The normalized matrix perturbation method

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©INAOE 2020

The author hereby grants to INAOE permission to reproduce and to distribute copies of this thesis document in whole or in part, fountain mention.

The Normalized Matrix

Perturbation Method

by

Braulio Misael Villegas Martínez

A dissertation

submitted to the Program in Optics,

Optics Department, in partial fulfillment of

the requirements for the

PhD Degree in Optics

at the

National Institute for Astrophysics, Optics and

Electronics

January 2020

Tonantzintla, Puebla

Advisors:

Dr. Héctor Manuel Moya Cessa,

Dr. Francisco Soto Eguibar

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Abstract

In 1926, Schr¨odinger supplied the theoretical basis for analyzing the time evolution of a quantum system through his famous equation that now bears his name. Soon after its advent, an intrinsic interest emerged to get exactly solvable models by using the Schr¨odinger equation. However, only a very small number of problems fulfil this specific requirement. The vast majority of physical systems are rather complicated to be treated exactly; one may try to find approximate solutions with the aid of perturbation methods

Among all available perturbative recipes, the Matrix Method inmediately stands out. Such as its name suggests, this scheme is devoted to seek perturbative solutions of time-dependent Schr¨odinger equation, ordered in power series of tridiagonal matri-ces. These solutions possess time-dependent factors which enable us to determine the temporal evolution of corrections. Surprisingly in this scheme, the existence of normalized solutions are completely ignored and neglected.

This thesis present the complement theoretical analysis of Matrix Method started in reference [18]. In this case, we focus in determine the general expression for the normalization constant, which in principle will allow us to obtain perturbative nor-malized solutions to any order correction. Henceforth, we shall refer to the complete approach as the Normalized Matrix perturbation Method. Further, we show that the treatment adopted to get the normalization constant is quite different from the usual intermediate normalization procedure used in the standard time-independent perturbation theory. In order to assess the efficacy and advantages of our approach, five examples, were analyzed leading to meaningful results in their approximative so-lutions and in fairly good agreement with their known exact or numerical soso-lutions.

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Resumen

En 1926, Schrödinger proporciona las bases teóricas para analizar la evolución tempo-ral de un sistema cuántico a través de su famosa ecuación que ahora lleva su nombre. Poco después de su advenimiento, surge un interés intr´ınseco para obtener modelos exactamente solubles mediante la ecuación de Schrö ecuación dinger. Sin embargo, solo un número muy pequeño de problemas relacionados con esta ecuación cumplen con este requisito. Debido que la mayor´ıa de los sistemas f´ısicos son bastante com-plicados de tratar de forma exacta, uno debe encontrar soluciones aproximadas con la ayuda de los métodos de perturbación.

Entre todas las recetas perturbativas disponibles, una que destaca de inmediato es el Método Matrix. Tal como sugiere su nombre, este esquema está dedicado a buscar soluciones perturbativas de la ecuación de Schrödinger dependiente del tiempo, orde-nadas en series de potencia de matrices tridiagonales. Estas soluciones, por supuesto, poseen factores dependientes del tiempo que nos permiten determinar la evolución temporal de las correcciones. Sorprendentemente, en este esquema, la existencia de soluciones normalizadas son completamente ignoradas y descuidadas.

En esta tesis, presentamos el análisis teórico complementario del Método Matrix ini-ciado en [18]. En este caso, nos enfocamos en determinar la expresión general para la constante de normalización, que en principio nos permitirá obtener soluciones perturbadas normalizadas para cualquier orden de corrección. En lo sucesivo, nos referiremos al enfoque completo como el Método Matricial de Perturbación Normal-izada. Además, mostramos que el tratamiento adoptado para obtener la constante de normalización es muy diferente al procedimiento de normalización intermedia uti-lizado en la teor´ıa de perturbaciónes independiente del tiempo. Con el fin de evaluar la eficacia y las ventajas de nuestro enfoque, se analizaron cinco ejemplos que con-dujeron a resultados significativos en sus soluciones aproximadas y en muy buen acuerdo con sus soluciones exactas o numericas conocidas.

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Acknowledgements

I dedicate this dissertation to my mother Balbina. Who made uncountable sacrifices so I would have access to an affordable quality of life and education since my early years. I thank her also for instilling a strong sense of personal responsibility, gen-erosity to help those less fortunate and the taste for hard work and good education. But, above all, for her always present patience and moral support at the various stages of my life, in particular for the encouragement and inspiration to pursue my dreams. You will always be my hero, thanks for trusting and never giving up on me.

I would like to express my heartfelt gratitude to my advisors Dr. H´ector Manuel Moya Cessa and Dr. Francisco Soto Eguibar for the time, support, encouragement and continuous guidance that they gave me during these past six years. I always will forever be grateful to my research advisor, Dr. Francisco Soto Eguibar. His insightful discussions, suggestions and constant feedback about the research was of very guidance help to pave a way to finished this thesis.

I also express my gratitude to the National Council on Science and Technology (CONACYT), for its financial support throughout my PhD studies.

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Contents

List of Figures

4.1 Probability density with quantum numbern = 10. The left-hand side (a) shows the exact solution while the right-hand side (b) presents the approximate one. A contour plot of their probability densities is depicted on (c), where it shows that for short times the approximate solution (red line) reproduces the exact one (black line) with highly accuracy whereas for t > 1.5 a very slight differences between them can be detected, but, it still yields to good approximation. These graphs were obtained by considering a perturbation strength equal to

λ= 0.1 and a frequency of ω = 1. . . 30

4.2 Probability density |ψ(x, t)|2 with α(0) = 3 andω= 2. The left-hand side (a) shows the exact solution while the right-hand side (b) presents the approximate one. A contour plot of their probability densities on (x, t) plane is depicted on (c), where it shows that the approximation presents a remarkable high accuracy with the exact result under the consideration of a perturbation strength equal to λ= 0.5. . . 31

4.3 Probability density with quantum number n = 20 at (a) t = 0.5 and (b) t = 1.5. The approximation given by PApro is highly accurate and compares favorably with the exact result, PExa, these solutions are get in the strong-coupling perturbation regime with a perturbed coefficient value of λ= 35. . . 33

4.4 Probability density of the cubic anharmonic oscillator versus x for a quantum number n = 20. One can notice from (a) that with

λ = 0.005, the first-order solution, PApro, does not differs substan-tially from numerical result,PN u, unlike toPRS whose behavior shows several discrepance. Once increases the perturbation parameter to (b)

λ= 0.01, PApro as wellPRS become completely inadequate to describe the numerical solution, being the differences more notorious for PRS. The numerical result was performed with t= 1 and ω= 1. . . 37

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List of Figures

4.5 Density probability of cubic anharmonic oscillator for a coherent state (a) and (b) a Schr¨odinger cat state. Notice in (a) that PApro gives a solution very close to PN u while PRS remains to far of reproduce it. This become more evident for the interference pattern generated by the Schr¨odinger cat state in (b), where PRS fails completely when it compared with PN u; on the contrary, PApro yields to reliable result with PN u. The above graphs were carried by using the parameters

λ= 0.005, ω= 1, α = 2 + 3i and θ =π/7. . . 41

4.6 Probability density of the repulsive harmonic oscillator with (a) a coherent state and (b) a Schr¨odinger cat state as initial states. In both cases the solutions present a parabolic behaviour; the values of the parameters are β = 6, ω = 1 and φ = π/5. Figures (c) and (d) depict the probability density distribution on the (x, t) plane for the same initial states. The solid, dotted and dashed lines represent the cases of β=2, 5 and 9, respectively. . . 48

4.7 Probability density of the linear anharmonic repulsive oscillator in the (x, t) plane when the initial state is a coherent state. The black line and the red dashed line indicate the exact and perturbative solutions. Graphs (a) and (b) show how the solutions behave for β = 1 at two different values of the perturbative parameter λ. For λ = 0.1, the second-order perturbative solution presents a remarkable high accu-racy with the exact result up to t = 3. After that time, a small but significantly discrepancy between them appear. For λ = 0.4, the ac-curacy of perturbative result is reduced over shorter time. Graphs (c) and (d) display the same time-range of convergence between the solutions for β = 8 and ω= 1. . . 56

4.8 Density probability of the linear anharmonic repulsive oscillator in the (x, t) plane for an Schr¨odinger-cat state as initial state withω = 1 and

φ=π/7. The black line and the red dashed line indicate the exact and perturbative solutions. Graphs (a) and (b) show how the solutions behave for β = 1 at two different values of perturbative parameter

λ. For λ = 0.1, the second-order perturbative solution presents a remarkable high accuracy with the exact result. Once increases the perturbation parameter toλ= 0.4, the accuracy of perturbative result is completely reduced. The two remaining graphs (c) and (d) display same features where a small but significantly discrepancy between solutions appear. . . 57

4.9 Field intensity versus propagation distance z using the exact solution (solid line), the third-order solution (red dashed line) and the small rotation method solution (blue dashed line), withα= 0.1 andω= 0.9, for the first three guides. . . 64

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List of Figures

4.10 Field intensity versus propagation distance z using the exact solution (solid line), third-order solution (red dashed line) and the small rota-tion method solurota-tion (blue dashed line), with α = 0.3 and ω = 0.9, for the first three guides. . . 65

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Chapter 1 Introduction

The main equation in non-relativistic quantum mechanics is the Schrödinger equa-tion, because its soluequa-tion, the wavefuncequa-tion, contains all the relevant information about the behavior of a physical quantum system [1–6]. Since its introduction the Schrödinger equation has been widely studied over several decades going to present day one of the most important cornerstones of modern physics, but despite all ef-forts that have been made in this equation, the successful physical systems where it has an exact solution are limited; the infinite well, the harmonic oscillator, the hydrogen atom [2–6] and the Morse potential [7] are typical set of potentials where an exact analytical solution is known. A considerable number of problems involved with the Schrödinger equation are often complex and cannot be solved exactly, then one is thus forced to analyse them by the use of perturbative methods [3–6, 8, 9] or numerical techniques [10–13]; when the approximation methods are correctly used, give us a very good understanding behavior of the phenomenon under study.

Time independent perturbation theory, also known as Rayleigh-Schr¨odinger per-turbation theory, has its roots in the works of Rayleigh and Schr¨odinger, but the mathematical foundations were only set by Rellich in the late thirties of the past century (see Simon [14] and references there in). This method has been applied with great success to solve a vast variety of problems such that, through its continuous implementation, a lot of techniques have been developed, which go from numerical methods [15] to those more mathematical and fundamental, as convergence prob-lems [16, 17].

In this thesis an alternative perturbative approach, called the Matrix Method [18–21], is studied and complemented. This new scheme, based on the implementation of triangular matrices, allows to solve approximately the time-dependent Schr¨odinger equation in an elegant and simple manner. The method has demonstrated that the corrections to the wavefunction and the energy can be contained in only one expres-sion, unlike the standard perturbation theory where it is needed to calculate them in separated ways [18–21]. Moreover, the Matrix Method may also be used when

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Chapter 1. Introduction

one may find a unitary evolution operator for the unperturbed Hamiltonian but it is not possible to find its eigenstates. Further, its approximated solutions not only present conventional stationary terms, but also time dependent factors which allows us to know the temporal evolution of the corrections. A remarkable feature is that the general expression to compute them does not distinguish if the Hamiltonian is degenerate or not [18–21]. Besides, the formalism offers an alternative to express the Dyson series in a matrix form. Another very important feature is that the Matrix Method can be extended to the Lindblad master equation [20]. Herein, the Matrix Method possess many attractive characteristics that cannot be found in the conven-tional treatments of the perturbation theory. It is also worth noting that our main goal over the Matrix Method is to extend and provide normalized approximated so-lutions in each step, at difference with the standard perturbation theory, where the solutions are not normalized.

The reminder of this thesis is organized as follows:

Chapter 2 provides a brief overview relative to the theoretical framework of the time-independent and time-dependent perturbation theories and emphasizes the ap-propriate procedure to get the perturbative order corrections in both cases. As is well known for the time-independent formalism, two expressions are obtained at each order, one for the energies and one for the wave functions. Meanwhile, the time-dependent perturbation method gives rise to a recursive formula to compute higher-order corrections. Immediately after these procedure are presented we jumps subsequently to the mathematical background of the Matrix method. Such approach is presented in Chapter 3 and will allow us to obtain a single solutions that contains both, energy and wave function corrections at same time. These correction to the solution are calculated recursively permit generate the Dyson series, and in this way obtain a new expression of it, but now in terms of a matrix series. Afterwards, we introduce the mathematical derivation to obtain the general expression to com-pute the normalization constant to any given order correction. Such development is predicated on the isolation of a multiplicative time-normalized factor in the general perturbative solution, ensuring that the perturbed solutions of the wavefunction are properly normalized for all t. Five models, which of majority an exact solution ex-ists: harmonic oscillator perturbed by linear potential, cubic anharmonic oscillator, repulsive harmonic oscillator, repulsive linear anharmonic oscillator and a binary waveguide array, are discussed in detail with the Normalized Perturbative Method in Chapter 4; there the main goal is to show the potential that our approach offers versus others like than the standard time independent perturbation theory. At the same time, we prove that the perturbation analysis used here works rather well for weak perturbations but also for its strong counterpart. Finally, chapter 5 is dedicated to the conclusions.

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Chapter 2 Perturbation Methods

2.1 The Rayleigh-Schr¨

odinger perturbation

the-ory

It is well known that a great majority of systems in nature cannot be solved exactly and we force to develop approximation analytic methods to deal with them. The chief among these approaches is the Rayleigh-Schr¨odinger perturbation theory which is appropriate when we have a time independent Hamiltonian, that may be broken down in two pieces, as follows

ˆ

H = ˆH0+λHˆp, (2.1)

where ˆH0 is called non-perturbed Hamiltonian, and it is usually assumed that incor-porates the dominant effects since possess known solutions, i.e. its eigenvalues and eigenfunctions

ˆ

H0|ψ(0)i=En(0)|ψ

(0)_i_, _(2.2)

are known. By other hand, the second part of (2.1) contains a term ˆHp which is not amenable to an analytical solution. However, this term can be modelled as

small disturbance to bring us perturbative solutions of the energy spectrum and in

the eigenfunctions of the full Hamiltonian ˆH, which in principle, it does not dif-fers significantly from those of the soluble part, H0. To keep track of the “size” of terms that appear in the development of this method, it is usual to associate the “perturbation”, ˆHp, with a dimensionless strength parameterλ, which is considered very small compared to unity, this mean, λ1. This perturbative parameter, as its name suggests, typically represents, for instance, the coupling-strength of interaction between the part which is soluble and the other which is not or even is introduced as an auxiliary expansion parameter to measure “the size” of corrections. In general,

λ represents any parameter, on which the full-system Hamiltonian ˆH depends. By translating the above into a mathematical language, it is well-known that the stan-dard perturbation theory [3–6, 9] produces the following perturbed expressions for

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Chapter 2. Perturbation Methods

the eigenvalues

En=En(0)+λE (1) n +λ

2_E(2)

n +· · ·, (2.3) and for the eigenfunctions

|ψni=|ψ(0)n i+λ|ψ (1) n i+λ

2_|_ψ(2)

n i+· · · , (2.4) where the super index (j) indicates the correction order, and so En(0) and |ψ(0)i are the eigenvalues and eigenfunctions of the unperturbed Hamiltonian ˆH0. The main problem of perturbation theory is to solve the eigenvalue equation given by

ˆ

H|ψni=

ˆ

H0+λHˆp

|ψni=En|ψni, (2.5) if we substitute the series expansions ofEnand|ψniinto Eq.(2.5), and the coefficients of like powers to λ on each side of equation are set equal to each other, we arrive at the following set of equations

ˆ

H0 ψ(0)_n

=E_n(0)ψ_n(0)

ˆ

H0 ψ_n(1)

+ ˆHp

ψ(0)_n

=E_n(0)ψ_n(1)

+E_n(1)ψ_n(0)

ˆ

H0 ψ_n(2)

+ ˆHp

ψ(1)_n

=E_n(0)ψ_n(2)

+E_n(1)ψ_n(1)

+E_n(2)ψ_n(0)

..

. (2.6)

Multiplying the above equations by Dψn(0)

and using the fact that eigenfunction of

unperturbed Hamiltonian operates over its left as Dψn(0)

ˆ

H0 = D

ψn(0)

E

(0)

n , due to the hermiticity property and from the normalization condition of the unperturbed eigenfunctions, Dψn(0)

ψ (0) n E

= 1, it follows that

E_n(0)=ψ_n(0)Hˆ₀ ψ(0)_n

E_n(1)=ψ_n(0)Hˆp

ψ(0)_n

E_n(2)=ψ_n(0)Hˆ_p ψ(1)_n

..

. (2.7)

It is clear to see that the first-order correction to the energy,En(1), is simply the expec-tation value of the perturbation with respect to the unperturbed state. Meanwhile, the second-order of energy requires know the first-order of wavefunction,

ψ (1) n E . So to proceed of its derivation, we use the identity operator in terms of the complete orthonormal set eigenfunctions of the unperturbed Hamiltonian as

ψ_n(1)

= ˆIψ(1)_n =X k ψ (0) k E D

ψ(0)_k

ψ (1) n E , (2.8)

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2.1. The Rayleigh-Schr¨odinger perturbation theory

where the inner product Dψ(0)_k

ψ (1) n E

can be obtained by multiplying both sides of

the second expression of Eq.(2.6) by Dψ_k(0)

to give

D

ψ_k(0) ψ (1) n E = D

ψ(0)_k

ˆ Hp ψ (0) n E

En(0)−E_k(0)

. (2.9)

Prior to the substitution of previous expression into Eq.(2.8), it is helpful to clar-ify a point which is basic to provide further understanding about the normaliza-tion procedure between the Rayleigh-Schr¨odinger perturbation theory and from the Matrix method, which it will be explained in next chapter. Reader can see into Eq.(2.9) that the denominator contains an average energy difference which is safe when pulled out the n =k term from the beginning; clearly, if En(0) =E_k(0) Eq.(2.9) diverges. However, this condition is not necessarily true in (2.8) where then-th term

in the sum can arise. To deal with this restriction, we must assume that

ψ (1) n E does

not include to ψ

(0) n

E

through a normalization procedure which typically requires that the overlap between the unperturbed and perturbed wavefunctions is taken as

D

ψn(0)

ψn

E

= 1. This procedure is termed as intermediate normalization and ensuring the unperturbed eigenfunction being orthogonal to all of the corrections, that mean,

D

ψn(0)

ψ

(m) n

E

= 0 for m≥1, instead of the conventional normalization hψn|ψni= 1. Upon substitution of (2.9) into (2.8), we have the required result

|ψ_n(1)i=X k6=n

Hpkn

En(0)−E_k(0)

ψ (0) k E , (2.10)

which correspond to 1st order correction of wavefunction and

E_n(2) =X k6=n

|Hpkn|

2

En(0)−E_k(0)

, (2.11)

to the 2nd order correction of energy, where we have defined

Hpkn =

D

ψ_k(0)

ˆ Hp ψ_n(0)

. (2.12)

The process for wavefunction is repeated until get 2nd order correction which is given by

|ψ(2)_n i=X k6=n

X

m6=n

HpkmHpmn

(En(0)−E_k(0))(En(0)−Em(0))

ψ (0) k E −X

k6=n

HpnnHpkn

(En(0)−E_k(0))2

ψ (0) k E . (2.13) It is worth noting that the previous expressions can be used only in the non-degenerate case, where we always have En(0) 6= E_k(0). In the degenerate case, a different and more complicated treatment is needed [3–6, 9].

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2.2 Time-dependent perturbation theory

In the previous section we have focused on the mathematical formalism to treat with physical systems whose total Hamiltonian ˆH is time-independent. However, the rules changes completely whether one deals with a Hamiltonian that depends on time, in particular when the temporal dependence resides into the perturbation. In such cases, one must therefore resort to the time-dependent perturbation theory. Indeed, this theoretical framework initiated and developed by Paul Dirac [22], it is a powerful tool to study the interaction and time evolution in quantum systems which cannot be solved exactly. The reader can consult the references [6, 9, 22, 23] which provides more details about this subject. In order to derive the works equations concerning to the dependent perturbation theory, we must begin by introducing the time-dependent Schr¨odinger equation (in all this work we will set_~= 1)

id

dt|ψ(t)i=

h

ˆ

H0 +λHˆp(t)

i

|ψ(t)i. (2.14)

The main goal is to find the solution of system state|ψ(t)i, in principle, this can be done in a very simple way by computing the perturbed propagator which it is useful to apply any initial state; once it is obtained, a perturbation expansion itself is carry out in terms of the small parameter λ. To do so, one must choose a proper frame to describes the dynamics of system (2.14). The interaction picture is suitable when the Hamiltonian of an system can be written as a sum of two parts, ˆH = ˆH0+ ˆV(t), where the first half is related with the time-independent term whose eigenvalues and eigenfunctions are known, whereas the second half is usually associated with a time-dependent potential term, in our case ˆV(t)→λHˆp(t) [24, 25]. Thus, the state vector of this picture is given by

|ψ_Int(t)i= ˆU₀†(t)|ψ(t)i, (2.15) the above is related to the Schr¨odinger picture, where the state vectors evolve in time but the operators not, through the inverse unitary transformation

ˆ

U0(t) = exp

−iHˆ0t

, (2.16)

which provides the evolution of system in the absence of perturbation. From Eq.(2.15), it is worth notice that if|ψ_Int(t)iis calculated, then one could readily determine|ψ(t)i

by inverting Eq.(2.15) to find

|ψ(t)i= exp−iHˆ0t

|ψ_Int(t)i. (2.17)

Inserting this into Eq.(2.14) we arrive to a Schr¨odinger equation where the depen-dence on ˆH0 is removed and only includes the effects of perturbation as follows

id

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2.2. Time-dependent perturbation theory

where we define

ˆ

H_Int(t) = expiHˆ0t

ˆ

Hp(t) exp

−iHˆ0t

(2.19)

Notice that Eq.(2.18) contains a mixture of the Heisenberg picture [24,25] version for ˆ

H_Int(t) where the operators change with time and the Schr¨odinger picture to deals with the time state vector|ψ_Int(t)ithrough a Schr¨odinger equation. The interaction picture is the main responsible of above features, since it is a hybrid representation of both frames.

Similar to the time-independent perturbation theory, we perform appropriate per-turbative expansion of |ψ_Int(t)i as a power series in λ

|ψ_Int(t)i=ψ(0)

Int(t)

+λψ(1)

Int(t)

+λ2ψ(2)

Int(t)

+. . . (2.20)

Plugging (2.20) into (2.18) and then equate separately term by term in powers of λ, we get

id dt

ψ(0)

Int(t)

=0,

id dt

ψ(1)

Int(t)

= ˆHInt(t)

ψ(0)

Int(t)

,

id dt

ψ(2)

Int(t)

= ˆH_Int(t)ψ(1)

Int(t)

,

..

. (2.21)

From (2.17) we can appreciate that both state vector in interaction picture and original state are equal when t = 0, i.e, |ψ_Int(0)i = |ψ(0)i. Then a expansion of (2.20) evaluated at t= 0 give us

|ψ_Int(0)i=|ψ(0)i=ψ(0)

Int(0)

+λψ(1)

Int(0)

+λ2ψ(2)

Int(0)

+. . . (2.22)

and if we equate separately the left and right-hand sides of this expression in powers of λ, it follows that

ψ(0)

Int(0)

=|ψ(0)i,

ψ(m)

Int (0)

=0, for m= 1,2,3.. (2.23)

The later is reasonable and it can be noticed in (2.21) where the zero order term is time independent. So, it is reasonable deduce that the solution of zero-order term is given by

ψ(0)

Int(t)

=|ψ(0)i, (2.24)

combining this result with the second expression of (2.21) gives us

id dt

ψ(1)

Int(t)

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whose formal solution is

ψ(1)

Int(t)

=−i

Z t 0

ˆ

H_Int(t1)dt1|ψ(0)i, (2.26)

which correspond to solution of the first-order term. Here, we have implemented correctly the initial condition ψ(1)

Int(0)

= 0 by setting the lower limit of integration equal to zero. Now, if we replace previous result for the next equation of order λ2_, we find

ψ(2)

Int(t)

=−i

Z t 0

ˆ

HInt(t1)dt1 ψ(1)

Int(t1)

=−

Z t 0

ˆ

H_Int(t1)dt1 Z t1

0 ˆ

H_Int(t2)dt2|ψ(0)i, (2.27)

the solution of second-order term gives rise to an iterated integral expression in chronology order. At this point the pattern is clear, and we can write the final result for |ψ(t)i as

|ψ(t)i=e−iHˆ0t h

1−iλ

Z t 0

dt1HˆInt(t1) + (−iλ)

2 Z t

0

dt1 Z t1

0

dt2HˆInt(t1) ˆHInt(t2) +. . .

+ (−iλ)n

Z t 0

dt1 Z t1

0

dt2· · · Z tn−1

0

dtnHˆInt(t1) ˆHInt(t2). . .HˆInt(tn) +. . .

i

|ψ(0)i,

(2.28)

this expansion of multiple integrations ordered in time is known as the Dyson series [26, 27].

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Chapter 3 The Normalized Matrix

Perturbation Method

3.1 The Matrix Method

In the same spirit as the time dependent perturbation theory, it is possible to develop a formulation to find a perturbative solution of (2.14), the main difference being that now the perturbation, ˆHp, does not depend on time. Strictly speaking, the formal solution of the time-dependent Schr¨odinger equation |ψ(t)i=e−iHtˆ _|_ψ₍₀₎_i_{, with the} complete Hamiltonian ˆH= ˆH0+λHˆp, can be expanded in a Taylor series and sorted in powers of λ; for example, an expansion for the propagator of system truncated up to first order term is

|ψ(t)i=

"

e−iHˆ0t₊_λ

∞

X

n=1

(−it)n

n! n−1 X

k=0 ˆ

H₀n−1−kHˆpHˆ0 k

#

|ψ(0)i. (3.1) The basic idea behind the Matrix Method [18–21] is obtain an analytic solution of previous equation through the following triangular matrix

M =

_ˆ

H0 Hˆp 0 Hˆ0

, (3.2)

whose diagonal entries are conformed by the unperturbed Hamiltonian and the upper triangle by the perturbation. One can find that if we multiply the matrix M by itself n-times, its upper element will contain exactly the same products of ˆH0 and

ˆ

Hp defined within summation in Eq.(3.1). In simple words, the matrix element M1,2 will give us the first order correction; based on this consideration, Eq.(3.1) is then transformed to

|ψ(t)i=he−iHˆ0t₊_λ₍_e−iM t₎ 1,2

i

|ψ(0)i, (3.3)

here the approximate solution has been split in two pieces; the first part belongs to the solution of the unperturbed system, that is well known, while the second part

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Chapter 3. The Normalized Matrix Perturbation Method

refers to the first order correction. In order to determine the solution of first order term, we have to keep in mind that the problem originally pose it must follows the same matricial convention, hence, the approximate solution (3.3) can be conveniently rewritten as

|ψ(t)i=ψ(0)

+λ ψP

1,2, (3.4)

where ψP

is a perturbed matrix defined as

ψP

=

|ψ1,1i |ψ1,2i

|ψ2,1i |ψ2,2i

. (3.5)

If we derived the equations (3.3) and (3.4) with respect to time and equate the corresponding coefficients of λ, we find

id dt

ψ(0)

= ˆH0e−i ˆ

H0t_|_ψ₍₀₎_i_,

id dt

ψP

=M e−iM t

|ψ(0)i 0 0 |ψ(0)i

, (3.6)

the first differential equation is trivial to solve and give us the zero-order solu-tion, ψ(0)

= e−iHˆ0t_|_ψ₍₀₎_i_{, whereas the integration of second one leads to} ψP

=

e−iM tI|ψ(0)i (I is the unity matrix) and which obeys the differential equation

id dt

ψP

=MψP

=

_ˆ

H0|ψ1,1i Hˆ0|ψ1,2i+ ˆHp|ψ2,2i 0 Hˆ0|ψ2,2i

, (3.7)

subject to the initial condition

ψP(0)

=

|ψ(0)i 0 0 |ψ(0)i

. (3.8)

One can notice from (3.7) that the matrix element|ψ2,1iis equal to zero, this makes sense, since M is a tridiagonal matrix and the resulting product of two upper trian-gular matrices is upper triantrian-gular too. The reader can easily check it by performing the matrices product on the right side of second equation (3.6). Further, the so-lution we are looking for is associated with the matrix element |ψ1,2i through the differential equation

id

dt|ψ1,2i= ˆH0|ψ1,2i+ ˆHpe

−iHˆ0t_|_ψ₍₀₎_i_, _(3.9)

where we have used the fact that|ψ1,1i=|ψ2,2i=e−i ˆ

H0t_|_ψ₍₀₎_i_{. Now if we make the}

transformation |φ1,2i = ei ˆ H0t|_ψ

1,2i, integrated the resulting expression and then it transform back to |ψ1,2i, we arrive to

|ψ1,2i=−ie−i ˆ H0t





t

Z

0

eiHˆ0t1_Hˆ

pe−i ˆ H0t1_dt

1 

(25)

3.1. The Matrix Method

which corresponds to first order correction. Note that above result is very similar to (2.26) and open the possibility to build a bridge between this approach and the Dirac time-dependent perturbation method described in chapter 2; for this make sense, we must of course assume at the beginning that ˆHp does not depend on time. Let us expand again the formal propagator of system, e−iHtˆ _{, in Taylor series and retaining} terms until λ2, we get

|ψ(t)i=

"

e−iHˆ0t₊_λ

∞

X

n=1

(−it)n

n! n−1 X

k=0 ˆ

H₀n−1−kHˆpHˆ0 k

#

|ψ(0)i

+λ2

∞

X

n=2

(−it)n

n! n−1 X

k=1 n−k

X

j=0 ˆ

H₀n−1−k−jHˆpHˆ0 j _ˆ

HpHˆ0 k−1

|ψ(0)i. (3.11) In analogy to the previous case, we consider now the matrix

M =





ˆ

H0 Hˆp 0 0 Hˆ0 Hˆp 0 0 Hˆ0



. (3.12)

If we multiply the matrix M by itself n-times, then one should find that all the information about the second order correction will be enclosed in the element M1,3 of our newly defined 3×3 triangular matrix, completely similar to (3.2). Therefore, the solution to second order correction can be determined by performing all of the algebraic steps outlined in the first order case to yields

|ψ(t)i=

h

e−iHˆ0t₊_λ₍_e−iM t₎

1,2+λ2(e−iM t)1,3 i

|ψ(0)i

=ψ(0)

+λ ψP

1,2+λ 2

ψP

1,3, (3.13)

where

|ψ1,3i=−e−i ˆ H0t

t

Z

0

dt1ei ˆ H0t1_Hˆ

pe−i ˆ H0t1

t1 Z

0

dt2ei ˆ H0t2_Hˆ

pe−i ˆ

H0t2_|_ψ₍₀₎_i_. _(3.14)

It becomes clear that the Matrix Method allows to transform the Taylor series of the formal solution of the time-dependent Schr¨odinger equation in a power series of the matrix M, which can be handled easily. Likewise, its iterative procedure allows us to find any kth-order correction in a simple and straightforward way through the following relation [18–21]

|ψ(t)i=

"

e−iHˆ0t₊

k

X

n=1

λn e−iM t₁_,n₊₁

#

|ψ(0)i

=ψ(0)

+ k

X

n=1

λn ψP

(26)

with the perturbed matrix defined as

ψP =   

|ψ1,1i . . . |ψ1,n+1i ..

. . .. ...

|ψn+1,1i . . . |ψn+1,n+1i 



, (3.16)

being the matrix element |ψ1,n+1i the relevant solution we are looking for, which is expressed in the form

|ψ1,n+1i= (−i) n

e−iHˆ0t

t Z 0 dt1 t1 Z 0

dt2· · · tn−1 Z

0

dtn ei ˆ H0t1_Hˆ

pe−i ˆ H0t1

×eiHˆ0t2_Hˆ

pe−i ˆ

H0t2_{. . . e}iHˆ0tn_Hˆ

pe−i ˆ

H0tn_|_ψ₍₀₎_i_. _(3.17)

This time-ordered series, restricted to the interval [0, t], is the fundamental piece to calculate the different correction terms; furthermore, we should point out that relationship (3.17) is the mathematical representation of the Dyson series [26, 27]. Albeit the Matrix method was originally conceived to deal with the effects of weak perturbations, holding in the limitλ→0. Above formalism can be readily extended to deal with its strong counterpartλ→ ∞. Let us consider a rescale of timeτ =tλin the time-dependent Schr¨odinger equation,iλ_dτd |ψi=Hˆ0+λHˆp

|ψi, whose formal solution, |ψ(τ)i= exph−iHˆp +1_λHˆ0

τi, looks very similar to the weak case; but here, it can be seen that the roles of unperturbed part and perturbation are reversed. Indeed, if we repeat the same line of steps as like the weak case, we obtain a Dyson series in matrix form for large values of λ as follows

|ψ(τ)i=

"

e−iHˆpτ₊

k

X

n=1 1

λn e

−iM t 1,n+1

#

|ψ(0)i

=ψ(0) + k X n=1 1 λn ψP

1,n+1, (3.18)

here the matrix element|ψ1,n+1i supplies all the information needed to calculate the different correction terms for the strong perturbation regime

|ψ1,n+1i= (−i)ne−i ˆ Hpτ

τ Z 0 dτ1 τ1 Z 0

dτ2· · · τn−1 Z

0

dτn ei ˆ Hpτ1_Hˆ

0e−i ˆ Hpτ1

×eiHˆpτ2_Hˆ 0e−i

ˆ

Hpτ2_{. . . e}iHˆpτn_Hˆ

0e−i ˆ

Hpτn_|_ψ₍₀₎_i_. _(3.19)

This duality on the Matrix Method gives us the possibility to analyze the solution of a quantum system in both regimes of the perturbative parameter λ. Despite one series

(27)

3.2. Normalization constant

is inverse to other, it is possible to link between them by considering the interchange ˆ

H0 ↔ Hˆp, setting λ = 1. The free choice of what part to system represents the perturbation is due to the symmetry of ˆHitself: this is duality in perturbation theory. It is important to remark that the perturbation theory presented in Chapter 2 is built up over weak coupling expansion; this regime is well known and it is characterized by the Eq.(2.28). A general scheme to prove the existence of its counterpart, a strong perturbation regime, has been attacked by several authors [28–31], but none of those analysis looked completely convincing, until the arrival of Frasca’s work based on the Navier-Stokes equation. Fraca’s perturbative formulation deals with small and large Reynolds numbers, and it has been successfully implemented in quantum mechanics [32–37]. In fact, the perturbation series (3.18) is similar to the reported by Frasca [36,37] but despite of they similarities, the matrix method gives a new way to cast the dual Dyson series in matrix form instead of usual integral representation. In addition, the approximative solutions of the Matrix method are not normalized, a normalization factor Nk allows eliminating the problems of divergence when t becomes to grow up at any kth-order corrections; being a property that marks the difference between our approach (as will see in next section) and the Frasca’s work, where Nk is intractable through him analysis.

3.2 Normalization constant

We have seen in the previous section that the approximated solutions of the Schr¨odinger equation are valid for the different regions of the perturbative parameter λ that go from a weak-to-strong coupling regime. Further, it is appropriate to mention that the expressions (3.15) and (3.18) can be written as a power series ofλ, along with the element|ψ1,n+1iof the perturbed matrix; however, both solutions are not normalized and it is convenient to get a normalization factor Nk that preserves their norm at any order. Keeping the above remark in mind, let us define the next normalized solution for the weak case

|Ψ(t)i=Nk(t)

ψ(0)

+ k

X

n=1

λn|ψ1,n+1i !

, (3.20)

where the corresponding value of Nk(t) may be easily determined by the normal-ization condition hΨ(t)|Ψ(t)i = 1 for all t. Then we deduce from the above that

Nk(t) =

"

1 + 2 k

X

n=1

λn<

ψ(0)|ψ1,n+1

+ k

X

m,n=1

λm+nhψ1,m+1|ψ1,n+1i #−12

, (3.21)

where <(z) means the real part of z. Here, the first contribution is due to inner product of the unperturbed eigenfunctions, ψ(0)_|_ψ(0)

(28)

arises from two single finite sums, one referent to the inner product of the zero-order term with thenth-order correctionψ(0)_|_ψ

1,n+1

, and the other respect to its complex conjugate

ψ1,n+1|ψ(0)

; as consequence, a purely real contribution is obtained of the sum of both over all k. The last part of the above equation is merely handled if we run m and n from 1 to k, such as presented in Table 3.1.

m/n 1 2 3 . . . k

1 λ2_h_ψ

1,2|ψ1,2i λ3hψ1,2|ψ1,3i λ4hψ1,2|ψ1,4i . . . λ1+khψ1,2|ψ1,k+1i

2 λ3_h_ψ

3 λ4_h_ψ

..

. ... ... ... ...

..

. ... ... ... . .. ...

k λ1+k_h_ψ

1,k+1|ψ1,2i λ2+khψ1,k+1|ψ1,3i λ3+khψ1,k+1|ψ1,4i . . . λ2khψ1,k+1|ψ1,k+1i

Table 3.1: This table displays the different terms of the double summation contained in the last term of Eq. (3.21), when m and n run from 1 to k

Notice that the double summation in (3.21) can be partitioned into two parts, one where m=n and which contains all diagonal terms, and the remainder part for those off-diagonal terms which can be represented in a double sum of the real part of hψ1,m+1|ψ1,j+1i as follows

k

X

m,n=1

λm+nhψ1,m+1|ψ1,n+1i= k

X

n=1

λ2nhψ1,n+1|ψ1,n+1i + 2

k−1 X

n=1 k>1

k

X

m=n+1

λn+m<(hψ1,n+1|ψ1,m+1i) ; (3.22) by applying the change of variable m=p−n into Eq.(3.22) and the resulting sum-mation into Eq.(3.21), we arrive to

Nk(t) =

"

1 + 2 k

X

n=1

λn<

ψ(0)|ψ1,n+1

+ k

X

n=1

λ2nhψ1,n+1|ψ1,n+1i +2

k−1 X

n=1 k>1

n+k

X

p=2n+1

λp<(hψ1,n+1|ψ1,p−n+1i) 

 

−1 2

,

(3.23)

which is the normalization constant [21] for the approximate analytical solution of the Schr¨odinger equation defined in Eq.(3.20). In an analogous fashion, the normalized solution for the strong perturbative expansion of Eq.(3.18) can be defined as

|Ψ(τ)i=Nk(τ)

ψ(0)

+ k

X

n=1 1

λn|ψ1,n+1i

!

(29)

where for this regime, Nk(τ) is given by

Nk(τ) =

"

1 + 2 k

X

n=1 1

λn<

ψ(0)|ψ1,n+1 + k X n=1 1

λ2nhψ1,n+1|ψ1,n+1i +2

k−1 X

n=1 k>1

n+k

X

p=2n+1 1

λp<(hψ1,n+1|ψ1,p−n+1i)



 

−1₂

,

(3.25)

In principle, the inclusion of factor Nk in our calculations can give a fairly good ap-proximation to the solution without convergence difficulties. An important remark on the proposed normalization procedure is that we have not invoked the usual inter-mediate normalization used in the standard perturbation theory, i.e. the imposition

ψ(0)_|_ψ 1,n+1

= 0 for all λ. To see this with more detail it necessary switching back to the expressions (3.10) and (3.14) for the first two order correction; from here, we show that both solutions can be handled through an alternative procedure to obtain the same results without integration schemes, and which consist of writing them in terms of the complete orthonormal set of eigenfunctions of the unperturbed Hamiltonian, ˆH0, through the completeness relation ˆI =

P

k

k(0) k(0)

. For the sake

of clarity, we just simplify the notation

ψ (0) k E

, used in Chapter 2, by k(0)

, with main purpose to avoid confusing it with |ψ1,n+1i. Inserting this identity operator inside of Eq.(3.10), together with the initial condition |ψ(0)i = n(0)

, we deduce that

|ψ1,2i=−i X

k

k(0) k(0) e−i

ˆ H0t

Z t 0

eiHˆ0t1_Hˆ

pe−i ˆ H0t1

n(0)

dt1

=−iX

k

Hpkne −iE(0)_k t

Z t 0

ei

E_k(0)−En(0)

t1

dt1. (3.26)

Notice that we have used the hermiticy condition, ˆH0 n(0)

= En(0)

n(0)

, of the unperturbed Hamiltonian. To treat with the integration for when the sum outside is k=n and k 6=n, we replaced it by

Z t 0

ei

E_k(0)−En(0)

t1

dt1 =   

 

tEn(1), when k =n

i "

ei(E

(0)

k −E

(0)

n )t −1

#

En(0)−Ek(0)

, when k 6=n

(3.27)

in laying down our formulation, we assume that the partition of summation when

k =n and k 6=n is due that all the procedure involved in the Matrix Method does not distinguish if the unperturbed part of the Hamiltonian is degenerated or not. So,

(30)

Eq.(3.17) provides a very general expression to compute the corrections and whence it follows that Eq.(3.26) is

|ψ1,2i=−itEn(1)e

−iEn(0)t

n(0)

−2iX

k6=n

Hpkne

−i₂tEn(0)+Ek(0)

sin

h

t 2

En(0)−E_k(0)

i

En(0)−E_k(0)

k(0)

,

(3.28)

being now the first-order correction expressed in terms of the eigenvalues ˆH0, where it also includes the first-order energy correction defined in (3.11). The above re-sult means that the first-order corrections to wavefunction and to the energy can be contained and written in only one expression, at difference of the standard per-turbation theory, where is needed to calculate them in a separate way. Further, the approximate solution not only present conventional stationary terms, but also time-dependent terms that allows one to figure out the temporal evolution of the corrections. Consequently, we deduce from the above solution that the derivation of second order is given by

|ψ1,3i=−ie−i ˆ H0t

t

Z

0

eiHˆ0t1_Hˆ

p

(

−it1En(1)e

−iEn(0)t1 n(0)

−2iX

k6=n

e−i

t₁

2

E_k(0)+En(0)

sin

h

t1 2

En(0)−E_k(0)

i

En(0)−E_k(0)

Hpkn

k(0)

)

dt1, (3.29)

where the expression inside of the curly brackets is the first order correction. Em-ploying again the identity operator ˆI and after some algebraic manipulation, one gets

|ψ1,3i=−e−iE (0) n t t2 2E 2(1) n +itE

(2) n n(0)

+ite−iE(0)n tX

k6=n

e−i2t

E_k(0)−En(0)

Hpkk−E

(1) n

Hpkn

En(0)−E_k(0)

k(0)

−ie−iE(0)n t_E(1)

n

X

k6=n

e−i

t

2

E(0)_k −En(0)

sin

h

t 2

E_k(0)−En(0)

i

En(0)−E_k(0)

2 Hpkn

k(0)

−2ie−i2tE (0)

n X

k6=n

X

q6=n

e−it2E (0) q sin h t 2

Eq(0)−En(0)

i

(Eq(0)−En(0))(En(0)−E_k(0))

HpqkHpkn

q(0)

+ 2iX

k6=n

X

q6=k

e−i

t

2

E(0)q +E_k(0)

sin

h

t 2

Eq(0)−E_k(0)

i

(Eq(0)−E_k(0))(En(0)−E_k(0))

HpqkHpkn

q(0)

.

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In particular we are able to determined up third-order correction by following the same strategy adopted above, to gives

|ψ1,4i=ite−iE (0) n t t2 6E 3(1)

n +tEn(1)En(2)+En(1)En(3)

n(0)

+itX

k6=n

e−iE(0)n t_E(1)

n |Hpkn|

2

En(0)−E_k(0)

2 k(0)

+iX

k6=n



te

−it

2(E (0)

n −Ek(0))−2

sinht₂(En(0)−E_k(0))

i

En(0)−E_k(0)





e−it2(E (0)

n −Ek(0))H_p

kk|Hpkn|

2

En(0)−E_k(0)

2 k(0)

+E_n(1)X

k6=n

isinhtE_n(0)−E_k(0)i+ 2 sin

t

2

E_n(0)−E_k(0)

_|_H

pkn|

2 k(0)

En(0)−E_k(0)

3

−iX

k6=n

X

q6=n

 



t−2eit2

En(0)−E(0)q

sin

h

t 2

En(0)−Eq(0)

i

En(0)−Eq(0)

 



HpknHpnqHpqk

q(0)

En(0)−Eq(0) En(0)−E_k(0)

+ 2iX

k6=n

X

q6=k

eit2

En(0)−Ek(0)

sin

h

t 2

En(0)−E_k(0)

i

En(0)−E_k(0)

2

Eq(0)−E_k(0)

HpnqHpqkHpkn q(0)

−2iX

k6=n

X

q6=k q6=n

eit2

E(0)n −Eq(0)

sin

h

t 2

En(0)−Eq(0)

i

HpnqHpqkHpkn

En(0)−Eq(0) Eq(0)−E_k(0) En(0)−E_k(0)

q(0) . (3.31) with

E_n(3) =X k6=n

X

q6=n



 

HpnkHpkqHpqn

En(0)−E_k(0) En(0)−Eq(0)

−

HpnnHpknHpnq

En(0)−E_k(0)

2 



. (3.32)

Let us multiply both sides of (3.28), (3.30) and (3.31) byn(0)

, this procedure gives

ψ(0)|ψ1,2

=−itE_n(1),

ψ(0)|ψ1,3

=−1

2 t 2_E2(1)

n + 2itE (2) n

+ 2iX

k6=n

ei2t(En−Ek)

sinht 2

En(0)−E_k(0)

i

En(0)−E_k(0)

2 |Hpkn| 2_,

ψ(0)|ψ1,4 =it t2 6E 3(1) n +tE

(1) n E

(2) n +E

(1) n E

(3) n

+ 2iE_n(1)X

k6=n

eit2

E(0)n −E(0)_k

sin

h

t 2

En(0)−E_k(0)

i

En(0)−E_k(0)

3 |Hpkn|

(32)

It can be seen clearly that the intermediate normalization, k

P

n=1

ψ(0)ψ₁_,n₊₁

= 0, is

impractical and does not work for the Matrix Method case due the inner product of the zero-order with first three correction terms are different from zero. In particular, these complex inner products have non-zero imaginary parts which should not be neglected if the called intermediate normalization is applied, for this reason, we have adopted other procedure to get a time-dependent normalization, which ensures real values at any power of λ. For example, if we get the normalization constant for the first-order correction of |Ψ(t)i when k= 1 into (3.23)

N1(t) =

1 + 2<

ψ(0)ψ₁_,₂

+λ2hψ1,2|ψ1,2i −1/2

, (3.34)

whose inner products are easy to evaluate with the information proportioned by Eq.(3.33); where <

ψ(0)_|_ψ 1,2

= 0, whereas the last term renders to

hψ1,2|ψ1,2i=t2En2(1)+ 4

X

k6=n

sin2h₂tEn(0)−E_k(0)

i

En(0)−E_k(0)

2 kHpknk 2

(3.35)

being now the normalization constant reduced to

N1(t) = 1 +λ2hψ1,2|ψ1,2i −1₂

. (3.36)

Let us assume now that k = 2 into Eq.(3.23), in this case, we arrive to the normal-ization constant of second order correction

N2(t) = n

1 +λ2

2<

ψ(0)|ψ1,3

+hψ1,2|ψ1,2i

+ 2λ3<(hψ1,2|ψ1,3i) +λ4hψ1,3|ψ1,3i

o−1₂

(33)

doing the inner products, one can find

2<

ψ(0)|ψ1,3

=− hψ1,2|ψ1,2i, 2<(hψ1,2|ψ1,3i) = 2t2En(1)E

(2) n −4tE

(1) n

X

k6=n

sin2h₂tE_k(0)−En(0)

i

En(0)−E_k(0)

3 |Hpkn| 2

+ 4tX

k6=n

cosh₂tE_k(0)−En(0)

i

sinh₂tE_k(0)−En(0)

i

En(0)−E_k(0)

2 |Hpkn|

2_H pkk

+ 4X k6=n

X

m6=n

sin2ht₂Em(0)−En(0)

i

En(0)−Em(0)

2

En(0)−E_k(0)

HpnmHpmkHpkn

+ 4X k6=n

X

m6=n

cosh₂t En(0)−E_k(0)

i

sinh₂tEm(0)−En(0)

i

En(0)−E_k(0) En(0)−Em(0) Em(0)−E_k(0)

×sin t 2

E_m(0)−E_k(0)

HpnmHpmkHpkn,

hψ1,3|ψ1,3i=

t4 4E

4(1) n +t

2

E_n2(2)−2t2E_n(2)X

k6=n

sin2h₂tEn(0)−E_k(0)

i

En(0)−E_k(0)

2 |Hpkn|

2

+ 2tE_n(2)X

k6=n

(

sinhtE_n(0)−E_k(0)i+ (Hpkk)

2 )

|Hpkn|

2

En(0)−E_k(0)

2

+E_n(1)X

k6=n

(_sinh_t_E(0)

n −E_k(0)

i

En(0)−E_k(0)

−2t

)

Hpkk|Hpkn|

2

En(0)−E_k(0)

2

−2tE_n(1)X

k6=n

X

m6=n

sin

h

t

Em(0)−En(0)

i

En(0)−Em(0)

2

En(0)−E_k(0)

HpkmHpnkHpnm

+ 2X k6=n

X

m6=n

sinhtEm(0)−En(0)

i

En(0)−Em(0)

2

En(0)−E_k(0)

HpmmHpkmHpnkHpmn

−4E_n(1)X

k6=n

X

m6=n

cosh₂tE_k(0)−En(0)

i

sinh₂t Em(0)−E_k(0)

i

Em(0)−E_k(0) En(0)−E_k(0) En(0)−Em(0)

2

×HpkmHpnkHpnm

+ 2XX

sinh₂tEm(0)−E_k(0)

i

(34)

−8X k6=n

X

m6=k

X

q6=n

cosh₂t E_k(0)−En(0)

i

sinh₂tEm(0)−E_k(0)

i

Em(0)−E_k(0) En(0)−E_k(0) Em(0)−En(0) En(0)−Eq(0)

×HpmmHpmnHpkmHpnkHpqnHpqk

−4E_n(1)X

k6=n

X

m6=n

sin

h

t 2

Em(0)−En(0)

i

En(0)−Em(0)

3

En(0)−E_k(0)

HpkmHpnmHpnk, (3.38)

Therefore, the normalization constant at this order correction can be recast as

N2(t) = h

1 + 2λ3<(hψ1,2|ψ1,3i) +λ4hψ1,3|ψ1,3i i−1₂

. (3.39)

We see that the procedure advocated here insures that inner products are always a real number, therefore, the isolation of a multiplicative time-dependent factor,

Nk(t), in both complete solutions (3.20) and (3.24) provides the correct normalization treatment to the Matrix Method.

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Chapter 4 Examples

Several examples have been presented in the articles where Matrix Method has been developed. In [20], the generalization to the case of the Lindblad master equation and its application to the problem for a lossy cavity filled with a Kerr medium is presented; in [18] the scenario of harmonic oscillator perturbed by a quadratic term is analyzed. However, in the latter case, the normalization process of the solutions was completely ignored.

This chapter will allow us to illustrate the efficiency of complete method and its range of applicability in comparison with other perturbative analysis, including the standard perturbation theory. To achieve this, some examples with analytical solu-tion will be treated making use of the normalized formalism introduced in Secsolu-tion 3.2 of Chapter 3. Then, their approximate results will be obtained and compared with their respective exact or numerical solutions.

4.1 Harmonic Oscillator with linear term in

po-tential.

4.1.1 Exact solution

Once said this, let us begin by examining the case of harmonic oscillator disturbed by linear anharmonic potential; this quantum model is described by the following Schr¨odinger equation

id

dt|ψ(t)i=

1 2pˆ

2 +ω

2

2 xˆ 2

+λxˆ

|ψ(t)i, (4.1)

where ˆp = −i_∂x∂ and ˆx = x are the momentum and position operators, whereas, the quantity λ is a dimensionless scale parameter which quantifies the perturbation strength of the linear anharmonic term. From a classical point of view, the above time dependent Schr¨odinger equation gives a physically description of a particle

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Chapter 4. Examples

with charge q located in a weak electric field of strength ε when λ = qε [9, 38–40]. Standard differential equation techniques have been tackled with great success to get an exact solution of this system, especially when a time dependence is involve in the anharmonic part [41–46]; here, we adopt an operator formalism to solve Eq.(4.1), both exactly and approximately. The momentum and position operators can be expressed in terms of the well-known raising and lowering operators ˆa† and ˆa as

ˆ

x=√1

2ω ˆa

†

+ ˆa,

ˆ

p=i

r

ω

2 ˆa

†₋_ˆ_a

. (4.2)

These ladder operators satisfy the commutation relations â,â† = 1,ˆn,â† = â†

and [ˆn,â] = −aˆ, being the number operator ˆn = â†â; then Eq.(4.1) turns into the equivalent form

id

dt|ψ(t)i=

ω

ˆ

n+1 2

+√λ

2ω ˆa

†

+ ˆa

|ψ(t)i. (4.3) here, the mass m of oscillator is set equal to 1.

To simplify the above Schr¨odinger equation and reach an exactly solvable form, it is convenient to perform the quantum transformation |φ(t)i= ˆD λ

ω√2ω

|ψ(t)i, where ˆ

D(α) = exp αˆa†−α∗ˆa

is the Glauber displacement operator [47, 48]. Indeed, the operator ˆD(α) acts as a displacement upon amplitudes of â and â†as ˆD(α)âDˆ†(α) = ˆ

a−α and ˆD(α)ˆa†Dˆ†(α) = ˆa†−α∗. So expression (4.3) is transformed into

id

dt|φ(t)i=

ω

ˆ

n+1 2

− λ

2

2ω2

|φ(t)i, (4.4)

which is nothing else than an harmonic oscillator displaced by a quantity ₂λ_ω22 and

with a quantized energy E0 = ω n+ 1₂− λ2

2ω2. If we now integrate the resulting

expression with respect to time and then transform it back to |ψ(t)i, one obtains

|ψ(t)i= exp

−it

2

ω− λ

2 ω2 ˆ D† λ ω√2ω

exp (−itωnˆ) ˆD

λ ω√2ω

|ψ(0)i. (4.5) In order to simplify the notation, one can insert the identity operator ˆI =eitωˆn_e−itωnˆ into the previous equation as follows

|ψ(t)i= exp

−it

2

ω− λ

2 ω2 ˆ D† λ

ω√2ω

exp (−itωnˆ) ˆD

λ

ω√2ω

eitωnˆe−itωˆn|ψ(0)i.

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4.1. Harmonic Oscillator with linear term in potential.

with the Hadamard formula [49, 50] eδAˆ_Be_ˆ −δAˆ _{= ˆ}_B₊_δh_A,_ˆ _B_ˆi₊ δ2 2!

h

ˆ

A,hA,ˆ Bˆii, . . .

, it is very easy to prove that

e−itωnˆDˆ

λ ω√2ω

eitωnˆ = ˆD

λe−itω ω√2ω

; (4.7)

and with the aid of the displacement operator property, ˆD(α) ˆD(β) = ˆD(α+β)ei=(αβ∗), Eq.(4.5) is simplified to

|ψ(t)i= exp

( − i 2 t

ω− λ

2

ω2

+ λ 2

ω3 sinh(ωt) )

ˆ

D(ξ(t))e−itωˆn|ψ(0)i, (4.8) where

ξ(t) =− √

2iλ ω3/2 exp

−iωt

2 sin ωt 2 . (4.9)

Further, using the factorized form ˆD(α) =e−|α|

2 2 eαˆa

†

e−α∗ˆa_{, the exact solution of the} linear anharmonic oscillator is found to be

|ψ(t)i= exp [−γ(t)] expξ(t)ˆa†exp [−ξ∗(t)ˆa] exp (−itωnˆ)|ψ(0)i, (4.10) with

γ(t) = i 2

t

ω− λ

2

ω2

+ λ 2

ω3 sin(ωt)

+ λ 2

ω3 sin 2 ωt 2 . (4.11)

This exact solution may be evaluated with any initial condition; here, for simplicity, we restrict our attention to two special examples of |ψ(0)i, such as the number state |ni which represents an eigenstate of the number operator ˆn with eigenvalue

n and the coherent state |αiwhich denotes a quasi-classical state produced by laser [47, 48, 51, 52]. The application of first initial condition can be done very simply if the completeness relation ˆI =P

k

|ki hk| is inserted into the exact solution as

|ψ(t)i= ˆI|ψ(t)i

=e−γ(t)

∞

X

k=0

e−itωn|ki hk|eξ(t)ˆa†e−ξ∗(t)ˆa|ni

=e−γ(t)

∞

X

k=0

e−itωn

k X p=0 n X q=0 (−1)q

s

n!k! (k−p)! (n−q)!

ξp₍_t₎_ξ∗q₍_t₎

p!q! δk−p,n−q|ki, (4.12)

if q =n−k+p with n≥k, we get

|ψ(t)i=e−γ(t)

∞

X

k=0

e−itωn

r

k!

n![−ξ

∗₍_t_)]n−k k

X

p=0

−|ξ(t)|2p

n!

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Chapter 4. Examples

it is easy to recognize the structure of the associated Laguerre polynomials [53, 54]

Ln−m m (x) =

m

X

r=0

(−x)rn!

r! (m−r)! (n−m+r)!; (4.14) then

|ψ(t)i=e−γ(t)

∞

X

k=0

e−itωn

r

k!

n![−ξ

∗

(t)]n−kLn−k k (|ξ(t)|

2

)|ki. (4.15) If we now consider the case p =k −n+q with k ≥n in the sum of Eq.(4.12) and after some algebra, leads to

|ψ(t)i=e−γ(t)

∞

X

k=0

e−itωn

r

n!

k![ξ(t)]

k−n_Lk−n n (|ξ(t)|

2

)|ki. (4.16) In fact, both solutions are written in the representation of displaced number states according to reported in the literature [55–61]. The wavefunction of the anharmonic system in the xrepresentation can be found multiplying both sides of Eq.(4.15) and Eq.(4.16) by hx|

ψn(x, t) =

ω

π

1/4

exp

−

γ+itωn+ωx 2 2 ∞ X k=0 Dkn √

2k_k_!Hk

√

ωx

, (4.17)

with

Dkn=

 

 q

k! n![−ξ

∗₍_t_)]n−k

Ln−k k (|ξ(t)|

2

), forn ≥k

q

n! k![ξ(t)]

k−n

Lk−n n (|ξ(t)|

2

), fork ≥n

(4.18)

where we have used that the harmonic oscillator eigenfunctions are

ψ(0)_n =hx|ni= √1

2n_n_!

ω

π

1/4

exp −ωx 2 2 Hn √

ωx, (4.19)

being Hn(x) = (−1)nex

2 _dn

dxn exp (−x2) the Hermite polynomials [62].

Let us suppose that the anharmonic system initial condition is |ψ(0)i = |αi, a coherent state. In this case, the exact solution of the time-dependent Schr¨odinger equation, ψα(x, t), can be obtained by using the position representation [63]

|xi=ω

π

1/4

exp ωx2 2 exp

−ˆa

†2 2 √

2ωxE, (4.20)

where

√

2ωx is a coherent state with amplitude √2ωx. The position eigenstate was introduced by Moya and Eguibar from starting point of the Caves approach to

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4.1. Harmonic Oscillator with linear term in potential.

define an squeezed state [63]. Afterwards, they just applied a squeeze in the vacuum state, and after then, displace it. Taking the expressions (4.20) and (4.10) we get

ψα(x, t) =

ω

π

1/4

exp

−γ(t) +ωx

2 2 D√ 2ωx e

−ˆa2

2 eξ(t)ˆa

†

e−ξ∗(t)ˆae−itωnˆ|αi. (4.21) As exp (−itωnˆ)|αi =|α(t)i with α(t) =αe−itω_{, and due the fact that the coherent} states are eigenstates of the annihilation operator with eigenvalues α(t) =αe−itω, we can cast the above as

ψα(x, t) =

ω

π

1/4

exp

−γ(t)−α(t)ξ∗(t) +ωx

2

D√

2ωx|e−ˆa

2 2 eξ(t)ˆa

†

|α(t)E. (4.22) Inserting the identity operator ˆI = exp

ˆ a2 2 exp

−aˆ2 2

,

ψα(x, t) =

ω

π

1/4

exp

−γ(t)−α(t)ξ∗(t) + ωx 2

2

(4.23)

×D√2ωx

exp

−ˆa

2

expξ(t)ˆa†exp

ˆ a2 2 exp

−ˆa

2

|α(t)i,

using that exp−aˆ2 2

ˆ

a†expˆa₂2= ˆa†−aˆ, we get

ψα(x, t) =

ω

π

1/4

exp

−γ(t)−α(t)ξ∗(t)− α

2₍_t₎

2 +

ωx2 2

D√

2ωx|eξ(t)(ˆa†−ˆa)|α(t)E.

(4.24)

Factorizing the exponential operator eξ(t)(â†−â) ase−ξ2(2t)eξ(t)â

†

e−ξ(t)ˆa and taking into account the overlap between coherent states √

2ωx|α(t)

= exph−ωx2₋ |α(t)|2 2

i

expα(t)√2ωx, we finally found

ψα(x, t) =

ω

π

1/4

exp

(

− 1

2

2γ(t) +α2(t) +ωx2+ξ2(t) +|α(t)|2+ 4α(t)<(ξ(t))

)

×exp

n√

2ω[α(t) +ξ(t)]x

o

. (4.25)

This is the exact solution of Schr¨odinger equation (4.3) in the coordinate represen-tation, when the initial state is a coherent state.

4.1.2 Perturbative solution

Let us proceed to solve the same quantum system but now through our normalized perturbative treatment. In this case, one must formulate that ˆHo = ω(ˆn+ 1/2)