variational energy: we can determine
its deviation from the exact result. Results for three particular anharmonic oscillators are
summarized below.
Two-Term Potentials
We applied the formalism summarized above for the anharmonic potential
V (r) = r2 + gm−2rm ,
where m = 3, 4, 6 (cubic, quartic, and sextic). In this Section, we present the conclusions
associated with each anharmonicity.
Cubic
The key expression for the approximate phase of the Approximant is
1
~
Φt(r) =
ã0 + ã1 gr + ã2 r
2 + ã3 g r
3
√
1 + b̃1 g r
+
1
4
log[1 + b̃1 g r] + D log
[
1 +
√
1 + b̃1 g r
]
,
where {ã0, ã1, ã2, ã3, b̃1} are parameters. If the constraints
b̃1 =
25
4
ã23 , ã2 =
125 ã23 + 12
150ã3
, −g2ã1 =
9375 a43 − 1000 a23 + 48
15000 a33
,
78 Chapter 8. Conclusions
are imposed, Φt reproduces all growing terms of the exact phase at large distances. Furthermore,
the constraint
ã0 =
2ã1
b̃3
+
D + 1
2
is imposed to have vanishing ∂rΦt at r = 0. As a result, Φt depends only on 1 free parameter.
Taking Ψ
(t)
0,0 as trial function, the single parameter can be chosen as optimal in the VM. Thus,
the variational energy of the ground state is obtained with absolute accuracy ∼ 10−5 (4- 5
s.d.) for D = 1, 2, 3, 6 and g = 0.1, 1.0, 10.0. This is confirmed by the calculation of the first
correction E
(t)
2 to the variational energy Evar, which is always of the order ∼ 10−6, while the
rate of convergence of the energy in Non-Linearization Procedure is
∣
∣
∣
∣
E
(t)
3
E
(t)
2
∣
∣
∣
∣
∼ 10−4. The relative
deviation of Ψ
(t)
0,0(r) from the exact wave function is bounded and small, it does not exceed
∼ 10−2 for any value of r. Hence, the Approximant leads to a locally accurate approximation
to the exact ground state wave function. Results of this accuracy were obtained for the low-lying
states: (0, 0), (0, 1), (0, 2), (1, 0). Undoubtedly, a similar accuracy would be reached for other
dimensions D and coupling constants g. For those states, the optimal parameters are smooth,
slow-changing functions of g andD. These results represent the first accurate calculations of the
energy of the low-lying states for the D-dimensional cubic anharmonic oscillator. Hence, these
results may serve as a benchmark for other calculations. To the best of the knowledge of the
author, there are not many examples in the literature in which a one-parametric Approximant
(trial function) leads to such accuracies for the energy and wave functions.
Quartic
For this anharmonicity, the prescription generates an approximate phase of the form
1
~
Φt(r) =
ã0 + ã2 r
2 + ã4 g
2 r4
√
1 + b̃4 g2 r2
+
1
4
log
[
1 + b̃4 g
2 r2
]
+
D
2
log
[
1 +
√
1 + b̃4 g2 r2
]
,
where {ã0, ã2, ã4, b̃4} are parameters, which are constrained
b̃4 = 9 ã24 , ã2 =
1 + 27 ã24
18ã4
.
These constraints guarantee that Φt reproduces exactly all two growing terms of the exact phase
at large distances. The remaining 2 free parameters are chosen to be optimal in the VM with
Ψ
(t)
0,0(r) as trial function. The variational energy of the ground state is obtained with absolute
accuracy ∼ 10−8 (8 - 9 s.d.) for D = 1, 2, 3, 6 and g = 0.1, 1.0, 10.0. This is confirmed by
the first correction E
(t)
2 , which is always of the order ∼ 10−8. The rate of convergence of the
79
energy in the Non-Linearization Procedure is
∣
∣
∣
∣
E
(t)
3
E
(t)
2
∣
∣
∣
∣
∼ 10−4. The relative deviation of Ψ
(t)
0,0(r)
from the exact wave function is bounded and very small, it does not exceed ∼ 10−6. Hence, the
Approximant led to a locally accurate approximation to the exact wave function. Results of
similar accuracy were obtained for the low-lying states: (0, 0), (0, 1), (0, 2), (1, 0). Clearly, the
same accuracy would be reached for other dimensions and coupling constants. For those states,
the optimal parameters are smooth and slow-changing functions of g and D. The variational
estimates energy have confirmed, and some times exceeded, the results found in the literature.
Sextic
According to the prescription, the approximate phase of the sextic case is given by
1
~
Φt(r) =
ã0 + ã2 r
2 + ã4 g
2 r4 + ã6 g
4 r6
√
1 + b̃4 g2 r2 + b̃6 g4 r4
+
1
4g2
log
[
c̃2 g
2 r2 +
√
1 + b̃4 g2 r2 + b̃6 g4 r4
]
+
1
4
log
[
1 + b̃4 g
2 r2 + b̃6 g
4r4
]
+
D
4
log
[
1 +
√
1 + b̃4 g2 r2 + b̃6g4r4
]
,
where {ã0, ã2, ã4, ã6, b̃2, b̃4} are parameters. which are constrained
b̃6 = 16 ã26 , b̃4 = 32 ã6 ã4 .
These constraints reduced the number of parameters to 4; they guarantee that Φt reproduces
exactly all growing terms of the exact phase at large distances. The remaining 4 free parameters
are chosen to be optimal in the VM with Ψ
(t)
0,0(r) as trial function. The variational energy
coincides with the exact one with not less than 8 s.d. for the states (0, 0), (0, 1), (0, 2), (1, 0) for
D = 1, 2, 3, 6 and g4 = 0.1, 1.0, 10.0. However, there are no doubts that it will be the same for
any integer D and g4 > 0. For those states, the optimal parameters are smooth slow-changing
as functions of g and D. The rate of convergence of energy in the Non-Linearization Procedure
remains
∣
∣
∣
∣
E
(t)
3
E
(t)
2
∣
∣
∣
∣
∼ 10−4. The relative deviation of Ψ
(t)
0,0(r) from the exact eigenfunction is a
bounded function and very small, it does not exceed ∼ 10−6. Hence, the Approximant led to
a locally accurate approximation of the exact wave function. Results of similar accuracy were
obtained for the first low-lying states. The variational estimates of the energy have confirmed,
and some times exceeded, the results found in the literature.
80 Chapter 8. Conclusions
Final Remarks
The reproduction of the growing terms at large distances of the phase reduces the number of
parameters that the Approximant contains. Hence, the dimensionality of the space of param-
eters is reduced. If the remaining free parameters are chosen to be optimal in the VM, highly
accurate estimates of the energy will occur. In addition, the Approximant will become a locally
accurate approximation of the exact wave function.
We noticed that the hierarchy of the energies of the first four low-lying states of the cubic,
quartic, and sextic anharmonic oscillators is the same: (0, 0), (0, 1), (0, 2), (1, 0) in arbitrary
D > 1 and non-vanishing coupling constant g. See Fig 8.1.
E
n
er
gy
(0, 0)
(1, 0)
(0, 1)
(0, 2)
E
E E
Figure 8.1: Schematic energy diagram for the first four bound states (nr, ℓ) for the potential
V (r) = r2 + gm−2rm, m = 3, 4, 6, at g > 0 in arbitrary dimension D.
The first two terms in the strong coupling expansion were estimated with not less than 9 d.d.
for the cubic, quartic, and sextic anharmonic oscillator. For the cubic potential, the first two
terms were calculated for the first time. For quartic and sextic, our estimates reproduced the
benchmarks known in the literature at D = 1, 2, 3. In D = 6, we established the benchmark.
It is clear that the formalism presented in this Part can be used to study some other systems
in quantum mechanics. For example, a similar formalism was used in the past to study the
low-lying states of the Yukawa potential leading to highly accurate results, see [47].
Part II
Hydrogen Atom in Constant Magnetic
Field
81

Chapter 9
Introduction
The hydrogen atom is the simplest atomic system in Nature. It contains a positively charged
proton and a negatively charged electron bound to the nucleus via the Coulomb force. Due to
the separation of variables in spherical coordinates (and in some other systems of coordinates),
the Schrödinger equation associated with this system was solved exactly in the seminal paper of
modern quantum mechanics [13]. It is not an exaggeration to say that it laid the groundwork for
subsequent studies of more complex atoms and molecules. In a non-relativistic consideration,
when the hydrogen atom is placed in a constant uniform magnetic field, separation of variables
no longer occurs for the Schrödinger equation, and it becomes non-solvable. The impossibility of
obtaining exact solutions makes necessary the use of approximate methods to find the spectrum
of the system. This feature makes the hydrogen atom in a constant uniform magnetic field an
appropriate system to study using the formalism presented in Part I.
9.1 Motivation
The study of the atomic properties in the presence of a magnetic field dates back to a series of
experiments carried out by Zeeman around 1896. In those experiments, he noticed the splitting
of the spectral lines of sodium when their source is placed under the influence of a magnetic
field.1
Early atomic models were incapable of describing adequately such phenomena. It was not
until the theory of quantum mechanics was established (1926) that a satisfactory theoretical
description occurred. Due to the success of this new theory, it was intermediately clear that a
study in this quantum mechanical framework could explain in full detail the splitting observed
by Zeeman. It was Schrödinger who presented the first results of quantum nature for the
1It is necessary to mention that around that time, the notions of proton and electron were absent. The
electron would be experimentally detected one year later, while the proton 15 years later.
83
84 Chapter 9. Introduction
splitting of the spectral lines of atoms with one electron of valence2 [68]. The splitting of a
given spectral line into components in the presence of a small magnetic field became known
as (linear) Zeeman effect. In stronger fields, the resulting splitting was called Paschen-Back
effect [69]. At ultra-strong fields, the splitting was known as Quadratic Zeeman effect. Each
regime was treated mainly in divergent perturbation theory (PT) and some other approximate
methods such as the adiabatic approximation and the Variational Method. Early attempts to
explore atomic systems in a magnetic field are summarized in the review paper by Garstang [70].
Some of those early attempts were motivated by astrophysics discoveries. This motivation dates
from the discovery by Hale in 1908 of magnetic fields in sunspots from the Zeeman splitting of
their spectral lines. Some other works were motivated by plasma and semiconductor physics,
where the effects of a large magnetic field can be present. In recent decades, the interest has
been renewed due to the discovery of white dwarfs and neutron stars. Their atmospheres,
rich in hydrogen, experience the strongest magnetic fields γ that can be found in Nature until
today. For example, a neutron star may have magnetic fields in the range 1012 − 1013 G, while
magnetars in the range 1013 − 1015 G.
Due to its simplicity, in comparison with other atoms, the hydrogen one has played a fun-
damental role in the understanding of the behavior of more complex atoms in magnetic fields.
In fact, the hydrogen atom is the most studied Coulombic system in weak, intermediate, and
strong magnetic fields. In the non-relativist consideration, there is a large number of works de-
voted to finding its spectrum approximately as a function of the magnetic field. Most of them
are devoted to study the energy levels, but no other relevant quantities like the quadrupole
moment or even the wave function.
9.2 Present Work
The purpose of Part II is to extend and apply the formalism presented in Part I to one of
the most relevant atomic systems in the presence of a constant uniform magnetic field: a hy-
drogen atom. Our ultimate goal is to design an adequate approximation of the ground state
wave function following the spirit of Part I. However, it should be noted that the Schrödinger
equation associated with the hydrogen atom in a magnetic field is effectively a two-dimensional
partial differential equation. It contrasts with the Schrödinger equation that corresponds to the
D-dimensional radial anharmonic oscillator, which was effectively one-dimensional due to the
separation of angular variables from the radial one. Despite this significant difference, as well
as of the obvious ones (potentials with different physical and analytical natures), we will show
that the formalism can be applied with success. In particular, the Riccati-Bloch and Gener-
alized Bloch equations can be derived to calculate PT in the weak coupling regime and study
asymptotic series. All the analytical information is collected to construct The Approximant : a
locally accurate and uniform approximation of the wave function of the ground state.
This Part may be read independently on the previous one, it is self-contained. However,
2 Just like the sodium atoms considered experimentally by Zeeman.
9.2. Present Work 85
sometimes we will make some analogies between the results presented here and those of Part I.

Chapter 10
Generalities
The Hamiltonian
Ĥ =
1
2me
(
p̂ +
e
c
A
)2
− e2
r
, r =
√
x2 + y2 + z2 , (10.0.1)
may describe a hydrogen atom in the presence of a constant magnetic field B = γ ẑ. In (10.0.1),
the mass of the proton is assumed to be infinite, mp = ∞ . Here me and −e are the mass and
charge of the electron, respectively, e corresponds to the charge of the infinitely massive proton
located at the origin of coordinates, and c is the speed of light. Besides, p̂ = −i~∇ is the
momentum operator of the electron, r is its distance to the origin. See Fig. 10.1 for the
geometrical setting.
It is straightforward to show that (10.0.1) is equivalent to
Ĥ =
1
2me
p̂2 +
1
mec
p̂ ·A +
1
mec
A · p̂+
e2
2mec2
A2 − e2
r
(10.0.2)
From now on, the symmetric gauge
A =
1
2
B× r , B = γ ẑ , (10.0.3)
is assumed1, it implies that the Hamiltonian (10.0.2) takes the form
Ĥ = − ~2
2me
∆ +
|e|γ
2mec
L̂z + V , ∆ = ∂2x + ∂2y + ∂2z . (10.0.4)
Here, we defined the potential
V = −e
2
r
+
e2γ2
8mec2
ρ2 , ρ =
√
x2 + y2 , (10.0.5)
1This choice of the vector potential leads to a constant magnetic field directed along the z-axis. The magni-
tude of this magnetic field is γ, measured in Gauss/Teslas.
87
88 Chapter 10. Generalities
depends on r and ρ only, L̂z is the projection of the momentum operator on the direction of
magnetic field,
L̂z = −i~ (x ∂y − y ∂x) , (10.0.6)
which is conserved. The Schrödinger equation associated to (10.0.4) is
Ĥψ = Eψ , (10.0.7)
with boundary conditions imposed in such a way that the wave function is normalizable
∫
ψ2 dV <∞ . (10.0.8)
The dependence of the potential on variables ρ and r hints that it is convenient to write the
Schrödinger equation (10.0.7) in the non-orthogonal system of coordinates (ρ, r, ϕ), see Fig.
10.1. In these coordinates it reads2
x
y
z
e
r
ϕ
ρ
p
Figure 10.1: System of coordinates (ρ, r, ϕ) at half-space z ≥ 0. The infinitely heavy proton
(p) is located at the origin.
− ~2
2me
[
∂2ρ +
2ρ
r
∂ρr + ∂2r +
1
ρ
∂ρ +
2
r
∂r +
1
ρ2
∂2ϕ
]
ψ − i|e|~γ
2mec
∂ϕψ
+
[
−e
2
r
+
γ2e2
8mec2
ρ2
]
ψ = E ψ . (10.0.9)
In Appendix E we present some relevant expressions involving differential operators written
in this system of coordinates. Needless to say, equation (10.0.9) is non-solvable for γ 6= 0:
energy and wave function can be found only in approximate form.
2Note that in this coordinates L̂z = −i~ ∂ϕ.
10.1. The Ground State: Riccati Equation: 89
Due to cylindrical symmetry of the system, any wave function is characterized by two quan-
tum numbers: the magnetic quantum number m (eigenvalue of L̂z) and the parity ν = ±1 with
respect to a reflection z → −z. It suggests considering the representation
ψ(ρ, r, ϕ) = zpρ|m|Ψ(ρ, r) eimϕ , m = 0,±1,±2, ... , p = ±1 (10.0.10)
for the states with definite parity, hence ν = (−1)p. Here z =
√
r2 − ρ2, see Fig. 10.1. The
function Ψ(ρ, r) satisfies the equation
− ~2
2me
[
∂2ρ +
2ρ
r
∂ρr + ∂2r +
2|m|+ 1
ρ
∂ρ +
2(|m|+ p+ 1)
r
∂r
]
Ψ
+
[
−e
2
r
+
γ2e2
8mec2
ρ2
]
Ψ = Em,p Ψ , (10.0.11)
where
Em,p = Em,p − |e|~γ m
2mec
(10.0.12)
is the energy with the linear Zeeman term subtracted. Note that the quantum number m now
plays the role of a parameter. From equation (10.0.11) one can explicitly see that Em,p should
be an even function with respect to the magnetic quantum number m,
Em,p = E−m,p , (10.0.13)
for any possible value of p. Therefore
Em,p = E−m,p +
|e|~γ m
mec
. (10.0.14)
Some general properties of the spectrum of the Schrödinger equation, such as the appearance
quasi-crossings and crossings of energy levels, its structure as well as analytical information,
can be found in the literature, see [70].
10.1 The Ground State: Riccati Equation:
The present study focuses on the ground state, which corresponds to quantum numbers m = 0
and ν = +. Thus, its wave function depends only on variables (ρ, r). At γ = 0, it corresponds
to the 1s0 state of the hydrogen atom. From now on, we present the (nodeless) ground wave
function and its energy dropping the labels of quantum numbers, presenting them as Ψ and E ,
respectively. It is usual to denote these states as 1s0 even for γ 6= 0. Following (10.0.11), the
equation that determines Ψ and E reads
− ~2
2me
[
∂2ρ +
2ρ
r
∂ρr + ∂2r +
1
ρ
∂ρ +
2
r
∂r
]
Ψ +
[
−e
2
r
+
γ2e2
8mec2
ρ2
]
Ψ = E Ψ . (10.1.1)
We now study the function Ψ(ρ, r) when it is written in the exponential representation, namely
Ψ(ρ, r) = e−Φ(ρ,r) . (10.1.2)
90 Chapter 10. Generalities
One can show that the phase Φ(ρ, r) satisfies a non-linear partial differential equation of second
order,
∂2ρΦ +
2ρ
r
∂ρrΦ + ∂2rΦ +
1
ρ
∂ρΦ +
2
r
∂rΦ − (∂ρΦ)
2 − 2ρ
r
(∂ρΦ)(∂rΦ) − (∂rΦ)
2
=
2me
~2
[
E +
e2
r
− γ2e2
8mec2
ρ2
]
. (10.1.3)
This equation is defined in the domain 0 ≤ ρ ≤ r and 0 ≤ r <∞, see Fig. 10.2.
ρ
=
r
r
ρ
Figure 10.2: Infinite wedge domain (shaded in gray) of the equation (10.1.3) in (ρ, r) space.
Note that (10.1.3) can be regarded as a generalization for two dimensions of the well-known
one-dimensional Riccati equation, cf. (3.3.4). It is worth mentioning that (10.1.3) is the key
equation of the present Chapter. Naturally, this equation can be solved exactly at γ = 0; it
will generate the 1s0 orbital. For γ 6= 0 it becomes non-solvable: its solution has to be found
in approximate form. We have already met in the past this situation for the D-dimensional
radial anharmonic oscillator. In the next Sections, we will show that the Riccati-Bloch and
Generalized Bloch equations may be found from equation (10.1.3).
10.2 Riccati-Bloch Equation
Let us define the dimensionless variables
s =
ρ
r0
, t =
r
r0
, (10.2.1)
where
r0 =
~2
mee2
≃ 5.29× 10−9cm (10.2.2)
10.3. Generalized Bloch Equation 91
is the Bohr radius. In these new variables (10.2.1), the Riccati equation (10.1.3) appears without
explicit dependence on c, e, ~, and me, namely
∂2sΦ +
2s
t
∂stΦ + ∂2tΦ +
1
s
∂sΦ +
2
t
∂tΦ − (∂sΦ)
2 − 2s
t
(∂sΦ)(∂tΦ) − (∂tΦ)
2 = ε +
2
t
− λ2s2
4
,
(10.2.3)
where
ε =
E
E0
, E0 =
mee
4
2~2
(10.2.4)
and
λ =
γ
γ0
, γ0 =
c|e|3m2
e
~3
. (10.2.5)
Note that E0 is the Rydberg constant while γ0 is the atomic unit of magnetic field, respectively;
E0 ≈ 2.18× 10−18J = 13.6 eV , (10.2.6)
γ0 ≈ 2.35× 105 T = 2.35× 109 G . (10.2.7)
Expressions (10.2.6) and (10.2.7) suggest that λ is the effective magnetic field measured in
atomic units γ0, which occurs instead of γ, while ε plays the role of energy measured in Rydbergs
(Ry).
Equation (10.2.3) is nothing but the dimensionless version of the Riccati equation, and we
call it Riccati-Bloch (RB) equation. Both equations coincide when we set ~ = 1, me = 1 and
e = −1. The RB equation governs the dynamics via the phase Φ in the (s, t)-space. Note that
(10.2.3) is analogous to equation (3.3.9).
10.3 Generalized Bloch Equation
Let us introduce in (10.1.3) different dimensionless variables
u =
ρ
ρ0
, v =
r
r0
, (10.3.1)
where
ρ0 =
mec|e|
~γ
. (10.3.2)
Note that ρ0 has explicit dependence on γ, while v coincides with t, see (10.2.1) and (10.3.1).
Introducing (10.3.1) into the Riccati equation (10.1.3), we obtain a two-dimensional Generalized
Bloch (GB) equation
λ2 ∂2uΦ +
2u
v
∂uvΦ + ∂2vΦ +
λ2
u
∂uΦ +
2
v
∂vΦ − λ2(∂uΦ)
2 − 2u
v
(∂uΦ)(∂vΦ) − (∂vΦ)
2
= ε +
2
v
− u2
4
.
(10.3.3)
92 Chapter 10. Generalities
cf. (3.3.20), see [44,45]. Note that the potential in the right hand side does not have any explicit
dependence on the parameters of the problem. The definitions of ε and λ are given in (10.2.4)
and (10.2.5), respectively. Just as it occurred for the RB equation, all variables/quantities
involved in (10.3.3) are dimensionless. Let us note that the variables u and s are remarkable
easy related
u = λ s , (10.3.4)
see (10.2.1), (10.2.5) and (10.3.1). This relation looks very similar to the one which occurred for
the anharmonic oscillator between the classical coordinate and the quantum one, see (3.3.24).
Chapter 11
Weak Coupling Regime
11.1 Riccati Bloch Equation
Just like we did for the D-dimensional radial anharmonic oscillator, we can use PT assuming a
weak coupling regime to solve the RB equation (10.2.3),
Φ(s, t) =
∞∑
n=0
λ2nΦn(s, t) , ε =
∞∑
n=0
λ2nεn . (11.1.1)
The zeroth order correction is
Φ0(s, t) = t , ε0 = −1 , (11.1.2)
it corresponds to the exact phase and energy of ground state (1s0) of the hydrogen atom at
γ = 0, respectively. It must be emphasized that if we set c = 1, me = 1 and e = 1, then the
effective coupling constant results in λ = γ~3. It implies that the PT developed for the ε in
powers of γ coincides with the semi-classical expansion in powers of ~3.1
For n > 0, one can show that corrections Φn and εn are determined by a linear partial
differential equation,
∂ssΦn +
2s
t
∂stΦn + ∂ttΦn +
(
1
s
− 2s
t
)
∂sΦn +
(
2
t
− 1
)
∂tΦn = εn − Qn , (11.1.3)
where
Qn = −
n−1∑
k=1
[
∂sΦk∂sΦn−k + ∂tΦk∂tΦn−k +
s
t
(∂sΦk∂tΦn−k + ∂tΦk∂sΦn−k)
]
(11.1.4)
1We already showed in the previous Part that a similar situation occurs for the radial anharmonic potential.
In that case, the semi-classical expansion of the energy is in powers of ~1/2.
93
94 Chapter 11. Weak Coupling Regime
is constructed from the previous (n − 1) corrections. The explicit analytical solution can be
found for equation (11.1.3), for example, the first order correction is
Φ1(s, t) =
1
24
s2t +
1
16
s2 +
1
24
t2 , ε1 =
1
2
. (11.1.5)
In general, the nth correction Φn is a polynomial in variables (s, t) with the following structure
Φn(s, t) =
n−1∑
j=0
n∑
k=j
(
a
(n)
j,k t + b
(n)
j,k
)
s2(n−k)t2(k−j) , a
(n)
0,n = 0 . (11.1.6)
For all order n, the coefficients a
(n)
j,k and b
(n)
j,k are rational real numbers. Interestingly, the equation
εn = 4 b
(n)
n−1,n−1 + 6 b
(n)
n−1,n (11.1.7)
holds for any n > 1. Hence the correction εn is also a rational number. Several corrections
can be easily constructed in this framework as a consequence of the polynomial nature of
the corrections Φn. They are determined by solving algebraic linear equations via elementary
algebra. Hence, the realization of PT is ultimately an algebraic procedure. As a consequence,
many terms in expansions (11.1.1) can be calculated explicitly. In Appendix F, we present the
first corrections of the Φ for n = 2, 3, 4. In turn, we present some corrections up to 100th order
for the energy ε; see Tables F.1 and F.2. It must be emphasized that it is the first time that the
coefficient ε100 has been calculated exactly. The corrections were computed using Mathematica
with absolute accuracy (in the form of rational numbers). The code designed for this purpose
is very compact: only 15 lines of commands are needed. The code is presented in explicit form
in Appendix F. The calculation of the first 50 corrections does not take more than one hour
of computations in a common laptop. To arrive up to order 100, it was necessary to leave the
laptop working for about two weeks.
Despite the simplicity of the calculation of PT corrections, it should be mentioned that,
according to the Dyson instability argument [50], both series (11.1.1) should be divergent. This
can be seen from the asymptotic behavior of εn for large n [71],
εn = 64
(−1)n+1
π
5
2
+2n
Γ
(
2n+
3
2
)
(
(1 + O(n−1)
)
, n→ ∞ . (11.1.8)
Following this formula, it is clear that the PT for ε is Borel-summable2. However, the Padé-
Borel procedure does not provide accurate results, even taking into account the first hundred
coefficients εn, see below. A similar situation occurs with conformal mapping [73].
From equations (11.1.8) and (11.1.7), we note that the coefficients b
(n)
n−1,n−1 and b
(n)
n−1,n also
grow factorially at large n. It is worth mentioning that the perturbative approach used to
solve the Riccati-Bloch equation is nothing but an example of the application of the so-called
Non-Linearization Procedure [8] (sometimes referred as logarithmic PT). In the previous Part,
it was used to solve the one-dimensional version of the RB equation. Here we have shown that
it can also be applied successfully to two-dimensional problems. A general description of this
method can be found in [8].
2See [72] for a detailed description of this method.
11.2. Generalized Bloch Equation 95
11.1.1 Behavior of the Phase Φ at Small Distances
For one-dimensional spectral problems, the exponential representation is convenient to study
the asymptotic behavior of the phase at some particular limits. For the D-dimensional radial
anharmonic oscillator, we showed that both RB and GB equations are suitable to construct
asymptotic series of the phase. This situation changes dramatically for the hydrogen atom in
a constant magnetic field. In this case, the phase is determined by a non-linear partial (RB
or GB) differential equation in two variables; it complexifies the construction of asymptotic
series. Interestingly, from the structure of corrections Φn, we can overcome this difficulty.
The structure of the Taylor series of the phase Φ at small s and t can be obtained from the
polynomial form of the correction Φn, see (11.1.6). Collecting the same powers in s and t,
coming from different corrections Φn, their formal sums will result in the asymptotic expansion
at small (s, t), namely
Φ(s, t) = t + σ1(λ
2) s2 + σ2(λ
2) t2 + σ3(λ
2) s2t + . . . , (s, t) → 0 . (11.1.9)
The functions σ1, σ2 and σ3 are given (formally) by
σ1(λ
2) =
∞∑
n=1
b
(n)
n−1,n−1λ
2n , σ2(λ
2) =
∞∑
n=1
b
(n)
n−1,nλ
2n , σ3 =
∞∑
n=1
a
(n)
n,n−1λ
2n . (11.1.10)
From equation (11.1.7), it is clear that
ε(λ2) = −1 + 4 σ1(λ
2) + 6 σ2(λ
2) . (11.1.11)
From Taylor series (11.1.9), once variables ρ and r are restored, one can immediately conclude
that the presence of a magnetic field does not break the cusp condition for the exact wave
function
C ≡
〈
ψ
∣
∣ δ(~r) ∂
∂r
∣
∣ψ
〉
〈ψ | δ(~r) |ψ〉 = − 1
r0
. (11.1.12)
In atomic physics, the parameter C is known as the cusp parameter [6]. For non-solvable
systems which consist of electrons and (infinitely massive) positive charges, it is usually used
to measure the local quality of the approximate wave function near the Coulomb singularities.
11.2 Generalized Bloch Equation
To solve the GB equation, we can develop PT in powers of λ2,
Φ(u, v) =
∞∑
n=0
λ2nφn(u, v) , ε =
∞∑
n=0
λ2nεn . (11.2.1)
The expansion for ε coincides with the one presented in (11.2.1). The zeroth order correction
φ0(u, v) is determined by the non-linear partial differential equation
2u
v
∂uvφ0 + ∂2vφ0 +
2
v
∂vφ0 − 2u
v
(∂uφ0)(∂vφ0) − (∂vφ0)
2 = ε0 +
2
v
− u2
4
. (11.2.2)
96 Chapter 11. Weak Coupling Regime
Its explicit solution in closed analytical form is given by
φ0(u, v) = A
(0)
0 (u) v + B
(0)
0 (u) (11.2.3)
where
A
(0)
0 (u) =
√
1 +
u2
12
, B
(0)
0 (u) =
1
2
log
(
1 +
u2
12
)
+ log
(
1 +
√
1 +
u2
12
)
. (11.2.4)
The correction φn(u, v) with n ≥ 1 obeys a linear partial differential equation, namely
2u
v
∂u,vφn + ∂2vφn +
2
v
∂vφn − 2u
v
(∂uφ0 ∂vφn + ∂uφn ∂vφ0) − 2(∂vφn ∂vφ0)
= εn − qn (11.2.5)
where
qn = ∂2uφn−1 +
(
1
u
− ∂uφ0
)
∂uφn−1
−
n−1∑
k=1
{
∂uφn−k−1∂uφk +
2u
v
∂uφn−k−1∂vφk + ∂vφn−k−1∂vφk
}
. (11.2.6)
It can be shown that all correction φn(u, v) is a polynomial in v with u-dependent coefficients,
φn(u, v) =
n∑
k=0
{
A
(n)
k (u) v + B
(n)
k (u)
}
v2(n−k) . (11.2.7)
In general terms, the functions A
(n)
k (u) and B
(n)
k (u) are determined by solving (ordinary) linear
differential equations of first degree. To write the first order correction (and next ones), it is
convenient to introduce the variable
w(u) =
√
1 +
u2
12
. (11.2.8)
Thus,
φ1(u, v) = A
(1)
0 (u) v3 + B
(1)
0 (u) v2 + A
(1)
1 (u) v + B
(1)
1 (u) (11.2.9)
where
A
(1)
0 = −(w − 1)(w + 1)
120w3
, (11.2.10)
B
(1)
0 = −6w3 − w2 − 9w − 6
120 (w + 1)w4
, (11.2.11)
A
(1)
1 = −(w − 1) (30w4 + 52w3 + 54w2 + 42w + 15)
120 (w + 1)w5
, (11.2.12)
B
(1)
1 = −(w − 1) (9w6 + 18w5 + 38w4 + 46w3 + 42w2 + 30w + 10)
80 (w + 1)w6
. (11.2.13)
11.2. Generalized Bloch Equation 97
The next corrections φ2, φ3, ..., can also be calculated explicitly. However, we omit to present
them since they involve cumbersome expressions. As we discuss in the next Section, some
interesting properties of functions A
(n)
k (u) and B
(n)
k (u) are related to their asymptotic behavior3
at large u and v.
11.2.1 Asymptotic Behavior of Φ at Large Distances
Using the corrections in PT obtained from the generalized Bloch equation, the asymptotic series
can be calculated in some particular directions in the domain (s, t).
Let us consider the line
s = α t (11.2.14)
where α ∈ (0, 1] is a parameter. One can see that along this line (keeping the value of α fixed),
the dominant asymptotic behavior of the nth correction to the phase
φn(u, v) = φn(λs, t)s=αt (11.2.15)
at large t comes from the term A
(n)
0 (λ(αt))t2n+1, see (11.2.7). Concisely,
φn(λ s, t)s=αt ∼ An
(αλ)2n−1
t2 + O
(
t0
)
, t→ ∞ , n = 0, 1, 2, ... , (11.2.16)
where An is a coefficient. The first three coefficients are
A0 =
1
2
√
3
, A1 = − 1
20
√
3
, A2 = − 23
2800
√
3
. (11.2.17)
From (11.2.16), one can note that the behavior of φ(λ s, t)s=αt is given by
φ(λ s, t)s=αt = λ
(
α
∞∑
n=0
Anα
−2n
)
t2 + 2 log(t) + . . . , t → ∞ . (11.2.18)
The logarithmic (subdominant) term comes from the function B
(0)
0 (u) displayed in (11.2.4). In
order to determine the sum in (11.2.18), we can define the generating function
A(α) = α
∞∑
n=0
Akα
−2n . (11.2.19)
One can check that it satisfies a non-linear differential equation, namely
(1− α2) (A′)2 + 4A2 − 1
4
α2 = 0 , A(1) =
1
4
. (11.2.20)
3At this stage, one can suspect that corrections φn play the role of generating functions, just like Gn(r) did
in the case of the radial anharmonic oscillator, see (4.4.2). Let us remind that, based on the functions Gn(r),
one can construct the asymptotic expansion of the phase at large distances.
98 Chapter 11. Weak Coupling Regime
The exact solution to this equation is given by
A(α) =
1
4
α2 . (11.2.21)
Hence, the behavior of the phase (11.2.15) at t→ ∞ results in
Φ(λ s, t)s=αt =
1
4
α2 λ t2 + 2 log(t) + O(t0) , t→ ∞ . (11.2.22)
Some other relevant limits in different directions can also be studied. For example, if s is fixed,
say s = s0, the asymptotic expansion at large t is of the form
Φ(s0, t) = C0(s0, λ) t + C1(s0, λ) log(t) + O(t0) , t→ ∞ . (11.2.23)
Here C1 and C2 are some unknown functions, except at λ = 0.
As we have seen, the perturbative solution of the GB equation in powers of λ2 can provide
information about the behavior of the exact phase at large distances. In contrast with the
solution of the GB equation of the radial anharmonic oscillator, further manipulations4 in
corrections φn are needed to obtain the asymptotic behavior of the phase.
11.3 Connection between RB and GB Equations
We have constructed two different representations for the phase Φ of the ground state wave
function. From one side
Φ =
∞∑
n=0
λ2nΦn(s, t) , s =
ρ
r0
, t =
r
r0
, (11.3.1)
and from the other
Φ =
∞∑
n=0
λ2nφn(u, v) , u =
ρ
ρ0
, v =
r
r0
. (11.3.2)
The definitions of r0 and ρ0 appear in (10.2.2) and (10.3.2), respectively. Due to uniqueness
of the phase Φ, it is clear that there must be a connection between corrections Φn(s, t) and
φn(u, v). In order to establish it, we use the polynomial structure of corrections (11.1.6) in
(11.3.1), thus
Φ =
∞∑
n=0
λ2n
n∑
j=0
n∑
k=j
(
a
(n)
j,k t + b
(n)
j,k
)
s2(n−k)t2(k−j) . (11.3.3)
4In the case of the D-dimensional anharmonic oscillator, it was enough to expand the first functions Gn(r)
at large distances to obtain the asymptotic behavior of the phase.
11.3. Connection between RB and GB Equations 99
Changing the order of summation and using the relation (10.3.4) between variables s and u, it
can be shown that (11.3.3) is equivalent to
Φ =
∞∑
n=0
λ2n
n∑
k=0
∞∑
j=n
(
a
(j)
k,n v + b
(j)
k,n
)
u2(j−n)v2(n−k) . (11.3.4)
Comparing expansions (11.3.2) and (11.3.4), we conclude that
φn(u, v) =
n∑
k=0
∞∑
j=n
(
a
(j)
k,n v + b
(j)
k,n
)
u2(j−n)v2(n−k) (11.3.5)
due to the uniqueness of the Taylor series. After further manipulations, (11.3.5) is expressed
as follows
φn(u, v) =
n∑
k=0
v2(n−k)
∞∑
j=0
(
a
(n+j)
k,n v + b
(n+j)
k,n
)
u2j . (11.3.6)
Comparing (11.2.7) and (11.3.6), we explicitly find that
A
(n)
k (u) =
∞∑
j=0
a
(n+j)
k,n u2j , (11.3.7)
and
B
(n)
k (u) =
∞∑
j=0
b
(n+j)
k,n u2j . (11.3.8)
Therefore, the meaning of the connection between the GB and RB equations is the following: the
coefficient functions A
(n)
k (u) and B
(n)
k (u) are nothing but generating functions of the coefficients
a
(n)
j,k and b
(n)
j,k , respectively.
5
5A similar statement was formulated for the D-dimensional radial anharmonic oscillator, see Chapter 2,
Section 4.3.

Chapter 12
The Approximant
The analytical information about the phase, obtained from the RB and GB equations, Taylor
and asymptotic series, is now used to design the Approximant : an approximation of the exact
(unknown) ground state wave function denoted by ψ(t). To do so, we follow (as long as pos-
sible) the prescription proposed and applied in the previous Part, where it was successful in
constructing the Approximant of some particular cases of the D-dimensional radial polynomial
anharmonic oscillator.
First, we assume the exponential representation for the Approximant in coordinates (ρ, r),
ψ(t)(ρ, r) = e−Φt(ρ,r) . (12.0.1)
Therefore, we can focus only on the construction of Φt(ρ, r). According to the prescription,
the approximate phase has to interpolate the expansions of small and large distances, i.e., the
expansions (11.1.9), (11.2.22), and (11.2.23). Following the reflection symmetry of the system
ρ → −ρ for fixed r, Φt(ρ, r) should be a function of ρ2. One of the simplest interpolations,
which accomplishes the prescription given above, is of the form
Φt(ρ, r) =
α0 + α1 r + α3 r
2 + a3 γ ρ
2 + α4 γ ρ
2 r
√
1 + β1 r + β2 r2 + β3 ρ2
+ log(1 + β1 r + β3 r
2 + β4 ρ
2)
(12.0.2)
where {α0, α1, α2, α3, α4, β1, β3, β4} are 8 free real parameters that later will be fixed via the
VM. Note that this formula is analogous to (6.1.2).
The appearance of a square root in the denominator is motivated by the generating function
φ0, see (11.2.3). The logarithmic term is included to mimic the appearance of logarithmic terms
in the exact wave function, see (11.2.4). It will generate a prefactor in the approximate wave
function, which ultimately is given by
101
102 Chapter 12. The Approximant
ψ
(t)
1s0
(ρ, r) =
1
1 + β1 r + β2 r2 + β3 ρ2
exp
(
−α0 + α1 r + α2 r
2 + α3 γ ρ
2 + α4 γ ρ
2 r
√
1 + β1 r + β2 r2 + β3 ρ2
)
.
(12.0.3)
This is the expression of the trial function that is going to be used in numerical calculations.
We have labeled this Approximant with 1s0, indicating the usual notation for the ground state
hydrogen atom in the absence of a magnetic field.
As was discussed in Chapter 1, the variational method allows us to fix the value of the
free parameters. We take the Approximant (12.0.3) and compute its associated variational
energy Evar: the parameter-dependent expectation value of the Hamiltonian (10.0.4). The
variational principle guarantees that the variational energy is an upper bound of the exact
energy, Evar ≥ Eexact. Then, we minimize with respect to the free parameters to obtain their
optimal configuration that provides the lowest upper bound.
12.1 Results
12.1.1 Energy
From now on we set c = ~ = e = me = 1 in numerical computations, see (10.2.4). It is not a
surprise that a numerical integration and minimization are needed to calculate and minimize
the variational energy associated with (12.0.3). A code in FORTRAN 77 was written for this
purpose. It uses the integration routine D01FCF from NAG-LIB as well as the minimization
routine MINUIT of CERN-LIB. Further details of the computational code can be found in
[54], [55], and [56]. Variational estimates of the energy are presented in Table 12.1 for some
representative magnetic fields in the range γ ∈ [0.01, 10 000] a.u.
Plots of optimal variational parameters as functions of log(1 + γ2) are shown in Appendix I,
see Fig. C.1. In general terms, they are smooth changing parameters in the whole domain of
magnetic fields studied.
Our variational calculations are compared with some benchmarks coming from literature.
However, for some magnetic fields, no estimate of the energy was found. Hence, to analyze
the accuracy of our results, we carried out some numerical calculations via the Lagrange-
Mesh method in the formulation presented in [58] using spherical coordinates {r, θ, ϕ}. The
complete description and application of this method to the present system - hydrogen atom
in a constant magnetic field - was already presented, see Section 2.5 in [75]. We implemented
the Lagrange-Mesh method in a notebook of Mathematica. For the whole range of magnetic
fields studied, the mesh was kept unchanged, it consisted of Nr = 80 radial functions and
12.1. Results 103
Table 12.1: Energy E and quadrupole moment Qzz of the ground state 1s0 of the hydrogen
atom in constant magnetic field calculated in Variational Method (results without superscript)
for γ ∈ [0.01, 10 000] with trial function (12.0.3). Benchmarks are included. Energy presented
in Ry.
γ E −Qzz γ E −Qzz
0.01 -0.999 950 005 51 0.000 248 5.0 2.239 209 0.506 493
-0.999 950 005 52a,e 0.000 249e 2.239 202a,e 0.506 331e
0.1 -0.995 052 960 5 0.023 270 10.0 6.504 427 0.445 22
-0.995 052 960 8a,e 0.023 2712e 6.504 405a,d 0.445 09d
0.5 -0.894 421 065 0.256 143 100.0 92.420 7 0.217 5
-0.894 421 075a,e 0.256 156 21e 92.420 3a,d 0.216 8d
1.0 -0.662 337 66 0.417 618 500.0 487.487 0.125 1
-0.662 337 79a,d 0.417 654d 487.485a,e 0.123 8e
2.0 -0.044 426 7 0.511 354 1 000.0 984.678 0.099 4
-0.044 427 8a,e 0.511 432e 984.675a,d 0.098 2d
γ0 0.000 001 0.513 561 10 000.0 9 971.74 0.049 3
0.000 000e 0.513 537e 9 971.72e,f 0.047 9e
γ0 = 2.065 211 858, see (12.1.2).
a Power series [74], d Lagrange Mesh [75], e Lagrange Mesh (present work), f Basis of Splines [76].
Nu = 200 angular ones1. Hence the approximate ground state function is an expansion in
terms of 80 × 200 = 16 000 functions. With this enormous mesh, we reproduced the results
obtained for all magnetic fields studied in [75]: 1, 10, 100, and 1 000 a.u. It seems that the
maximum accuracy in energy - 19 exact s. d. - is obtained for magnetic fields γ . 1 a.u. In
general terms, for γ . 1 000 a.u. the method reaches in energy an accuracy of not less than 10
exact s. d. The accuracy decays dramatically once we study fields beyond γ ≈ 1 000 a.u.
For example, it reaches only 6 exact digits at γ = 10 000 a.u. Using this method, we have
confirmed and extended the results shown in [75].
Making a comparison between our results with benchmarks, we concluded the relative de-
viation of the variational estimate of the energy to the exact one is very small in the whole
domain of considered magnetic fields,
∣
∣
∣
∣
Evar − Eexact
Eexact
∣
∣
∣
∣
. 10−6 , γ ∈ [0.01, 10 000] . (12.1.1)
To the best of the author’s knowledge, there is no other variational calculation with a compact
1 u = cos θ.
104 Chapter 12. The Approximant
trial function that provides such an accurate estimate of the energy. This may be seen from
Table 12.2, where we show a collection of estimates of the ground state energy at γ = 1 a.u.
In this Table, we have introduced the result obtained from the Padé-Borel summation applied
to the first 100 coefficients in the perturbation expansion (11.1.1) of ε. This result provides 11
exact s.d. for the energy at γ ≤ 1a.u. However, the accuracy of the energy calculated by this
procedure decays dramatically for γ & 2 a.u.
Table 12.2: The ground state energy E1s0 in Ry of the hydrogen atom in constant magnetic
field γ = 1. The results ranked by accuracy. Infinite massive proton assumed.
Reference E1s0 Method
[77] Wallis & Bowlden, 1958 -0.346 Variational Method
[78] Yafet et al., 1956 -0.523 Variational Method
[79] Galindo & Pascual, 1976 -0.690 Rational Approximants
[80] Turbiner, 1984 -0.61 Variational Method
[81] Potekhin & Turbiner, 2001 -0.63 Adiabatic Approximation
[82] Kobori & Ohyama, 1997 -0.654 Variational Method
[83] Pokatilov & Rusanov, 1969 -0.659 1 Variational Method
[84] Larsen, 1968 -0.661 Variational Method
[85] Lee et al., 1973 -0.661 6 Variational Method
[86] Makado & McGill, 1986 -0.662 2 Variational Method
[87] Cabib et al., 1971 -0.662 4 Variational Method
[88] Arteca et al., 1983 -0.662 7 Regularization of PT
[89] Chen & Goldman, 1992 -0.662 35 Expansion of the Wavefunction
[90] Praddaude, 1972 -0.662 33 Power Series
[81] Potekhin & Turbiner, 2001 -0.662 332 Variational Method
[91] Vieyra & Olivares, 2008 -0.662 332 Variational Method
[92] Rösner et al., 1984 -0.662 338 Multi-Configuration Hartree-Fock
[93] Ivanov, 1988 -0.662 337 4 Finite Difference Method
Present Work, 2020 -0.662 337 7 Variational Method
[94] Xi et al., 1992 -0.662 337 78 Spline-basis
[76] Wang & Hsue, 1995 -0.662 337 79 Spline-basis
[73] Le Guillou & Zinn-Justin, 1983 -0.662 337 79 Borel-Leroy Summation
[95] Wunner & Ruder, 1982 -0.662 337 79 Expansion of the Wavefunction
[96] Fonte et al., 1990 -0.662 337 793 Variational Rayleigh-Ritz
Present Work, 2020 -0.662 337 793 46 Padé-Borel Summation
[97] Stubbins et al., 2004 -0.662 337 793 466 Variational Rayleigh-Ritz
[74] Kravchenko et al., 1996 -0.662 337 793 466 Method of Moments
[75] Baye et al., 2008 -0.662 337 793 466 315 9 Lagrange Mesh
Present Work, 2020 -0.662 337 793 466 316 071 2 Lagrange Mesh
[97] Stubbins et al., 2004 -0.662 337 793 466 316 071 220 596 ... Multiconfiguration
[98] Nakashima & Nakatsuji, 2010 -0.662 337 793 466 316 071 220 596 ... Free Complement
We also paid attention to the critical magnetic field γ0, for which the ground state energy
12.1. Results 105
vanishes2,
E(γ0) = 0 . (12.1.2)
The value of γ0 in the Lagrange-Mesh method, results in
γ0 = 2.065 211 858 a.u. , (12.1.3)
with E ∼ 10−10Ry. For this magnetic field, the variational trial function (12.0.3) provides
E ∼ 10−6Ry.
The local deviation between the Approximant and the exact wave function can be estimated
by studying the vicinity of the Coulomb singularity3 via the cusp parameter (11.1.12). A
straightforward calculation shows that the cusp parameter C(t), derived from to the Approxi-
mant (12.0.3), is given by a simple formula
C(t) = α1 +
(
1− α0
2
)
β1 , (12.1.4)
see (11.1.12). The results for C(t) are presented in Table 12.3. It can be seen that C(t), calculated
with optimal parameters as entry, satisfies the cusp condition accurately for γ . 500 a.u. with
an error of . 16%. For the largest magnetic field studied, γ = 10 000 a.u. the deviation
increases up to 70%. The cusp parameter C was also estimated in the Lagrange Mesh Method.
In general, this calculation provides a parameter C with not less than 6 exact s. d. in the whole
range of magnetic fields considered.
Table 12.3: Estimate of the cusp parameter for the ground state 1s0 coming from the Approx-
imant (12.0.3). Only s. d. presented. Atomic units used.
γ C(t) γ C(t)
0.01 1.000 002 5.0 0.997
0.1 0.999 97 10.0 1.002
0.5 0.999 7 100.0 1.065
1.0 0.999 3 500.0 1.159
2.0 0.996 5 1 000.0 1.23
γ0 0.996 7 10 000.0 1.7
12.1.2 Quadrupole Moment
It is well-known that the hydrogen atom in its ground state acquires a quadrupole moment at
γ > 0, see [81]. Based on the cylindrical symmetry of the system, it can be easily shown that
2Note that γ0 defines a normalizable zero mode of the Hamiltonian (10.0.4).
3Located at r = 0, see (10.0.5).
106 Chapter 12. The Approximant
0 3 6 9 12 15 18
0.1
0.2
0.3
0.4
0.5
0.6
0 3 6 9 12 15 18
0.5
1
1.5
2
Figure 12.1: Left: Plot of the absolute value of the quadrupole moment (−Qzz) in (a.u)2 as a
function log(1+ γ2). Variational estimate (see 12.0.3) of the quadrupole moment as well as the
one coming from Padé-Borel summation presented. In the Lagrange-Mesh method, the mini-
mum of the quadrupole moment −Qzz = 0.524 52 (a.u)2 is reached at γ ≅ 2.968 69a.u. Right:
Expectation values 〈z2〉 and 〈ρ2〉 calculated by means of Approximant (12.0.3) as functions
of log(1 + γ2). The intersection of the curves occurs at γ ≈ 2.806a.u.; at this magnetic field
〈z2〉 = 〈ρ2〉 ≈ 0.524. For γ & 700a.u. the value of 〈z2〉 is at least two orders of magnitude
larger than 〈ρ2〉. Atomic units used.
the quadrupole moment tensor is diagonal and only contains one single independent element,
say
Qzz = 〈ρ2〉 − 2 〈z2〉 . (12.1.5)
Once optimal parameters are found for the Approximant (12.0.3), the Qzz component can be
estimated for different values of γ using the square of the Approximant as a compact analytical
expression for the electronic charge distribution. Plots of Qzz and expectations values, 〈z2〉 and
〈ρ2〉, as functions of γ are presented in Fig. 12.1. The numerical results are shown in Table
12.1. In the literature, there are not many accurate benchmarks for Qzz. Hence, in order to
estimate the deviation of Qzz (calculated via the Approximant (12.0.3)) with respect to the
exact value, we use the Lagrange-mesh method to obtain highly accurate estimates of Qzz that
can be regarded as the exact values, see Table 12.1. These estimates indicate that for small
magnetic fields γ . 2, we reproduce not less than 4 exact s.d. In turn, for γ & 2 we reproduce
exactly 2-3 s.d.
In addition to those calculations, the quadrupole moment is estimated using the PT calculated
for Φn, see (11.1.1). In this manner, we can construct the expansion of the quadrupole moment
in powers of γ2,
Qzz(γ) =
∞∑
n=0
Q(n)
zz γ
2n . (12.1.6)
12.1. Results 107
Up to O(γ8) this expansion4 looks like
Qzz(γ) = −5
2
γ2 +
615
32
γ4 − 2564987
11520
γ6 +
19550772149
5529600
γ8 + . . . . (12.1.7)
Due to the polynomial nature of corrections Φn with rational coefficients, see (11.1.6), correc-
tions Q
(n)
zz in the expansion (12.1.6) will be rational numbers as well. The calculation of the
first Q
(n)
zz indicates that series (12.1.6) are alternating with very fast growing coefficients as n
grows. It is an indication that (12.1.6) could be a divergent series. The largest coefficient that
we calculated was Q
(26)
zz whose (rounded) value is
Q(26)
zz = −8.188× 1047 . (12.1.8)
Making a comparison of the first 26 coefficients Q
(n)
zz in the expansion (12.1.6) with those εn
in (11.1.1), one can notice that the series for the quadrupole moment diverges faster than for
the energy. With a motivation coming from the results obtained for the energy, we tested the
applicability of the Padé-Borel summation to the first 26 terms of the series (12.1.6), assuming
the same growth of Q
(n)
zz as in (11.1.8) for εn. In general, the Padé-Borel summation provides
accurate results for γ . 2 a.u: it confirms not less than 3 correct s. d. of Qzz calculated either
by means of the Approximant or the Lagrange-Mesh method. The accuracy of the Padé-Borel
summation decays dramatically for magnetic fields larger than γ = 2 a.u. The plot of the Qzz
as a function of γ, calculated in this scheme, is presented in Fig. 12.1.
4First ten coefficients are presented explicitly in Appendix, see F.3.

Chapter 13
Conclusions
Following the formalism presented for the D-dimensional radial anharmonic oscillator in Part
I, a compact wave function Approximant for the ground state of a hydrogen atom in a constant
magnetic field was constructed. PT, asymptotic expansions at small and large distances, the
choice of coordinates (ρ, r) to work with, as well as the RB and GB1 equations, were basic
ingredients to construct the Approximant. The main result of the study is summarized in the
following expression for the Approximant
ψ
(t)
1s0
(ρ, r) =
1
1 + β1 r + β2 r2 + β3 ρ2
exp
(
−α0 + α1 r + α2 r
2 + α3 γ ρ
2 + α4 γ ρ
2 r
√
1 + β1 r + β2 r2 + β3 ρ2
)
,
where {α0, α1, α2, α3, α4, β1, β3, β4} are 8 free parameters. To fix their value, the Variational
Method using the Approximant ψ
(t)
1s0
as trial function was employed. It led to a variational
estimate of the ground state energy with a very small relative deviation, of order ∼ 10−6 in
the whole studied range of magnetic fields γ ∈ [0.01, 10 000] a.u. These results reproduced
with high accuracy the benchmark energies (some of them are established by this work in the
Lagrange-Mesh method).
As an additional calculation, the quadrupole moment was computed for the same range
using the square of the Approximant as a compact expression of the electronic density. Results
coincide with benchmarks (coming from the Lagrange-Mesh method) in not less than 3 s. d.
for γ . 100 a.u. It was explicitly demonstrated that the cusp condition of the exact hydrogen
atom holds even in the presence of a constant magnetic field. Our Approximant provides a cusp
parameter with a very small deviation from the exact value. For the weak and intermediate
magnetic fields γ . 500 a.u., its relative deviation is about 15%.
Due to the success of the Approximant when describing the ground state, it is clear that it can
be modified to study excited states. Multiplying the Approximant by a suitable polynomial,
1Which leads to the true semi-classical expansion.
109
110 Chapter 13. Conclusions
some particular families of excited states can be studied, just like it was done in [91]. For
example,
z ψ
(t)
1s0
(ρ, r)
where z =
√
r2 − ρ2, is a suitable Approximant to describe the excited state 2p0. However, it
should be mentioned that the general structure of the nodal surfaces, in the form of a prefactor,
is still unknown.
Due to the algebraic nature of PT in the framework of the Non-Linearization Procedure, the
first 100 energy corrections (in powers of the effective magnetic field) with absolute accuracy
were obtained: corrections are presented for the first time as rational numbers. Using these
corrections, it was summed up the (divergent) PT for the energy via the Padé-Borel summation
technique. It led to highly accurate results (not less than 11 s.d.) for small magnetic field γ . 1
a.u.
Further Studies
• PT can be constructed in the framework of the Non-Linearization Procedure to estimate
the accuracy of the variational energy and wave function. In this context, the Approx-
imant plays the role of the unperturbed wave function. This approach was successfully
applied to the D-dimensional radial anharmonic oscillator: corrections in PT of the vari-
ational energy were computed. As a result, its deviation with respect to the exact value
was estimated, see Eq. (7.2.27). One could try to follow the same procedure for the
hydrogen atom in a magnetic field by considering corrections to the approximate phase
and the energy,
Φexact = Φ
(0)
t + Φ
(1)
t + Φ
(2)
t + . . . , Eexact = Evar + E
(t)
2 + E
(t)
3 + . . .
where Φ
(0)
t ≡ Φt, see Eq. (12.0.2). In the Non-Linearization procedure, it can be eas-
ily shown that the first order correction to the phase, denoted by Φ
(1)
t , obeys a two-
dimensional linear partial differential equation (PDE), namely
∆Φ
(1)
t − 2∇Φ
(0)
t · ∇Φ
(1)
t = Evar − V (1) .
Here we have denoted V (1) = V − V0, where
V = −1
r
+
γ2
8
ρ2 , V0 ≡ ∆Ψ(t)
Ψ(t)
.
If the solution for ∇Φ
(1)
t is found, one could calculate the first correction to Evar using
that
E
(t)
2 = −
∫ (
∇Φ
(1)
t
)2
ψ(t)ψ(t) dV
∫
ψ(t)ψ(t) dV
,
see [8]. If V (1) is subdominant in comparison to V , i.e. the ratio of V (1) to V is bounded at
large distances, then PT developed in the Non-Linearization Procedure will be convergent.
111
• We did not take into account the information coming from the strong coupling regime
(γ → ∞) in the design of the Approximant. For the ground state energy, the first two
terms of the strong coupling expansion are well-known,
E = γ − log2 γ + . . . ,
see [99]. It would be interesting to use this expansion in the RB and GB equations and
study the strong coupling regime for the phase. This piece of missing information could
lead to an improvement of the Approximant.2
• In this work, an infinitely massive and static proton was assumed. In the Schrödinger
equation, finite mass effects can be incorporated making a pseudoseparation of variables of
the center of mass and the relative coordinates, see [100] for details. Naturally, taking into
account the finite mass of the proton, the spectrum of the hydrogen atom in a constant
magnetic field will be modified. It turns out that when the (finitely) massive proton is
fixed in the space (hydrogen atom at rest), the Schrödinger equation that describes this
situation looks very similar to Eq. (10.0.1). The only difference is that the reduced mass
of the hydrogen atom replaces the electron mass me in Eq. (10.0.1), see [101]. As a
consequence, the Approximant that we have presented in this Part is also appropriate
to describe the ground state of the finite massive static hydrogen atom in a constant
magnetic field, see [11]. The study of finite mass effects was beyond of the purpose of the
present Thesis, it will be done elsewhere.
2For example, an extra parameter, say q, may be inserted in the form of a degree of the prefactor of the
Approximant. In this manner
ψ
(t)
1s0
(ρ, r) =
1
(1 + β1 r + β2 r2 + β3 ρ2)q
exp
(
−α0 + α1 r + α2 r
2 + α3 γ ρ
2 + α4 γ ρ
2 r
√
1 + β1 r + β2 r2 + β3 ρ2
)
.
The insertion of this extra parameter surely will lead to higher accuracy for both the energy and the cusp
parameter.

General Conclusions
113

115
• Part I
For the cubic, quartic, and sextic radial anharmonic oscillator potentials, whose cor-
responding Schrödinger equations do not allow finding exact solutions, highly accurate
approximate eigenfunctions of the low-lying states are written for arbitrary coupling con-
stant and space dimension. Associated energies lead to unprecedented relative accuracies
of (at least) order . 10−5 for any coupling constant in any dimension. These energies
are in complete agreement with the highly accurate results of the Lagrange-Mesh calcu-
lations, performed in the present thesis. Those results can be generalized for the radial
anharmonic oscillator with arbitrary anharmonicity. It manifests a solution of the spectral
problem of the radial anharmonic oscillator.
• Part II
A locally accurate ground state wave function (the Approximant) for the hydrogen atom
subjected to a constant uniform magnetic field is found. This approximate function leads
to high accuracy for the ground state energy (with relative accuracy . 10−5), as well as
for the quadrupole moment. In the whole domain of accessible magnetic fields in Nature,
the obtained energies are in complete agreement with the highly accurate ones provided
by the Lagrange-Mesh method and the Padé-Borel summation3. In a non-relativistic
consideration with a static proton, the Approximant can be regarded as the solution of
the Schrödinger equation, which does not admit exact solutions.
3The domain of applicability of the Padé-Borel sum occurs for weak and intermediate fields.

Appendices
117

Appendix A
The WKB (Semi-Classical)
Approximation
We present an extension of the WKBmethod that can be found in elementary books on quantum
mechanics, e.g. [1,2]. In particular, we focus the discussion on the radial Schrödinger equation
in arbitrary dimension D = 1, 2, ..., which describes an S-state1 of an arbitrary radial potential
− ~2
2M
(
∂2r +
D − 1
r
∂r
)
Ψ + V (r)Ψ = EΨ . (A.0.1)
If we assume that the potential fulfills
(i) V (r) ≥ 0
(ii) V (0) = 0
(iii) V (r → ∞) → +∞
we can guarantee the existence of bound states. We adopt the exponential representation for
the wave function to find them,
Ψ(r) = eiS(r)/~ . (A.0.2)
The reason why we include ~ explicitly will be clear at the end of this Appendix. For now, it
is clear that the physical units of S are those of ~, i.e. units of action. In this representation
(A.0.1) becomes a Riccati-type equation
i~S ′′(r)− S ′(r)
(
S ′(r)− i~ (D − 1)
r
)
= 2M(V (r)− E) (A.0.3)
cf. (3.3.4). For our purposes, it is enough to study this equation in the so-called non-classical
domain in r (which is characterized by the inequality V (r) > E) beyond the largest turning
1Characterized by zero angular momentum ℓ = 0.
119
120 Appendix A. The WKB (Semi-Classical) Approximation
point. To solve (A.0.3) in PT we consider the expansion
S(r) =
∞∑
n=0
Sn(r)~
n (A.0.4)
keeping E fixed. It is straight forward to show that equations
S ′
0 = ±i
√
2M(V − E) , (A.0.5)
and
S ′
1 =
i(D − 1)
2r
+
i
2
S ′′
0
S0
(A.0.6)
determine the first 2 corrections. In general, S ′
n is given by the expression
S ′
n =
iS ′′
n−1 +
i(D−1)
2r
S ′
n−1 −
∑n−1
k=1 S
′
kS
′
n−k
2S ′
0
, (A.0.7)
as we can see, the calculation of S ′
n is an algebraic procedure. To guarantee the normalizability
of the wave function (A.0.2) we need to take the positive sign in (A.0.5). Thus, it will decay
exponentially at large distances. Taking this into account, it is clear that the wave function is
of the form
Ψ(r) =
1
r
D−1
2 (2M(V (r)− E)1/4)
exp
(
−1
~
∫ r√
2M(V (r)− E) dr + O(~)
)
, (A.0.8)
as long as r → ∞. Some remarks should be pointed out:
(i) Contrary to the function S(r), the energy E is considered fixed as a function of ~. Of
course, this is not always true. For most of the potentials, the energy E should be
~-dependent.
(ii) The integrand that occurs in (A.0.8) has a physical meaning, it is (almost) the classical
action presented in the form ∼
∫
p dq.
Further properties and results can be obtained from the WKB approximation. To do so, it is
necessary to study the behavior of the wave function in the classical domain(s) and then match
the solutions for each region (the non-classical and classical one). Some of those results are
well known, for example, the celebrated Bohr-Sommerfeld from early quantum mechanics. A
review of those properties is found in [1].
Appendix B
First PT Corrections and Generating
Functions
B.1 Cubic Case
The expansions in powers of v for the first three corrections Yn(v), n = 1, 2, 3 for the cubic
anharmonic oscillator potential (7.2.1), see (4.1.15) and (4.1.18) are presented.
• At v → 0
Y1(v) = ǫ1v +
2 ǫ1
D + 2
v3 − 1
D + 3
v4 +
ǫ1
(D + 2)(D + 4)
v5 + . . . ,
Y2(v) = ǫ2 v +
ǫ21 + 2 ǫ2
D + 2
v3 +
6 ǫ21 + 4 ǫ2
(D + 2)(D + 4)
v5 − 2 ǫ1
(D + 3)(D + 5)
v6 + . . . ,
Y3(v) = ǫ3 v +
2(ǫ1 ǫ2 + ǫ3)
D + 2
v3 +
2(ǫ31 + 6 ǫ1 ǫ2 + 2 ǫ3)
(D + 2)(D + 4)
v5 − 2 ǫ2
(D + 3)(D + 5)
v6 + . . . .
(B.1.1)
In these formulas we have denoted ǫn =
εn
D
.
• At v → ∞
Y1(v) =
1
2
v2 +
1
4
(D + 1) − ε1
2
v−1 +
1
8
(D2 − 1)v−2 + . . . ,
Y2(v) = − 1
8
v3 − 1
16
(3D + 4)v +
ε1
4
− 1
32
(6D2 + 6D + 16 ε2 − 1)v−1
+
1
8
(3D − 2) ε1v
−2 + . . . ,
Y3(v) =
1
16
v4 +
1
32
(5D + 8)v2 − 3 ε1
16
v +
1
64
(15D2 + 26D + 16 ε2 + 10)
− 1
32
(15D ε1 + 16 ε3)v
−1 + . . . . (B.1.2)
A straightforward application of the formulas (4.2.3) and (4.4.2) allows us to calculate the
121
122 Appendix B. First PT Corrections and Generating Functions
generating functions. In particular, G3 and G4 are given by
G3 = − (2M)1/2
g2
(
ε1
2
log
[
w − 1
w + 1
])
,
G4 = − (2M)1/2
g2
(
5 + (5 + 12D)w + (1− 6D(D + 1))(w + 1)w2 (3w2 − 2)
48(w − 1)(w + 1)2w3
(1− 16 ε2 − 6D(D + 1))
32
log
[
w − 1
w + 1
])
, (B.1.3)
where w = (1 + gr)1/2. Explicit formulas for the first corrections εn in PT at D = 1 have been
presented in (7.2.5).
B.2 Quartic Case
We present the corrections ε4,6 and Y4,6 in explicit form for the quartic anharmonic potential
(7.3.1) in the expansion (7.3.6) and (7.3.7),
ǫ2 =
1
4
(D + 2) ,
ǫ4 = − 1
16
(D + 2)(2D + 5) ,
ǫ6 =
1
64
(D + 2)(8D2 + 43D + 60) ,
(B.2.1)
Y2(v) =
1
2
v3 + ǫ2v ,
Y4(v) = −1
8
v5 − 1
16
(3D + 8)v3 + ǫ4v ,
Y6(v) =
1
16
v7 +
1
32
(5D + 16)v5 +
1
16
(3D2 + 17D + 25)v3 + ǫ6v ,
(B.2.2)
with ǫ2n =
ε2n
D
.
We also present two generating functions for the phase, see expansion (4.4.1),
G4 =
5
24g2w3
+
D (1 + w + w2)
4g2(w + 1)w2
+
D2
8g2w
,
G6 = − 5
16g2w6
− D (15 + 30w + 20w2 + 16w3 + 20w4 + 30w5 + 15w6)
48g2(w + 1)2w5
− D2 (4 + 8w + 8w2 + 12w3 + 18w4 + 9w5)
32g2(w + 1)2w4
− D3 (1 + 3w2)
48g2w3
,
(B.2.3)
B.3. Sextic Case 123
where w =
√
1 + g2v2. Two remarks in a row: (i) in the variable w all generating functions are
rational functions, (ii) the polynomial structure in variable D of generating functions becomes
evident.
B.3 Sextic Case
We present explicitly the first corrections ε8,12 and Y8,12 for the sextic anharmonic oscillator
(7.4.1). See (7.4.3) and (7.4.4).
ǫ4 =
1
8
(D + 2)(D + 4) ,
ǫ8 = − 1
128
(D + 2)(D + 4)(9D2 + 72D + 152) ,
ǫ12 =
1
1024
(D + 2)(D + 4)(81D4 + 1404D3 + 9624D2 + 31152D + 40384) .
(B.3.1)
Y4(v) =
1
2
v5 +
1
4
(D + 4)v3 + ǫ4v ,
Y8(v) = − 1
8
v9 − 1
16
(3D + 16)v7 − 1
16
(3D2 + 27D + 64)v5
− 1
32
(4D3 + 49D2 + 204D + 288)v3 + ǫ8v ,
Y12(v) =
1
16
v13 +
1
32
(5D + 32)v11 +
5
64
(3D2 + 34D + 104)v9
+
1
32
(8D3 + 125D2 + 688D + 1344)v7
+
1
256
(55D4 + 1038D3 + 7708D2 + 26784D + 36800)v5
+
1
256
(36D5 + 783D4 + 7040D3 + 32768D2 + 78912D + 78336)v3
+ ǫ12v, (B.3.2)
where it has been defined ǫ4n = ε4n
D
.
Now we present explicitly two generating functions G4, G6 in the expansion (4.4.1),
G4 =
r2 (5 + w2)
12w3
+
Dr2 (1 + w + w2)
4w2(w + 1)
+
D2r2
16w
,
G6 =
5− 3w2
4g2w6
+
D (15 + 15w − 4w2 + 2w3 + 6w4 + 6w5)
24g2(w + 1)w5
+
D2 (2 + 2w + w2 + 3w3 + 3w4)
16g2(w + 1)w4
+
D3 (1 + 3w2)
96g2w3
, (B.3.3)
124 Appendix B. First PT Corrections and Generating Functions
where w =
√
1 + g4r4. Two remarks: i) in the variable w, generating functions are rewritten
as rational ones; ii) The polynomial structure in variable D of these generating functions is
evident.
Appendix C
Plots of Variational Parameters:
Anharmonic Oscillator
In this Appendix, we present the plots of the optimal parameters in the Variational Method for
the Approximants of the ground state of the cubic, quartic, and sextic anharmonic oscillator.
The number of equation of each Approximant is indicated in the title of each Section.
C.1 Cubic: (7.2.24)
2 4 6 8 10
g
-10
-5
0
5
10
0 0.4 0.8
0.45
0.55
0.65
2 4 6 8 10
g
0.45
0.5
0.55
0.6
0.65
0 2 4 6 8 10
g
0.2
0.4
0.6
0.8
Figure C.1: Variational parameters ã0, ã2 and ã3 of the ground state as functions of g for
D = 2, 3, 6.
125
126 Appendix C. Plots of Variational Parameters: Anharmonic Oscillator
C.2 Quartic: (7.3.26)
✷ ✹ ✻ ✽ ✶
❣
✁
-✶

✶
✷
✸
❛
✂
✵
(a)
✵ ✵✁ ✵✂ ✵✄ ✵☎
✵✆
✵✆✆
✵✝
✞ ✷ ✹ ✻ ✽ ✶✞
❣
✟
✞✠✡
✞✠✻
✞✠☛
✞✠✽
✞✠☞
✶✠
❛
✌
✟
(b)
✵ ✷ ✹ ✻ ✽ ✶✵
❣

✵✁✵✂
✵✁✶
✵✁✶✂
✵✁✷
✵✁✷✂
✵✁✄
✵✁✄✂
❛
☎
✆
(c)
Figure C.2: Ground state: Variational parameters ã0 (a), ã2 (b) and ã4 (c) vs the coupling
constant g for D = 1, 2, 3, 6. Parameters (7.3.28), which allow us to reproduce the first two
terms G0, G2 in the expansion (4.4.1) shown by solid (black) line which is horizontal for ã2 (b)
and ã4 (c).
C.2. Quartic: (7.3.26) 127
✷ ✸ ✹ ✺ ✻
❉
-✶
✵
✶
✷
✸
❛

✁
(a)
✶ ✷ ✸ ✹ ✺ ✻
❉✵✺
✵✻
✵✁
✵✂
✵✄
❛
☎
✆
(b)
✶ ✷ ✸ ✹ ✺ ✻
❉✵✵✺
✵✶
✵✶✺
✵✷
✵✷✺
✵✸
✵✸✺
✵✹
❛
✁
✂
(c)
Figure C.3: Variational parameters ã0(a), ã2(b), ã4(c) vs D for fixed g2 = 0.1, 1, 10. D-
independent parameters (7.3.28), which allow us to reproduce the first two terms G0, G2 in
the expansion (4.4.2), shown by solid (black) horizontal lines, see (7.3.28). In (a) the horizontal
lines correspond to ã0 =
1
3g2
at g2 = 0.1(i), 1(ii), 10(iii).
128 Appendix C. Plots of Variational Parameters: Anharmonic Oscillator
C.3 Sextic: (7.4.27)
✷ ✹ ✻ ✽ ✶
❣✁
-✸
-✷
-✶

❛
✂
✵
(a)
✵ ✷ ✹ ✻ ✽ ✶✵
❣✵✁✷
✵✁✷✂
✵✁✄
✵✁✄✂
✵✁✹
❛
☎
✆
(b)
✵ ✷ ✹ ✻ ✽ ✶✵
❣✵✁✶
✵✁✶✷
✵✁✶✹
✵✁✶✻
✵✁✶✽
❛
✂

(c)
✵ ✷ ✹ ✻ ✽ ✶✵
❣✵✁✷✂
✵✁✄
✵✁✄✂
✵✁✹
✵✁✹✂
✵✁✂
❜
☎

(d)
✵ ✷ ✹ ✻ ✽ ✶✵
❣
✵✁✵✂
✵✁✶
✵✁✶✂
✵✁✷
✵✁✷✂
❜
✄
☎
(e)
✵ ✷ ✹ ✻ ✽ ✶✵
❣✵✁✶✂
✵✁✷
✵✁✷✂
✵✁✄
✵✁✄✂
✵✁✹
❝
☎
✆
(f)
Figure C.4: Ground state (0, 0): Variational parameters ã0 (a), ã2 (b), ã4 (c), b̃4 (d), b̃6 (e),
c̃2 (f) vs the coupling constant g4 in domain g4 ∈ [0, 10] for D = 1, 2, 3, 6.
C.3. Sextic: (7.4.27) 129
✷ ✸ ✹ ✺ ✻
❉✵
-✶
-✷
-✸
❛

✁
(a)
✶ ✷ ✸ ✹ ✺ ✻
❉✵✷
✵✷✺
✵✸
✵✸✺
✵✹
❛
✁
✂
(b)
✶ ✷ ✸ ✹ ✺ ✻
❉✵✶
✵✶✷
✵✶✹
✵✶✻
✵✶✁
❛
✂
✄
(c)
✶ ✷ ✸ ✹ ✺ ✻
❉✵✷✺
✵✸
✵✸✺
✵✹
✵✹✺
✵✺
❜
✁
✂
(d)
✶ ✷ ✸ ✹ ✺ ✻
❉✵
✵✵✺
✵✶
✵✶✺
✵✷
✵✷✺
❜
✁
✂
(e)
✶ ✷ ✸ ✹ ✺ ✻
❉✵✶
✵✷
✵✸
✵✹
❝
✁
✂
(f)
Figure C.5: Ground state (0, 0): Variational parameters ã0 (a), ã2 (b), ã4 (c), b̃4 (d), b̃6 (e),
c̃2 (f) vs dimension D for g4 = 0.1, 1.0, 10.0.
Appendix D
General Aspects of the Two-Term
Radial Anharmonic Oscillator
Let us consider the general two-term radial anharmonic oscillator potential,
V (r) = r2 + gm−2rm , m > 2 . (D.0.1)
cf. (3.2.1), where a2 = am = 1 and a3, a4, ..., am−1 = 0. Expansion of ε in powers of λ has the
form
ε(λ) = ε0 + λm−2εm−2 + λ2(m−2)ε2(m−2) + ... , (D.0.2)
see (4.1.5). Not surprisingly, for even potentials (m = 2p, p = 1, 2, ...) the PT coefficients
ε2n(p−1), n = 1, 2, ..., are polynomials in D factorized as follows
ε2n(p−1)(D) = D(D + 2)(D + 4) . . . (D + 2p− 2)R(n−1)(p−1)(D) (D.0.3)
where R(n−1)(p−1)(D) is a polynomial inD of degree (n−1)(p−1), see [29]. We emphasize that, in
the framework of the Non-Linearization Procedure, any PT coefficient ε2n(p−1)(D) is calculated
by linear algebra means. Series (D.0.2) is divergent due to a Dyson instability argument. It is
evident that for larger D and m, the index of divergence is larger, see e.g. [93].
For potential (D.0.1) generating functions G0(r) and G2(r) can be written in closed form,
G0(r)
(2M)1/2
=
2
m+ 2
r2
{
√
1 + (gr)m−2 +
m− 2
4
2F1
(
1
2
,
2
m− 2
;
m
m− 2
;−(gr)m−2
)}
, (D.0.4)
g2G2(r)
(2M)1/2
=
1
4
log
[
1 + (gr)m−2
]
+
D
m+ 2
log
[
1 +
√
1 + (gr)m−2
]
, (D.0.5)
where 2F1 is the hypergeometric function.
130
Appendix E
System of Coordinates {ρ, r, ϕ}
A general description of the non-orthogonal system of coordinates {ρ, r, ϕ} is presented in this
Appendix. Here we show some relevant expressions for the differentiation of a given function.
E.1 Definition
The system of coordinates {ρ, r, ϕ}, shown schematically in Fig. 10.1, is defined by the change
of variables
ρ =
√
x2 + y2 (E.1.1)
r =
√
x2 + y2 + z2 (E.1.2)
ϕ =



arctan
(y
x
)
x > 0 , y > 0
2π + arctan
(y
x
)
x > 0, y < 0
π
2
sgn(y) x = 0
π + arctan
(y
x
)
x < 0
(E.1.3)
where {x, y, z} denote Cartesian coordinates. The new system of coordinates is defined in the
domain
0 ≤ ρ ≤ r , 0 ≤ r <∞ , 0 ≤ ϕ ≤ 2π . (E.1.4)
The change of variables presented in (E.1.1), (E.1.2) and (E.1.3) is unambiguously defined for
the half-space z ≥ 0.
131
132 Appendix E. System of Coordinates {ρ, r, ϕ}
Conversely, the Cartesian coordinates may be retrieved from (E.1.1), (E.1.2) and (E.1.3),
x = ρ cosϕ , (E.1.5)
y = ρ sinϕ , (E.1.6)
z =
√
r2 − ρ2 . (E.1.7)
E.2 Differentiation
The Non-Linearization Procedure requires some expressions that involve differential operators;
we present them below. The basic expressions come from the partial derivatives, for which a
straightforward calculation via the chain rule indicates that
∂x = cosϕ∂ρ +
ρ cosϕ
r
∂r − sinϕ
ρ
∂ϕ , (E.2.1)
∂y = sinϕ∂ρ +
ρ sinϕ
r
∂r +
cosϕ
ρ
∂ϕ , (E.2.2)
∂z =
√
r2 − ρ2
r
∂r . (E.2.3)
Using these expressions, one can easily check that the Laplacian operator ∇2 takes the form
∇2 = ∂2ρ +
2ρ
r
∂ρr + ∂2r +
1
ρ
∂ρ +
2
r
∂r +
1
ρ2
∂2φ . (E.2.4)
Another relevant quantity comes from the dot product of two gradients. Let us consider func-
tions f and g written in variables (ρ, r, ϕ), explicitly
f = f(ρ, r, ϕ) , g = g(ρ, r, ϕ) . (E.2.5)
A direct calculation using (E.2.1), (E.2.2), and (E.2.3), leads to
∇f · ∇g = ∂ρf ∂ρg + ∂r f∂rg +
ρ
r
(∂rf ∂ρg + ∂ρf ∂rg) +
1
ρ2
∂φf ∂φg . (E.2.6)
Appendix F
Perturbation Theory for Hydrogen
Atom in Constant Magnetic Field
F.1 Computational Code: Non-Linearization Procedure
Below it is presented in explicit form the Mathematica Notebook which realizes the Non-
Linearization procedure for the hydrogen atom in presence of a uniform and constant magnetic
field. The program is optimized to run it directly from the terminal/console.
(*This program calculates PT series of the enery and phase \
of hydrogen atom in magnetic field in its the ground state \
using the Non -Linerization procedure in variables (rho=s,r) *)
(* Load Module: module load Mathematica /11.2 *)
(* Usage: math -noprompt -run < file.txt *)
(*----------------------------------------------------*)
(* Autor: Juan Carlos de Valle 2019( revisited) *)
(*----------------------------------------------------*)
(*----------------------------------------------------*)
(* Order (Input Data) *)
m = 100;
(*----------------------------------------------------*)
(* Initial Data *)
f[0] = r;
(*----------------------------------------------------*)
(* Constraint *)
Do[
Subscript[Overscript[a, n], 0, 2*n + 1] = 0,
{n, 1, m}
];
(*----------------------------------------------------*)
(* ******************************************** *)
(* MAIN PROGRAM *)
(* ******************************************** *)
(* Definitions *)
f[n_] :=
Expand[Sum[( Subscript[Overscript[A, n], k, j]*r \
+ Subscript[Overscript[B, n], k, j])*s^(2*(n - k))*r^(2*\
(k - j)),
{j, 0, n - 1}, {k, j, n}]];
Subscript[E, 0] = -1^2;
Q[1] := (1/4)*s^2;
Q[n_] :=
-Expand[Sum[D[f[k], s]*D[f[n - k], s] +
D[f[k], r]*D[f[n - k], r] + (s/r)*(D[f[k], s]*D[\
f[n - k], r] +
D[f[k], r]*D[f[n - k], s]), {k, 1, n - 1}]];
eq[n_] :=
Expand[D[f[n], s, s] + ((2*s)/r)*D[f[n], s, r] \
+
D[f[n], r, r] + (1/s - (2*s)/r)*D[f[n], s] +
2*(1/r - 1)*D[f[n], r] - Subscript[E, n] + Q[n]\
];
(*----------------------------------------------------*)
(* OUTPUT *)
(*----------------------------------------------------*)
Energy ={};
(* Solving Algebraic Equations *)
Do[
sol = SolveAlways[eq[i] == 0, {r, s}];
Apply[Set , sol[[1]], {1}];
PrintTemporary[i];
AppendTo[Energy ,{i,Subscript[E, i]}];
Export["Coefficients.dat", Energy],
{i, 1, m}
];
133
134 Appendix F. Perturbation Theory for Hydrogen Atom in Constant Magnetic Field
F.2 Corrections in Perturbation Theory
We present explicitly the first corrections φn, n = 2, 3, 4 in the expansion (11.1.1). Additionally,
in Tables F.1 - F.2 high order corrections εn are shown. Finally, the first 10 coefficients in the
expansion of the quadrupole moment, see (12.1.6), are displayed in Table F.3.
−φ2(s, t) =
1
1152
s4t+
1
1440
s2t3
+
11
4608
s4 +
13
1440
s2t2 +
1
2880
t4
+
193
5760
s2t+
1
120
t3
+
193
3840
s2 +
337
5760
t2 (F.2.1)
φ3(s, t) =
1
27648
s6t+
1
11520
s4t3 +
1
60480
s2t5
+
7
55296
s6 +
163
138240
s4t2 +
131
241920
s2t4 +
1
181440
t6
+
61
11520
s4t+
8063
1209600
s2t3 +
53
201600
t5
+
803
92160
s4 +
33311
806400
s2t2 +
2927
604800
t4
+
90877
691200
s2t+
2027
43200
t3
+
90877
460800
s2 +
188173
691200
t2 (F.2.2)
−φ4(s, t) =
5
2654208
s8t+
110592
s6t3 +
163
29030400
s4t5 +
1
2419200
s2t7
+
163
21233664
s8 +
293
2211840
s6t2 +
9833
58060800
s4t4 +
727
29030400
s2t6 +
1
9676800
t8
+
8819
13271040
s6t+
1663979
812851200
s4t3 +
24733
40642560
s2t5 +
167
20321280
t7
+
10577
8847360
s6 +
13945163
1083801600
s4t2 +
22721
2822400
s2t4 +
5989
22579200
t6
+
27927329
650280960
s4t+
29335139
451584000
s2t3 +
4828099
1016064000
t5
+
816005783
13005619200
s4 +
1349713153
4064256000
s2t2 +
146213807
2709504000
t4
+
16222576613s2t
16257024000
+
141801871t3
338688000
+
16222576613s2
10838016000
+
36642046037t2
16257024000
(F.2.3)
F.2. Corrections in Perturbation Theory 135
Table F.1: First ten coefficients εn (in rational form) of the series expansion for ε calculated in
the Non-Linearization Procedure, see (11.1.1).
n (−1)n+1εn
0 1
1 1/2
2 53/96
3 5581/2304
4 21577397/1105920
5 31283298283/132710400
6 13867513160861/3538944000
7 5337333446078164463/62426972160000
8 995860667291594211123017/419509252915200000
9 86629463423865975592742047423/1057163317346304000000
10 6127873544613551793091647103033033/1776034373141790720000000
Table F.2: Coefficients ε10n, n = 1, 2, ..., 10, of the series expansion for ε calculated in Non-
Linearization Procedure, see (11.1.1). Coefficients rounded to 3 displayed d. d.
n −ε10n n −ε10n
1 3.450× 109 6 5.655× 10140
2 2.160× 1029 7 1.410× 10173
3 3.215× 1053 8 6.046× 10206
4 3.720× 1080 9 3.127× 10241
5 6.263× 10109 10 1.479× 10277
136 Appendix F. Perturbation Theory for Hydrogen Atom in Constant Magnetic Field
Table F.3: First ten coefficients Q
(n)
zz of the series expansion in powers of γ2 for Qzz, see (12.1.6).
n (−1)n+1Q
(n)
zz
0 0
1 5/2
2 615/32
3 2564987/11520
4 19550772149/5529600
5 16195465214963/221184000
6 1664517172621593361/867041280000
7 3895122983853951241069/62426972160000
8 12095392486876132744589230813/4894274617344000000
9 4348730023196338444920601655427467/37000716107120640000000
10 646955302702442553065579733086802864719/97681890522798489600000000
Appendix G
Computational Code: Hydrogen Atom
in Magnetic Field
In this Appendix, the computational Hinb.f which performs the variational method using
(12.0.3) as the trial function is explicitly shown. It uses the integration routine D01FCF from
NAG-LIB as well as the minimization routine MINUIT of CERN-LIB. The input is data (such
as magnetic field, domain of integrations, etc.) is written in data sheet as shown below.
Magnetic field (in units of B_0): H (change of vars in rho)
2.35D0
Number of regions in rho (nrho) nrho+1 limits (NUM / DEN)
2 2
Limits for rho NUM (according to Number of Regions)
0.0 1.24 7.86
Limits for rho DEN
0.0 1.24 7.86
Number of regions in z (nz) nz+1 limits (NUM / DEN)
3 3
Limits for z (NUM) (according to Number of Regions)
0.0D0 0.78 2.85 15.4
Limits for z (DEN)
0.0D0 0.78 2.85 15.4
Default Acc. NUM/DEN , Factor , Flag for corr., acc. for corr , screen =1/ file=0
1.D-10 1.D-10 1.0D0 1 1.0D-02 1
SET TITLE
’Variational Energy for H in a magnetic field ’
PARAMETERS
1 ’a0=’ 2.2425240 0.1 2.0 3.99
2 ’a1=’ 1.0104802 0.1 0.9 1.2
3 ’a2=’ -0.2722381 0.01 -0.43 0.51
4 ’a3=’ 0.0411529 0.01 0.0 0.4
5 ’a4=’ 0.0545618 0.01 -0.1 0.2
6 ’a5=’ 0.0369813 0.01 0.0 0.057
7 ’b1=’ 0.0268046 0.01 0.0 0.03
8 ’b2=’ 0.0466748 0.01 0.0 0.1
9 ’b3=’ 0.0000000 0.01 0.00 0.06
10 ’b4=’ 0.0223345 0.01 0.0 0.8
11 ’c =’ 0.0833333 0.00 0.0 0.24
12 ’d =’ 2.0000000 0.0 0.9 2.8
set strategy 2
minimize
minimize 2000
stop
The main body of the program is presented in explicit form in the following pages.
137
138 Appendix G. Computational Code: Hydrogen Atom in Magnetic Field
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Appendix H
Lagrange-Mesh Method for Hydrogen
Atom in Magnetic Field: Code
Below it is presented in explicit form the Mathematica Notebook which implements the La-
grange Mesh Method for the hydrogen atom in presence of a uniform and constant magnetic
field. The program is optimized to run it directly from the terminal/console.
(*This program caculates the energy , wave function , \
quadrupo lemoment and cusp parameter of the first \
bound states of the Hydrogen atom in magnetic field *)
(* Load module: module load Mathematica /11.2 *)
(* Usage: math -noprompt -run < file.txt *)
(* Author: Juan Carlos del Valle Nov 19 2019( revisited )*)
(* ******************************************** *)
(* Initial Data Here*)
(* ******************************************** *)
(*----------------------------------------------------*)
(* Magnetic Quantum Number *)
m=0;
(*----------------------------------------------------*)
(*Basis *)
Nr=80;
Ntheta =200;
(*----------------------------------------------------*)
(* Scaling Parameter *)
h=0.005;
(*----------------------------------------------------*)
(* Potential *)
V[r_ ,u_]:= - 2/r + 1/4 gamma^2 (1-u^2)r^2
(*----------------------------------------------------*)
(* Parity [+1 ,-1]*)
p=1;
(*----------------------------------------------------*)
(* Magnetic Field *)
gamma =4000;
(*----------------------------------------------------*)
(* Tolerance in Float Numbers *)
dx =0.000000000000000001;
(*----------------------------------------------------*)
(* ******************************************** *)
(* MAIN PROGRAM *)
(* ******************************************** *)
(* Matrix Elements *)
sols1=NSolve[LaguerreL[Nr,x]==0,x,WorkingPrecision - >50];
k=0;
sols1 /.{ r__Rule}:>Set@@@ ({r}/.var:x->Subscript[var ,++k]);
Clear[k]
(* R a c e s de los polimonios de Legendre *)
sols2=NSolve[LegendreP[Ntheta+Abs[m],Abs[m],u]==0&&\
0<=u<1,u,WorkingPrecision - >50];
k=0;
sols2 /.{ r__Rule}:>Set@@@ ({r}/.var:u->Subscript[var ,++k]);
Do[Subscript[u, -i]=- Subscript[u, i],{i,1,Ntheta /2}]
(* Radial Kinetic Elements *)
Do[For[i=1,i<j,i++,t[i,j]=( -1)^(i-j) (Subscript[x, i]\
+Subscript[x, j])/(( Subscript[x, i] Subscript[x, j])^(1/2)\
(Subscript[x, i]-Subscript[x, j])^2);t[j,i]=t[i,j]],{j,1,Nr}];
Do[t[i,i]=(4+(4 Nr+2) Subscript[x, i]-Subscript[x, i]^2)\
/(12 Subscript[x, i]^2) ,{i,1,Nr}];
Clear[i]
(* Angular Kinetic Elements *)
Do[For[i=1,i<j,i++,s[i,j]=( -1)^(i-j) (2(1- Subscript[u, i]^2)\
^(1/2)(1 - Subscript[u, j]^2)^(1/2))/( Subscript[u, i]-\
Subscript[u, j])^2;s[j,i]=s[i,j]],{j,1,Ntheta /2}];
Do[s[i,-j]= -( -1)^(i+j) (2(1- Subscript[u, i]^2)^(1/2)\
(1- Subscript[u, -j]^2)^(1/2))/( Subscript[u, i]-\
Subscript[u, -j])^2 ,{i,1,Ntheta /2},{j,1,Ntheta /2}];
Do[s[i,i]=1/3 (Ntheta+Abs[m])( Ntheta+Abs[m]+1)+(2(m^2 -1))\
/(3(1- Subscript[u, i]^2)) ,{i,1,Ntheta /2}];
143
144 Appendix H. Lagrange-Mesh Method for Hydrogen Atom in Magnetic Field: Code
Clear[i]
(* BLOCK FOR HAMILTONIAN *)
f[n_ ,k_]:=(k-1)Nr+n
(* Kinetic Elements *)
T[a_ ,b_,c_,d_]:=2 1/2 (t[a,c]KroneckerDelta[b,d]+1/\
Subscript[x, a]^2 (s[b,d]+ ps[b,-d]) KroneckerDelta[a,c])
Do[Subscript[TT, f[a,b],f[c,d]]=T[a,b,c,d],{a,1,Nr},\
{b,1,Ntheta /2},{c,1,Nr},{d,1,Ntheta /2}]
(* Potential Elements *)
V[a_ ,b_,c_,d_]:=V[h Subscript[x, a],Subscript[u, b]]\
KroneckerDelta[a,c]KroneckerDelta[b,d]
Do[Subscript[VV, f[a,b],f[c,d]]=V[a,b,c,d],{a,1,Nr},\
{b,1,Ntheta /2},{c,1,Nr},{d,1,Ntheta /2}]
(* Building the Hamiltonian and Diagonalization *)
Ham=Table[Table[ 1/h^2 Subscript[TT, i,j]+ Subscript\
[VV , i,j],{j,1,Nr Ntheta /2}] ,{i,1,Nr Ntheta /2}];
epsilon =-1+ Eigenvalues[IdentityMatrix [( Ntheta
Nr)/2]+Ham ,-1,Method ->{"Arnoldi","Criteria"\
->"RealPart",MaxIterations - >10000000}][[1]];
(* Coefficients for EigenFunctions *)
CF=Eigenvectors[Ham+IdentityMatrix [( Ntheta Nr)/2],-1,\
Method ->{"Arnoldi","Criteria"->"RealPart",MaxIterations\
- >10000000}];
Do[Subscript[c, (1-r)*Nr+s,r]= Rationalize[CF [[1]][[s]],dx]\
, {r,1,Ntheta /2}, {s,Nr*(r-1)+1,r*Nr}];
(* Quadrupole Moment *)
z2=h*h*2* Sum[Subscript[c, i, j]^2* Subscript[x, i]^2*\
Subscript[u, j]^2,{i,1,Nr},{j, 1,Ntheta /2}];
r2=h*h*Sum[Subscript[c, i, j]^2* Subscript[x, i]^2*\
(1- Subscript[u, j]^2), {i, 1,Nr},{j, 1,Ntheta /2}];
Q=z2-r2;
Export["B" <> ToString[N[gamma ]] <> "_cf.dat", CF [[1]]];
(*Cusp Parameter *)
f[r_, i_] := ((-1)^i*( LaguerreL[Nr, r]/(r - Subscript\
[x, i]))* Exp[-(r/2)])/ Subscript[x, i]^2^( -1)
df[y_, i_] := D[(( -1)^i*( LaguerreL[Nr, r]/(r - \
Subscript[x, i]))* Exp[-(r/2)])/ Subscript[x, i]^2^(-1) , r]\
/. r -> y
CuspN=Sum[f[0, i]*df[0, n]* Subscript[c, i, j]*\
Subscript[c, n, j],{i,1,Nr}, {j,1,Ntheta /2}, {n,1,Nr}];
CuspD=Sum[f[0, i]*f[0, n]* Subscript[c, i, j]* Subscript\
[c, n, j], {i,1,Nr}, {j,1,Ntheta /2}, {n,1,Nr}];
Cusp= CuspN/(h*CuspD);
(* OUTPUT *)
fileName = "B"<>ToString[N[gamma]]<>".dat";
file = OpenWrite[fileName , PageWidth -> Infinity ];
WriteString[file ," *******************************\n "];
WriteString[file ,"Hydrogen Atom in Magnetic Field \n "];
WriteString[file ,"    in Lagrange Mesh Method     \n "];
WriteString[file ,"********************************\n "];
WriteString[file ,"                                \n "];
WriteString[file ,"  Q.  number m = "<>ToString[m]"\n" ];
WriteString[file ,"                                \n "];
WriteString[file ,"................................\n "];
WriteString[file ,<>ToString[Nr]<>"x"<>ToString[Ntheta]"\n"];
WriteString[file ,"    Scaling " <>ToString[N[h]]<>"\n"];
WriteString[file ,"................................\n "];
WriteString[file ,"                                \n "];
WriteString[file ,"Energy = "<>ToString[epsilon]<>"\n\"];
WriteString[file ,"                                \n "];
WriteString[file ,"QM="<>ToString[NumberForm[Q,16]]<>"\\ n"];
WriteString[file ,"                                \n "];
WriteString[file ,"  Cusp Par. = "<>ToString[Cusp]"\n "];
WriteString[file ,"................................\n "];
Close[file]
Appendix I
Plots of Variational Parameters:
Hydrogen Atom in Magnetic Field
Plots of the optimal variational parameters {α0,1,2,3,4, β1,2,3} for Approximant (12.0.3) as func-
tions of log(1 + γ2) are shown below.
0 3 6 9 12 15 18
2
3
4
5
6
0 3 6 9 12 15 18
1
2
3
4
0 3 6 9 12 15 18
2
4
6
8
10
12
0 3 6 9 12 15 18
0.05
0.10
0.15
0.20
0.25
Figure I.1: Optimal variational parameters {α0, α1, α2, α3} of the Approximant (12.0.3) as
functions of log(1 + γ2).
145
146 Appendix I. Plots of Variational Parameters: Hydrogen Atom in Magnetic Field
0 3 6 9 12 15 18
0.2
0.4
0.6
0.8
1.0
1.2
0 3 6 9 12 15 18
2
4
6
8
0 3 6 9 12 15 18
4
8
12
16
20
0 3 6 9 12 15 18
0.5
1.0
1.5
2.0
Figure I.2: Optimal variational parameters {α4, β1, β2, β3} of the Approximant (12.0.3) as func-
tions of log(1 + γ2).
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