Solution of the Time-dependent Schr˜odinger Equation via ... · Time-dependent Schr˜odinger...

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Thesis for the degree of Doctor of Philosophy Solution of the Time-dependent Schr¨odinger Equation via Complex Quantum Trajectories By: Yair Goldfarb Advisor: Prof. David J. Tannor December 20, 2007 Submitted to the scientific council of The Weizmann Institute of Science Rehovot, Israel

Transcript of Solution of the Time-dependent Schr˜odinger Equation via ... · Time-dependent Schr˜odinger...

Page 1: Solution of the Time-dependent Schr˜odinger Equation via ... · Time-dependent Schr˜odinger Equation via Complex Quantum Trajectories By: Yair Goldfarb Advisor: Prof. David J. Tannor

Thesis for the degree ofDoctor of Philosophy

Solution of theTime-dependent Schrodinger Equation

viaComplex Quantum Trajectories

By: Yair GoldfarbAdvisor: Prof. David J. Tannor

December 20, 2007

Submitted to the scientific council of

The Weizmann Institute of ScienceRehovot, Israel

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Acknowledgements

This thesis summarizes the end results of almost five years of work. The Acknowledge-

ments, on the other hand, pay tribute to the people who have accompanied me throughout

this entire time. From this perspective the Acknowledgements are, in my opinion, of equal

importance to the rest of the manuscript. Hence, the reader will forgive me if I break the

first rule of scientific writing and allow myself temporarily not to be brief.

When I first asked David Tannor to meet me in order to discuss the possibility

of starting a PhD in his group, his reply began with the words: “I’ll be delighted to

see you...”. This response which was given without any prior acquaintance, summarizes

David’s approach to his students — first of all one gets all the credit in the world. With

all the ups and downs that characterize long term human relations such as supervisor-

student interaction, I see David as personally responsible for what has been the most

intensive, productive and enjoyable period of my life so far. The fact that not even once

did I feel reluctant at the end of a weekend to go back to the quantum ditches (that is,

if I wasn’t already in the office) is a clear indication of this.

Special thanks are due to Ilan Degani and Jeremy Schiff with whom I have col-

laborated. Ilan’s help in breaking the “quantum force blockade” was invaluable and was

the single most important break-through in this work. Jeremy’s ability to think concep-

tually while remaining at the same time totaly familiar and fluent with all the nuts and

bolts of the mathematical numerical procedures makes him unique among the academics

I have met. I wish to thank Ilya Averbukh and Eli Pollak, my PhD committee mem-

bers. Even though they have given me a hard time, all their remarks proved to be of

great importance and straight to the point. Many thanks to the members of the ‘Tannor

group’, past and present, especially Shlomo Sklarz and Erez Boukobza for their help,

company, friendship and long hours of political debates. Without Erez my PhD would

have been a much grayer experience.

I would also like to thank those that helped without knowing. First and foremost Ron

Lifshitz, my M.Sc advisor, who took me as his student under problematic circumstances

and guided me through my first scientific endeavor. I am very proud to present myself

as one of his first students. Responsible of my “spiritual” well being I wish to thank

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Assi Ben-Porat my Tai-Chi teacher and Ze’ev Erlich my Aikido teacher who are a

constant source of inspiration. My regular training with them was crucial in helping me

overcome the more frustrating parts of doing a PhD. I wish to thank three of my friends,

Alex Ben-Ari, Fabio Nudelman and Ori Ben-Zvi. For five years they have been my

patient verbal punching bags when it came to complaining about being stuck for months

at a time with my work. Literally holding my hand during these years, I am grateful to

Noa Sela, Livnat Bazini, Olga Vigini, Sarit Rohkin and Kineret Muller.

Last but far from being least I wish to express my love to the Goldfarb clan, Giladush,

Adar, Eran, Be’erit, David and Dvora. Without whom I would have spent all this

time being lonely. Special thanks to my parents who have been pushing me forward by

constantly asking me when will I finish and get a real job. Finally, I have an answer.

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Abstract

Ever since the advent of Quantum Mechanics, there has been a quest for a trajectory

based formulation of quantum theory that is exact. In the 1950’s, David Bohm, building

on earlier work of Madelung and de Broglie, developed an exact formulation of quantum

mechanics in which trajectories evolve in the presence of the usual Newtonian force plus

an additional quantum force. In recent years, there has been a resurgence of interest in

Bohmian Mechanics as a numerical tool because of its apparently local dynamics, which

could lead to significant computational advantages for the simulation of large quantum

systems. However, closer inspection of the Bohmian formulation reveals that the non-

locality of quantum mechanics has not disappeared — it has simply been swept under

the rug into the quantum force. In the first part of the thesis we present several new

formulations, inspired by Bohmian mechanics, in which the quantum action, S, is taken

to be complex. The starting point of the formulations is the complex quantum Hamilton-

Jacobi equation. Although this equation is equivalent to the time-dependent Schrodinger

equation it has been relatively unexplored in comparison with other quantum mechan-

ical formulations. In all the formulations presented, we propagate trajectories that do

not communicate with their neighbors, allowing for local approximations to the quantum

wavefunction. Importantly, we show that the new formulations allow for the description

of nodal patterns as a sum of the contribution from several crossing trajectories. The

new formulations are applied to one- and two-dimensional barrier scattering, thermal rate

constants and the calculation of eigenvalues.

In the second part of the thesis we explore a new mapping procedure developed for

use with the mapped Fourier method. The conventional procedure uses the classical

action function to generate a coordinate mapping that equalizes the spacing between

extrema and nodal positions of the specified wavefunction. The new procedure utilizes

the Miller-Good transformation to construct a mapping that scales the amplitude of the

wavefunction as well. The mapped Hamiltonian that is obtained through this procedure

is of simpler form and its matrix representation more easily calculated than in current

methods. We also present a preliminary attempt to extend conventional coordinate map-

ping procedures to a two-dimensional non-separable potential. This is part of an ongoing

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effort to find separable mapping procedures that might be directly generalized to non-

separable multidimensional problems.

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Contents

I Solution of the Time-dependent Schrodinger Equation viaComplex Quantum Trajectories 1

1 Introduction 2

2 Preliminaries and Motivation 5

2.1 Hydrodynamic formulation of quantum mechanics vs. Ehrenfest’s theorem 5

2.2 Complex quantum Hamilton-Jacobi equation . . . . . . . . . . . . . . . . 7

3 Unified derivation of trajectory methods 9

3.1 Real time propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Imaginary time propagation . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Method of zero-velocity complex action (ZEVCA) 14

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2.1 Tunneling probabilities . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2.2 Energy eigenvalues in 1D and 2D . . . . . . . . . . . . . . . . . . 18

4.2.3 Thermal rate constants . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Method of Bohmian mechanics with complex action (BOMCA) 32

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Root search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.3.1 Tunneling probabilities in 1D . . . . . . . . . . . . . . . . . . . . 36

5.3.2 Tunneling amplitudes in 2D . . . . . . . . . . . . . . . . . . . . . 40

5.3.3 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6 Interference with real trajectories 47

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7 Method of complex time-dependent WKB 51

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

7.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7.2.1 Time-independent vs. Time-dependent WKB . . . . . . . . . . . 53

7.2.2 Integrating along classical trajectories . . . . . . . . . . . . . . . . 54

7.2.3 Initial conditions and complex classical trajectories . . . . . . . . 56

7.2.4 Complex root search and superposition . . . . . . . . . . . . . . . 56

7.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

7.3.1 First Order approximation, N = 1 . . . . . . . . . . . . . . . . . . 58

7.3.2 Second Order approximation, N = 2 . . . . . . . . . . . . . . . . 63

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8 Remarks on the relationship between BOMCA and other methods 69

8.1 BOMCA and the modified deBroglie-Bohm approach to quantum mechanics 69

8.2 Unified derivation, BOMCA and the derivative propagation method . . . 70

8.3 BOMCA and complex time-dependent WKB . . . . . . . . . . . . . . . . 72

8.4 BOMCA and generalized Gaussian wavepacket dynamics . . . . . . . . . 73

II Miller-Good transformation for themapped Fourier method 75

9 Miller-Good transformation for the mapped Fourier method 76

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

9.2 Review of the Mapped Fourier Grid Hamiltonian . . . . . . . . . . . . . 79

9.2.1 The Fourier grid Hamiltonian (FGH) . . . . . . . . . . . . . . . . 79

9.2.2 Phase space analysis, or why mapping ? . . . . . . . . . . . . . . 81

9.2.3 The mapping procedure . . . . . . . . . . . . . . . . . . . . . . . 82

9.3 Review of the Miller-Good method . . . . . . . . . . . . . . . . . . . . . 86

9.4 The Miller-Good transformation for the mapped Fourier method (MIGMAF) 88

9.4.1 The exact Miller-Good method . . . . . . . . . . . . . . . . . . . 88

9.4.2 The mapped MIGMAF Hamiltonian . . . . . . . . . . . . . . . . 89

9.4.3 FGH method applied on MIGMAF mapped Hamiltonian . . . . . 90

9.5 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

9.5.1 Comparison of the approximate MG method and the exact MG

method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

9.5.2 MIGMAF vs. Coordinate Mapping . . . . . . . . . . . . . . . . . 97

9.5.3 Two dimensional mapped Fourier method . . . . . . . . . . . . . 102

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9.6 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

A Review of the classical Hamilton-Jacobi equation 107

B Review of Bohmian mechanics 110

C BOMCA - Analytically solvable examples 114

D Trajectory crossing and the uniqueness of the action field 120

E List of publications 125

Bibliography 126

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Part I

Solution of the Time-dependent

Schrodinger Equation via Complex

Quantum Trajectories

1

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Chapter 1

Introduction

The first semiclassical method, the WKB method[1], was published almost simultaneously

with the publication of the Schrodinger equation in 1926. Since then, semiclassical meth-

ods have continued to attract great interest for two primary reasons. First, semiclassical

methods give insight into classical-quantum correspondence. Second, for large systems

they hold the promise of significant computational advantages relative to full quantum

mechanical calculations. In particular, in recent years much progress has been made in

the chemical physics community in developing time-dependent semiclassical methods and

quantum trajectory methods that are accurate and efficient for multidimensional sys-

tems. From a physical point of view, trajectory methods try to evade the non-locality

imbedded in quantum mechanics (QM). Mathematically speaking, trajectory methods

aim at casting the time-dependent Schrodinger equation (TDSE), which is a PDE, in

terms of ODEs related to classical equations of motion. This transformation has signifi-

cant computational advantages that can ease the inherent difficulty of multi-dimensional

quantum calculations.

One approach that has shown significant progress is the use of Bohmian mechanics

(BM) as a numerical tool[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In the 1950’s,

David Bohm, building on earlier work by Madelung[16] and de Broglie[17], developed

an exact formulation of QM in which trajectories evolve in the presence of the usual

Newtonian force plus an additional quantum force[18]. Bohm’s formulation was originally

developed as an interpretational tool, to recover a notion of causality in QM. In 1999 two

groups working independently, Rabitz et al.[2] and Wyatt et al.[3], demonstrated that

the Bohmian formulation can also be used a numerical tool to do quantum calculations.

However, closer inspection of the Bohmian formulation reveals that the non-locality of

QM has not disappeared — it has simply been swept under the rug into the quantum

force. Particularly disturbing is the fact that for simple cases such as Gaussian wave

packet dynamics of the free particle and the harmonic oscillator, where classical-quantum

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correspondence should be perfect, the quantum force is not only non-vanishing but is the

same magnitude as the classical force[19].

This observation suggests exploring a complex version of the Bohmian formulation

that is based on the complex quantum Hamilton-Jacobi (HJ) equation (CQHJE). The

CQHJE is an exact formulation of quantum mechanics. It is obtained by inserting the

ansatz

ψ(~x, t) = exp

[i

~S(~x, t)

](1.1)

into the TDSE where S(~x, t) is complex and contains both the amplitude and the phase

of the wavefunction. The result is

∂S

∂t+

1

2m(~∇S)2 + V (~x, t) =

i~2m

~∇2S, (1.2)

where we recognize on the LHS of eq.(1.2) the form of the classical HJ equation for

complex S. Of the many formulations of QM, the CQHJE has received relatively little

attention. The equation seems to have appeared for the first time in Pauli’s Handbuch der

Physik article in 1933 as the starting point for the derivation of WKB theory[20]. In recent

years it was rediscovered independently by Leacock and Padgett[21] and others[22, 23,

24, 19]. Despite the long history of this equation, there has been little, if any, discussion

of its structure vis-a-vis the real quantum HJ equation that appears in the hydrodynamic

formulation of QM. Furthermore, the CQHJE has been largely overlooked as a possible

starting point for studying classical-quantum correspondence, or for its possible utility

as a numerical method for solving the TDSE. The complexification of the conventional

Bohmian formulation via eq.(1.2) has far reaching consequences which we discuss below.

At this stage, suffice it to say that the CQHJE suggests that to lowest order QM can be

thought of as classical mechanics with an additional geometrical phase and normalization

(coming from the RHS of eq.(1.2)).

In this part of the thesis we present several new trajectory formulations that share

several points in common. First, they have the same starting point, the CQHJE. Second,

the new formulations have the same approach for solving eq.(1.2). Since the classical

HJ equation is equivalent to Newton’s second law of motion expressed in terms of an

action field S(~x, t) it is tempting to try to solve eq.(1.2) using classical (or classical-like)

trajectories. This approach was originally applied in the context of the conventional

Bohmian formulation by Wyatt et al.[3] and Rabiz et al.[2]. A third point that is shared

by some of the formulations presented below is related to the nodal problem in Bohmian

formulations. The conventional Bohmian formulation breaks down in the vicinity of nodes

of the wavefunction. Since nodes in QM arise from interfering amplitudes, it is only

natural to attempt to solve the nodal problem by applying the superposition principle,

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that is to decompose the wavefunction into two nodeless parts and to propagate each part

separately using trajectories [12, 13]. However, in current methods the decomposition

and the trajectory propagation are separate steps. A somewhat more natural strategy

is to let the trajectories themselves determine the decomposition. Consider the rule,

familiar from semiclassical mechanics[25], that if different initial conditions lead to the

same final position the amplitudes are superposed. Such a procedure implicitly defines a

decomposition of the wavefunction, each component of which is expected to be nodeless.

We show below that indeed, superposing the contributions of complex crossing trajectories

yields accurate interference patterns in close agreement with the exact QM.

Part I of the thesis is organized as follows. In chapter 2 we argue that the CQHJE is a

superior starting point than the hydrodynamic formulation of QM for exploring classical-

quantum correspondence. In chapter 3 we present a general formulation for solving the

CQHJE (equivalently, the TDSE) along trajectories. The generality of the formulation

is a result of leaving the velocity field of the trajectories unspecified. Particular choices

of the velocity field yield the Zero-Velocity Complex Action (ZEVCA) method (chapter

4) and BM with Complex Action (BOMCA) method (chapter 5). Chapter 6 describes

preliminary results of interference patterns obtained using real classical trajectories. In

chapter 7 we present a modification of a complex time-dependent WKB method referred

to as the Complex Trajectory Method (CTM)[26]. We show a simple way to derive

the equations of motion of CTM and more importantly, how to incorporate interference

patterns in this method. Chapter 8 explores the relationship between BOMCA and other

formulations such as CTM, Generalized Gaussian Wavepacket Dynamics (GGWPD)[27,

28] and others. Four appendices accompany part I. In App. A we review the derivation

of the classical HJ equation and its relation to Newton’s second law of motion. In App.

B we review conventional BM and its use as a numerical tool. App. C is dedicated to

two analytically solvable examples of the BOMCA equations of motion. App. D explores

the issue of trajectory crossing in classical mechanics, BM and BOMCA.

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Chapter 2

Preliminaries and Motivation

In this chapter we argue that the CQHJE equation provides a superior starting point

than the hydrodynamic formulation of QM for understanding classical-quantum corre-

spondence and doing semiclassical calculations. We present in section 2.1 the equations

of the hydrodynamic formulation. We show that the ~ dependent term in these equations

gives rise to a quantum force even for Gaussian wavepackets that is no smaller than any

other term in these equations. Moreover, neglecting that ~-dependent terms represents a

classical limit opposite to that of the Ehrenfest’s theorem. This contradicts the common

wisdom in which the ~ dependent term is viewed as a mere perturbation. The picture

that emerges from the CQHJE is very different (section 2.2), and suggests that to a first

approximation QM is classical mechanics with complex trajectories, with the addition of

a geometrical phase and a normalization factor but no additional force. This provides

the motivation for solving the CQHJE using complex trajectories.

2.1 Hydrodynamic formulation of quantum mechan-

ics vs. Ehrenfest’s theorem

The hydrodynamic formulation of QM is obtained by inserting the ansatz

ψ(~x, t) = A(~x, t) exp

[iS(~x, t)

~

], (2.1)

into the TDSE

i~∂ψ(~x, t)

∂t= − ~

2

2m~∇2ψ(~x, t) + V (~x, t)ψ(~x, t). (2.2)

A(~x, t), S(~x, t) are real functions representing the amplitude and phase respectively.

V (~x, t) is the potential function, m is the mass and ~ is Planck’s constant divided by

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2π. By separating the result into its real and imaginary parts two PDE’s are obtained

∂A

∂t+

~∇S

m~∇A +

1

2mA~∇2S = 0, (2.3)

∂S

∂t+

1

2m(~∇S)2 + V (~x, t) =

~2

2m

~∇2A

A. (2.4)

The first equation can be shown to be a continuity equation (see App. B). The second

equation is referred to as the (real) quantum HJ equation (RQHJE) for S; it differs

from the classical HJ equation (the LHS of eq.(2.4)) by the addition of a real “quantum

potential”

QR ≡ − ~2

2m

~∇2A

A. (2.5)

The quantum potential is the only ~-dependent term in eqs.(2.3) and (2.4), hence it is

usually argued that it contains all the QM in what is otherwise a classical set of equations.

Leading textbooks state that the classical limit is obtained by setting QR → 0 without

noticing that this contradicts Ehrenfest’s theorem[29, 30, 31]. According to Ehrenfest’s

theorem, the quantum correction is proportional to 〈x2〉 − 〈x〉2, hence the narrower the

wavepacket the smaller the quantum correction. On the other hand, taking QR → 0

is equivalent to~∇2AA

→ 0, hence the wider the wavepacket the smaller the quantum

correction.

In the 1950’s, the hydrodynamic equations became the basis for BM[18, 32, 33]. Tak-

ing the derivative with respect to x of eq.(2.4), defining a velocity field ~v = d~xdt

, and using

the relation ddt

= ∂∂t

+ ~v · ~∇, Bohm obtained:

md~v

dt= −~∇V − ~∇QR, (2.6)

where FR = −~∇QR is the so called “quantum force”. The designation of a quantum po-

tential and a quantum force strengthens the impression that the classical limit is obtained

by setting these equal to zero. However, it is easily checked that in the quintessential

example of classical-quantum correspondence, Gaussian wavepacket dynamics, the quan-

tum potential and quantum force are no smaller than any of the other terms in the

problem[19]. We demonstrate this for the case of a one-dimensional Gaussian wavepacket

ψ(x, t) = exp

{−αt(x− xt)

2 +i

~pt(x− xt) +

i

~γt

}, (2.7)

propagating in a harmonic oscillator (HO) potential, V (x) = 12mω2x2, where xt and pt

are the average position and momentum of the wavepacket. Explicit expressions for αt,

xt, pt and γt are given in App. C by eqs.(C.15a-d) for a coherent state. Comparing ψ(x, t)

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with ansatz (2.1) one can identify A(x, t) and consequently the quantum potential and

quantum force

QR = −~2

mat{2at(x− xt)

2 − 1}, (2.8)

FR =4~2

ma2

t (x− xt), (2.9)

where at ≡ <(αt). Focusing specifically on a coherent state where αt = α0 = mω2~

we see that FR = mω(x− xt), which is equal in magnitude to the classical force. The

contradiction between the standard classical limit of the hydrodynamic equations and

Ehrenfest’s theorem combined with the observation that the quantum force (and quantum

potential) cannot be regarded as a perturbation has led us to explore a complex version

of the hydrodynamic formulation, eq.(1.2).

2.2 Complex quantum Hamilton-Jacobi equation

As we stated in the Introduction, the CQHJE is obtained by inserting ansatz 1.1 into the

TDSE, where S(~x, t) is complex and contains both the amplitude and the phase of the

wavefunction. The LHS of eq.(1.2) has the structure of the classical HJ equation, but for

complex S. The RHS can be viewed as a complex quantum potential

QC = − i~2m

~∇2S. (2.10)

Note that there is no longer an additional continuity equation of the type of eq.(2.3), and

thus eq.(1.2) is not a hydrodynamic formulation in the usual sense.

Despite the superficial similarity of QR in eq.(2.5) with QC in eq.(2.10) the two are

very different. We demonstrate this using the example of the one-dimensional Gaussian

in a HO potential (eq.(2.7)). Comparing ψ(x, t) with ansatz (1.1) one finds that

QC =~2

mαt, FC = −∂QC

∂x= 0. (2.11)

The complex quantum force defined as the spatial derivative of the complex quantum

potential is zero!

To gain additional insight into the complex quantum potential we make a Taylor

expansion around xt and write

ψ(x, t) = exp

[i

~S(x, t)

]= exp

{i

~

[S0(t) + Sx(t)(x− xt) +

1

2Sxx(t)(x− xt)

2 + ...

]}

(2.12)

7

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where

S0(t) = S[x(t), t], Sx(t) =∂S

∂x

∣∣∣∣x=xt

, Sxx(t) =∂2S

∂x2

∣∣∣∣x=xt

. (2.13)

Comparing eq.(2.12) with eq.(2.7) we identify Sxx(t) with αt, within multiplicative con-

stants. As shown in reference [34] the equation of motion for γt is

dγt

dt=

p2t

2m− V (xt) +

i~2m

αt, (2.14)

hence QC is identified with the non-classical term in the equation for dγdt

. In the con-

text of Gaussian wavepackets this term is responsible for two things: a time-dependent

geometrical phase and the wavepacket normalization. Since the CQHJE is an exact refor-

mulation of the TDSE the HO example suggests that to lowest order QM can be thought

of as classical mechanics with an additional geometrical phase and normalization but no

additional forces. This suggests a higher degree of locality then in the RQHJE. From

a practical viewpoint, the fact that the LHS of eq.(1.2) has the form of the classical

HJ equation but for a complex S suggests that if QC were zero this equation could be

solved exactly by propagating classical complex trajectories (see App. A). These two

observations, the suggestion of a higher degree of locality in comparison with the RQHJE

and the emergence of complex trajectories are our motivation to try to solve the CQHJE

using trajectories. In chapter 3 we present a general formulation to this end.

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Chapter 3

Unified derivation of trajectory

methods

In this chapter we present a unified framework for solving the CQHJE using trajectories.

Within this framework independent trajectories are propagated using a velocity field that

is not specified explicitly. Particular choices of the velocity field yield the ZEVCA and

BOMCA methods (see chapters 4 and 5 respectively). This chapter is divided into two

sections. In section 3.1 we derive the solution of the CQHJE in real time. In section 3.2

we show how the real time solution can be extended by a simple transformation to yield

the solution of the imaginary time Schrodinger propagator.

3.1 Real time propagation

We start with the unified framework for solving the CQHJE using trajectories in real

time for the multidimensional case. Using the notation

O~n(~x, t) = O(n1,...,nd)(~x, t) ≡ ∂n1

∂xn11

...∂nd

∂xndd

O(~x, t), (3.1)

we can rewrite the CQHJE (eq.(1.2)) as

∂S

∂t+

1

2m

d∑j=1

(S~ej)2 + V (~x, t) =

i~2m

d∑j=1

S2~ej, (3.2)

where d is the dimensionality of the system and the ~ejs are the standard basis vectors

(unit vectors of dimension d with value one at position j and zero elsewhere). The

classical HJ equation can be conveniently solved by integrating along trajectories that

satisfy the classical equations of motion. The classical trajectories fulfill the relation

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d~xdt

=~∇Sm

= ~v(~x, t) and from a mathematical point of view, they are the characteristics of

the classical HJ equation (see App. A). Here we use an analogous approach to solve the

CQHJE by integrating along some family of trajectories. We define a family of trajectories

by choosing a velocity field ~v(~x, t), which can be done in an infinite number of ways. As

opposed to the classical case, the velocity field is not necessarily predetermined by the

PDE we are trying to solve. Solutions of

d~x

dt= ~v(~x, t) (3.3)

determine trajectories, ~x(t), parameterized by their initial position ~x(0). Our aim is

to solve eq.(3.2) along these trajectories. To this end we apply the Lagrangian time

derivative on S(~x, t)

dS

dt=

(∂

∂t+ ~v · ~∇

)S =

∂S

∂t+

1

m

d∑j=1

vjS~ej, (3.4)

where vj is the jth component of ~v. Substituting ∂S∂t

from eq.(3.2) into eq.(3.4) yields

dS

dt=

i~2m

d∑j=1

S2~ej− 1

2m

d∑j=1

(S~ej)2 − V (~x, t) +

1

m

d∑j=1

vjS~ej. (3.5)

Equations (3.3) and (3.5) are not a closed set since they depend on partial derivatives

of the phase, S~ej, S2~ej

, j = 1, ..., d. However, a closed set of ODE’s that describes the

propagation of the phase can be obtained in the following way. We apply the Lagrangian

time derivative on the factors S~w in eq.(3.5) where ~w is an arbitrary vector of length d

with positive integer (or zero) components. This yields

dS~w

dt=

∂S~w

∂t+

1

m

d∑j=1

vjS~w+~ej. (3.6)

∂S~w

∂tis obtained by applying the operation ∂w1

∂xw11

∂w2

∂xw22

... ∂wd

∂xwdd

onto eq.(3.2) which results in

∂S~w

∂t+

1

2m

d∑j=1

[(S~ej

)2]

~w+ V~w =

i~2m

d∑j=1

S2~ej+~w. (3.7)

Inserting ∂S~w

∂tfrom eq.(3.7) into eq.(3.6) yields

dS~w

dt=

1

2m

d∑j=1

{i~S2~ej+~w −

[(S~ej

)2]

~w+ 2vjS~w+~ej

}− V~w. (3.8)

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The first term in the triangular brackets shows that the equation for an arbitrary S~w

depends on higher order partial derivative terms of the form S2~ej+~w. Hence, a closed

set of ODE’s can be obtained only by an infinite hierarchy of equations. However this

infinite set is formally exact and allows for the propagation of independent trajectories.

A similar approach has been used in conventional BM formulations[6, 7, 8].

In practice we truncate the hierarchy of phase partial derivatives at some order N by

setting to zero higher order phase partial derivatives, that is S~w = 0 for∑d

j=1 wj > N

(where wj are the components of ~w). We then write the equations of motion (eq.(3.8)) for

all possible S~w with∑d

j=1 wj ≤ N . The equations of motion for the individual trajectories

are given bydxj

dt= S~ej

/m, j = 1, ..., d,. The initial conditions for the propagation are

specified by a choice of the initial position ~x(0) ∈ Cd for which

S~w[~x(0), 0] = −i~[ln ψ(~x(0), 0)]~w. (3.9)

The number of equations of motion for the phase and its partial derivatives for a system

of d dimensions and truncation order N can be shown to be d +(

d+NN

). We summarize

the equations of motion of the unified derivation

d~x

dt= ~v[~x(t), t], (3.10)

dS~w

dt=

1

2m

d∑j=1

{i~S2~ej+~w −

[(S~ej

)2]

~w+ 2vjS~w+~ej

}− V~w ford∑

j=1

wj < N, (3.11)

S~w = 0 ford∑

j=1

wj > N. (3.12)

Note that by solving the equations of motion for ~w = ~0 we obtain S[~x(t)]. Inserting the

result into ansatz (1.1) yields the wavefunction

ψ[~x(t)] = exp

{i

~S[~x(t)]

}(3.13)

at position ~x(t). Since a significant part of the thesis deals with one-dimensional systems,

we write the equations of motion explicitly for this case

dx

dt= v[x(t), t], (3.14)

dSn

dt=

i~2m

Sn+2 − 1

2m(S2

1)n + vSn+1 − Vn; n = 0, 1, ..., N, (3.15)

SN+1 = SN+2 = 0, (3.16)

11

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where (S21)n =

∑nj=0

(nj

)Sj+1Sn−j+1 and we have used the notation

On ≡ ∂nO

∂xn|[x(t),t]. (3.17)

The initial conditions are

Sn[x(0), 0] = −i~∂n ln[ψ(x, 0)]

∂xn

∣∣∣∣x(0)

. (3.18)

As a final remark we note that the partial derivatives of the phase allow for the con-

struction of the wavefunction in the vicinity of position ~x(t) by using a Taylor expansion.

For example, for the one-dimensional case

ψ(x, t) ≈ exp

{i

~

N∑n=0

Sn

n![x− x(t)]n

}. (3.19)

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3.2 Imaginary time propagation

Equations (3.10)-(3.12) plus eq.(3.13) yield the solution of the Schrodinger equation or

equivalently the solution of the Schrodinger real time propagator

exp

[−iHt

~

]ψ(~x, 0) = ψ(~x, t), (3.20)

where H is the Hamiltonian operator

H = − ~2

2m~∇2 + V (~x). (3.21)

Several applications such as the calculation of thermal rate constants require the solution

of the imaginary time Schrodinger propagator. This propagator takes the form

exp

[−Hτ

2

]ψ(~x, 0) = ψ(~x, τ), (3.22)

where ψ(~x, τ) is called the “thermal wavefunction”. ψ(~x, τ) is the solution of a Schrodinger-

like equation

−2∂ψ

∂τ(~x, τ) = − ~

2

2m~∇2ψ(~x, τ) + V (~x)ψ(~x, τ), (3.23)

where τ plays the role of time. The mapping of τ to pure imaginary time (hence the

term—“imaginary time propagation”)

t = −i~2

τ (3.24)

transforms eq.(3.23) to the familiar form of the TDSE

i~∂ψ

∂t(~x, t) = − ~

2

2m~∇2ψ(~x, t) + V (~x)ψ(~x, t), (3.25)

where for simplicity we allow a slight abuse of notation ψ(~x, τ) → ψ(~x, t). At this stage

we insert into eq.(3.25) an ansatz identical to eq.(1.1)

ψ(x, t) = exp

[i

~S(x, t)

], (3.26)

and use the same procedure as described in section 3.1. This yields equations identical to

eqs.(3.10)-(3.12) with the only difference being t → t. From these equations plus eq.(3.26)

the thermal wavefunction is obtained.

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Chapter 4

Method of zero-velocity complex

action (ZEVCA)

4.1 Introduction

The simplest choice for the velocity field in eq.(3.10) is

v(~x, t) = 0 =⇒ ~x(t) = ~x(0). (4.1)

The resulting trajectories are straight lines, hence the name — “Zero-Velocity Complex

Action method”[35]. Since ZEVCA yields an approximation of the wavefunction at a fixed

position it is a cousin of grid methods. However, we compute S by solving a hierarchy of

spatial derivatives of S along a stationary trajectory; information from the neighborhood

of a trajectory is incorporated only through the initial conditions. This property makes

the approximation local from the quantum mechanical point of view, and hence a cousin

of semiclassical methods. In reference [15] section 7.2, Wyatt considers the solution of the

global hydrodynamic equations of QM on fixed grid points (Eulerian grid) but dismisses

its usefulness as a numerical tool. The ZEVCA formulation shows how to obtain useful

output of a local propagation with as little as a single grid point.

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4.2 Applications

For a variety of quantum mechanical applications, such as the calculation of thermal rate

constants and tunneling probabilities, knowledge of the wavefunction in all of configura-

tion space is unnecessary: these quantities can be calculated by data at a single position

or a small interval of configuration space. For such calculations, ZEVCA has a significant

numerical advantage since it produces local information at a predetermined position. The

first ZEVCA application we present is the calculation of tunneling probabilities (section

4.2.1); ZEVCA is well suited for this calculation since tunneling probabilities can be cal-

culated from a time integral of the probability current at a fixed position. The second

application is the calculation of the lowest energy eigenvalue of a bound potential in one

and two dimensional systems (section 4.2.2). In both these applications we calculate

the results by propagating just a single trajectory. In the third application we calculate

thermal rate constants in a one-dimensional system by propagating only two trajectories.

4.2.1 Tunneling probabilities

In this section we calculate tunneling probabilities using a single trajectory propagation of

the ZEVCA real time Schrodinger propagator. The system we examine is the scattering of

a one-dimensional Gaussian wavepacket from an Eckart barrier potential centered around

x = 0. The potential is given by

V (x) =D

[cosh(βx)]2, (4.2)

where D is the barrier height and 1/β gives an estimate of the barrier width. The initial

wavefunction is a Gaussian wavepacket

ψ(x, 0) = exp

[−αc(x− xc)

2 +i

~pc(x− xc) +

i

~γc

], (4.3)

where 1/√

αc relates to the Gaussian width and γc = − i~4

ln(2αc

π) takes care of the nor-

malization. xc and pc are the average position and momentum respectively. The initial

conditions of eqs.(3.14)-(3.16) are obtained by inserting eq.(4.3) into eq.(3.18)

S0(x, 0) = iαc~(x− xc)2 + pc(x− xc) + γc, (4.4)

S1(x, 0) = 2iαc~(x− xc) + pc,

S2(x, 0) = 2iαc~,

Sn(x, 0) = 0, n ≥ 3.

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The final tunneling probability T for an initial wavefunction centered at xc ¿ 0 and

having pc ≥ 0 is given by

T = limt→∞

T (t), (4.5)

where

T (t) =

∫ ∞

0

|ψ(x′, t)|2dx′ (4.6)

is the time-dependent tunneling probability. The integration begins at x = 0 since this is

the position of the maximum of the barrier. T (t) can be expressed by a time integration

of the probability current at x = 0. We show this by first writing the quantum mechanical

continuity equation∂ρ(x, t)

∂t= −∂J(x, t)

∂x, (4.7)

where ρ ≡ |ψ|2 is the probability amplitude and the probability current J is given by

J =~m=

(ψ†

∂ψ

∂x

). (4.8)

By inserting ρ(x, t) into eq.(4.6) we can write

T (t) =

∫ ∞

0

ρ(x′, t)dx′ (4.9)

=

∫ ∞

0

dx′∫ t

0

∂ρ(x′, t′)∂t′

dt′

= −∫ ∞

0

dx′∫ t

0

∂J(x′, t′)∂x′

dt′

= −∫ t

0

dt′∫ ∞

0

∂J(x′, t′)∂x′

dx′ =∫ t

0

J(0, t′)dt′.

where in the third stage we have used eq.(4.7) and in the fourth stage we changed the

order of the integration and performed the spatial integration. Inserting ansatz (1.1) into

eq.(4.8) reveals that

J =|ψ|2m<(S1) =

1

mexp

[−2=(S0)

~

]<(S1), (4.10)

where we use the notation defined by eq.(3.17). Inserting eq.(4.10) into the final result

of eq.(4.9) yields

T (t) =1

m

∫ t

0

exp

{−2=[S0(0, t′)]

~

}<[S1(0, t

′)]dt′. (4.11)

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Note that S1 is readily obtained by the propagation of set (3.14)-(3.16). To calculate T (t)

we need to set N and solve eqs.(3.14)-(3.16) with initial conditions (4.4) at x(0) = 0.

Inserting S0(0, t) and S1(0, t) into eq.(4.11) completes the derivation of the tunneling

probability.

We turn to the numerical results. The parameters we insert into eqs.(4.2) and (4.3)

are D = 40, β = 4.3228, αc = 30π, xc = −0.15, and pc =√

2mE where E = 20, m = 30

and ~ = 1. All quantities here and henceforth are given in atomic units. In fig.4.1

we illustrate the potential and the wavefunctions. Note that the initial wavefunction is

located close to the barrier maximum, for reasons that we discuss below.

−1 0 1 20

1

2

3

4

5

x

|ψ(x

,t)|2

t=0t=1Potential

Figure 4.1: Plot of an initial Gaussian wavefunction propagating in an Eckart barrier. Att = 1 where the wavefunction is clearly divided into a reflected part and a transmittedpart. The depiction of the barrier’s width is in proportion to that of the wavefunction,whereas the height of the barrier has no physical meaning. The arrow indicates thedirection of the average average momentum. The parameters of the system are given inthe text.

In figures 4.2(a) and 4.2(b) we depict the approximations to |ψ(0, t)|2 and T (t) for a

series of values of the truncation order N . Note that |ψ(0, t)|2 is equal to the exponential

term in eq.(4.11). The relative error between the exact tunneling probability and the

asymptotic value of T (t) (see eq.(4.5)) for N = 2, N = 6, and N = 10 is roughly 20%,

4% and 0.5% respectively. Clearly, the numerical results converge quickly to the exact

quantum mechanical result. In figures 4.3(a) and 4.3(b) we plot |ψ(0, t)|2 and T (t) for a

set of N ’s where we take pc = 0. Although the relative error in the wavefunction ψ(0, t)

still converges uniformly, the relative error in T (t) does not: the errors are 1.5%, 6% and

0.8% for N = 2, N = 4, and N = 6 respectively. We have verified that the parameters we

17

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use in this example yield results that are attributed to tunneling effect and not to classical

transmission that originates from high energy components of the Gaussian wavepacket.

The calculation of tunneling probabilities using the ZEVCA formulation has a sig-

nificant restriction on its use. Since the ZEVCA trajectories remain fixed, the formula-

tion is very sensitive to the initial conditions inserted in the equations of motion. The

choice of x(0) must satisfy two properties: (1) The derivatives of the potential (Vn[x(0)],

n = 1, ..., N) must have at least one term that is significantly different from zero. (2)

The initial wavefunction at x(0) (ψ[x(0), 0]) needs to be significantly different from zero.

These restrictions ensure that the initial conditions “capture” the wavefunction and the

potential’s surroundings at x(0). These restrictions have prevented us from choosing an

xc in the negative asymptotic region of the Eckart barrier since such a choice would have

produced ψ(0, 0) → 0.

4.2.2 Energy eigenvalues in 1D and 2D

The ZEVCA imaginary time propagator (see section 3.2) allows for a simple calculation

of the first energy eigenvalue of a bound potential. As in the pervious application, the

calculation requires just a single trajectory propagation. We start with a short deriva-

tion that demonstrates how the first eigenvalue may be calculated using imaginary time

propagation.

An arbitrary bound potential defines a set of eigenfunctions φj(x), j = 1, 2...,∞that satisfy Hφj(x) = Ejφj, where Ej are the energy eigenvalues and H is given in

eq.(3.21). The eigenfunctions can be used as a basis set for the expansion of an arbitrary

wavefunction ψ(x)

ψ(x) =∞∑

j=1

ajφj(x), (4.12)

where∑∞

j=0 |aj|2 = 1. If we define a time scale τ1 = 2π~E1

then the operation of the

imaginary time propagator for τ À τ1/~ on an initial wavepacket ψ(x, 0) yields

limτÀτ1/~

exp

[−Hτ

2

]ψ(x, 0) = lim

τÀτ1/~exp

[−Hτ

2

] ∞∑j=1

ajφj(x) (4.13)

= limτÀτ1/~

∞∑j=1

ajφj(x) exp

[−Ejτ

2

]

= a1φ1(x) exp

[−E1τ

2

].

In the first stage we inserted eq.(4.12), in the second stage we applied the imaginary time

propagator on the eigenfunctions and in the last stage we applied the limit. Inserting the

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0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

t

|ψ(0

,t)|2

E=20, pc=34.6

Exact QMN=2N=6N=10

(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

t

T(t

)

Exact QMN=2N=6N=10

(b)

Figure 4.2: ZEVCA numerical results for |ψ(0, t)|2 ((a)) and the transmitted probabilityT (t) ((b)) for a series of values of the truncation order N . The system corresponds tofig.4.1 where the exact parameters are given in the text. The relative error between theexact tunneling probability and the asymptotic value of T (t) (see eq.(4.5)) for N = 2,N = 6, and N = 10 is roughly 20%, 4% and 0.5% respectively.

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0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

t

|ψ(0

,t)|2

E=0, pc=0

Exact QMN=2N=4N=6

(a)

0 0.2 0.4 0.6 0.8 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

t

T(t

)

Exact QMN=2N=4N=6

(b)

Figure 4.3: Same as fig.4.2 expect that pc = 0. The relative errors between the the exacttunneling probability and the asymptotic value of T (t) (see eq.(4.5)) for N = 2, N = 4,and N = 6 are roughly 1.5%, 6% and 0.8% respectively.

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relation τ = 2i~ t (eq.(3.24)) into the final result of eq.(4.13) and equating it with eq.(3.26)

yields

a1φ1(x) exp

[−iE1t

~

]= exp

[i

~S(x, t)

], (4.14)

hence,

S(x, t) = −E1t− i~ ln[a1φ1(x)]. (4.15)

Taking the partial derivative with respect to t of the last equation yields

E1 = −∂S(x, t)

∂t, (4.16)

where we recall that this relation holds for τ À τ1/~ =⇒ |t| À τ1. In section 3.2 we

showed that eqs.(3.14)-(3.16) hold for the imaginary time propagator with the substi-

tution of t → t. Hence, from inserting n = 0 into eq.(3.15) and using eq.(4.16) we

obtain

E1 = −∂S

∂t= −dS

dt= −

[i~2m

S2 − 1

2m(S1)

2 − V

], (4.17)

where we took v = 0 (eq.(4.1)). Since eq.(4.17) holds for any choice of x(0) we need to

propagate eqs.(3.14)-(3.16) at only a single position for a sufficiently long (imaginary)

time t to calculate E1.

Liu and Makri have used a Bohmian related formulation, the Trajectory Stability

Method (TSM)[8], to calculate energy eigenvalues [36]. TSM emerges from conventional

Bohmian mechanics by constructing a hierarchy of equations of motion for spatial deriva-

tives of the phase and the amplitude of the wavefunction. In reference [36], Liu and

Makri use TSM for imaginary time propagation at constant-position characteristics, as

we do here. But the modification of TSM for imaginary time propagation is non-unique;

moreover, producing constant-position characteristics is quite an elaborate procedure that

must be repeated at every time step. In contrast, the ZEVCA transformation from the

Schrodinger real time propagator to the imaginary time propagator is unique and the

fixed characteristics are obtained for all time simply by choosing dxdt

= 0. As we demon-

strate, the energy eigenvalues obtained using ZEVCA are of the same accuracy as using

TSM while the formulation is decidedly simpler.

For the sake of comparison, we first consider two of the one-dimensional potentials that

were studied in reference [36]. The first of these is a quartic potential V (x) = 12x2 + x4.

The parameters of the Gaussian initial wavepacket (eq.(4.3)) are αc = 0.5, xc = 1, pc = 0

and m = 1. The fixed trajectory that we propagate is at x(0) = 0. The results are

very robust with respect to the choice of the initial parameters. In fig.4.4 we depict E1

(eq.(4.17)) as a function of τ (t = − i~2τ) for a series of values of the truncation order N .

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The relative error between the exact energy eigenvalue (E1 = 0.8038) and the ZEVCA

approximations are roughly 38%, 4% and 0.6% for N = 2, N = 4 and N = 8 respectively.

For N = 16 the relative error reaches 0.1%. We see a clear convergence to the exact

quantum mechanical result as a function of N .

0 2 4 6 80

0.2

0.4

0.6

0.8

τ

E1

Quartic potential

Exact E1

N=2N=4N=8

Figure 4.4: A comparison between the exact lowest energy eigenvalue of a quarticpotential and the results obtained using a the ZEVCA imaginary time propagator ineq.(4.17) with series of values of the truncation parameter N . The potential function isV (x) = 1

2x2 + x4, where the numerical parameters appear in the text. The relative error

between the exact energy eigenvalue (E1 = 0.8038) and the ZEVCA approximations forN = 2, N = 4 and N = 8 is roughly 38%, 4% and 0.6% respectively.

The second example is a Morse potential V (x) = D[1−exp(−αx)]2. The parameters of

the potential and the mass are D = 0.1745, α = 1.026 and m = 1837/2. These parameters

correspond to the vibration of an H2 molecule. The parameters of the initial Gaussian

wavepacket (eq.(4.3)) are αc = 4.5924, xc = 0.1 and pc = 0. The fixed trajectory is

positioned at x(0) = 0. As in the previous example, the results are robust with respect to

the choice of the initial Gaussian parameters and the position of the fixed trajectory, x(0).

In fig.4.5 we depict E1 as function of τ for a series of values of the truncation order N .

The relative error between the exact energy eigenvalue (E1 = 0.0098565) and the ZEVCA

approximations are roughly 1.4%, 0.08% and 2 · 10−4% for N = 2, N = 4 and N = 6

respectively. In this example we see a faster convergence to the exact result as a function

of N than in the quartic case. The reason is that the parameters of the Morse potential

correspond to a small perturbation from a HO potential. For the HO it is possible to

show that the ZEVCA approximation for N = 2 yields the exact quantum result. For

anharmonic oscillators, the truncation at N = 2 incorporates only Vj[x(0), 0], j = 0, 1, 2

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in eqs.(3.14)-(3.16), which is equivalent to making a harmonic approximation to the

potential. Note that the results obtained for N = 2 for both the quartic potential and

the Morse potential correspond to the energy eigenvalue of the harmonic approximation

to the two potentials respectively. Since the parameters of the Morse potential correspond

to a smaller perturbation from a harmonic oscillator potential than the parameters for

the quartic potential, the convergence with N is faster.

The third example corresponds to a two-dimensional coupled potential of the form:

V (x, y) = 12x2 + 2y2 + x2y4. The initial wavefunction is a Gaussian wavepacket

ψ(x, y, 0) =

(4αxαy

π2

)1/4

exp[−αxx

2 − αyy2], (4.18)

where αx = 0.7, αy = 0.35 and m = 1. The fixed trajectory is positioned at x(0) = y(0) =

0. As in the one-dimensional cases, the results are robust with respect to the choice of the

initial Gaussian parameters and the position of the fixed trajectory. In fig.4.6 we depict

E1 as function of τ for a series of values of the truncation order N . The relative error

between the exact energy eigenvalue (E1 = 1.5682) and the ZEVCA approximations is

roughly 4.3%, 0.3% and 0.032% for N = 2, N = 4 and N = 8 respectively.

4.2.3 Thermal rate constants

Exact quantum thermal rate constants can be expressed as the time integral of the flux-

flux auto correlation function. Miller, Schwartz and Tromp[37] showed that for a symmet-

ric potential centered about x = 0 the flux-flux auto correlation function can be written

as

Cff (t) =

(~

2m

)2∂2

∂x∂x′|〈x′|e− iHt

~ e−βH2 |x〉|2

∣∣∣∣∣x=x′=0

, (4.19)

where H is the Hamiltonian of the system (eq.(3.21)), β = 1/kBT and kB is Boltzmann

constant. The thermal rate constant is given by

k(T ) = Q−1

∫ ∞

0

Cff (t)dt, (4.20)

where Q = (m/2π~2β)1/2 denotes the reactant partition function per unit volume. Denot-

ing Cfs(t) ≡∫ t

0Cff (τ)dτ as the “flux-side” (FS) auto correlation function, the thermal

rate constant equals

k(T ) = Q−1 limt→∞

Cfs(t). (4.21)

The derivation of the flux-side auto correlation function will be the objective of our

numerical calculations.

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0 200 400 600 8009.5

9.6

9.7

9.8

9.9

10

x 10−3

τ

E1

Morse potential

Exact E1

N=2N=4N=6

Figure 4.5: A comparison between the exact lowest energy eigenvalue of a Morse potentialand the results obtained by using the ZEVCA imaginary time propagator in eq.(4.17)with series of values of the truncation parameter N . The Morse potential parameterscorrespond to the vibration of an H2 molecule (the numerical parameters appear in thetext). The plot is a zoom of a complete graph that initiates at τ = 0, E1 ' 4.5 · 10−3.The relative error between the exact energy eigenvalue (E1 = 0.0098565) and the ZEVCAapproximations is roughly 1.4%, 0.08% and 2 · 10−4% for N = 2, N = 4 and N = 6respectively.

Evaluation of the second derivative in eq.(4.19) by finite differences requires two in-

dependent propagations of the combined operator O(t) ≡ e−iHt~ e−

βH2 . This can be seen

by writing this derivative explicitly

∂2

∂x∂x′|〈x′|O(t)|x〉|2

∣∣∣∣x=x′=0

= (4.22)

= lim∆x1→0,∆x2→0

1

∆x1∆x2

(|〈∆x2|O(t)|∆x1〉|2 − |〈0|O(t)|∆x1〉|2 − |〈∆x2|O(t)|0〉2 + |〈0|O(t)|0〉|2

).

One propagation is performed on a spatial eigenstate localized at x = 0, |0〉, and a second

propagation is performed on a spatial eigenstate localized at an arbitrarily small ∆x1,

|∆x1〉. Each propagation is composed of first constructing the “thermal wavefunction”

(applying the imaginary time propagator e−βH2 ) on the initial wavefunction and then

applying the Schrodinger propagator e−iHt~ . During the Schrodinger operation, the wave-

function at x = 0 and at an arbitrarily small x = ∆x2 is required for each of the two

propagations, yielding a total of four quantities (the terms in the brackets of eq.(4.22)).

Note that β here and τ in section 3.2 play the same role as “time” of the imaginary

propagator.

24

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0 5 10 151

1.1

1.2

1.3

1.4

1.5

1.6

τ

E1

2D coupled potential

Exact E1

N=2N=4N=8

Figure 4.6: A comparison between the exact lowest energy eigenvalue of a 2D coupledpotential V (x, y) = 1

2x2 +2y2 +x2y4 and the results obtained by using the ZEVCA imag-

inary time propagator with series of values of the truncation parameter N . The relativeerror between the exact energy eigenvalue (E1 = 1.5682) and the ZEVCA approximationsis roughly 4.3%, 0.3% and 0.032% for N = 2, N = 4 and N = 8 respectively.

The general numerical process we described for calculating the FS auto correlation

function requires the wavefunction only in the close vicinity of the potential maximum at

x = 0 (since ∆x1 → 0, ∆x2 → 0). Hence, the solution of the combined propagator in all

space, as given by conventional grid methods, is unnecessary. The need for only local data

is nicely compatible with the ZEVCA numerical scheme we present here. In the ZEVCA

scheme the propagated wavefunction is obtained in a desired region of coordinate space

using only two trajectories.

The numerical results we obtain for the FS correlation function are compared with

two calculations of the exact result. The first calculation is obtained by applying a

split operator scheme for both parts of the combined operator O(t) and calculating the

correlation function via eq.(4.22). The second calculation yields the asymptotic value of

the FS correlation function by a straightforward numerical integration. It can be shown

that

limt→∞

Cfs(t) = Qk(T ) =1

2π~

∫ ∞

0

[1− ρ(E)]e−βEdE, (4.23)

where ρ(E) is the reflection coefficient from the barrier and on the LHS we applied

eq.(4.21). Since we analyze an Eckart barrier, ρ(E) is given analytically in Eckart’s

original paper[39] leaving only the numerical integration to be preformed. The potential

barrier is a symmetric Eckart barrier given by eq.(4.2) with parameters corresponding to

25

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the H+H2 reaction: m = 1061, D = 0.0156 and β = 1.3624.

Approximation of the combined operator

From eq.(4.22) we see that we need to calculate terms of the form 〈x′′|O(t′)|x′〉. The

spatial eigenstate |x′〉 in position space can be approximated by a properly normalized

Gaussian wavefunction

|x′〉 ≈(α

π

) 12exp[−α(x− x′)2] = ψ(x, 0; x′), (4.24)

where we take the width of the wavefunction to be much smaller than the width of the

potential barrier√

α À b. The wavefunction defines a complex phase field

S(x, 0; x′) = −i~ ln[ψ(x, 0; x′)] = i~α(x− x′)2 − i~2

ln(α

π

), (4.25)

where we have used ansatz (1.1).

The first part of the calculation is to obtain the thermal wavefunction at position

x′′. This is performed by first setting an order of truncation N , and then propagating

eqs.(3.14)-(3.16) on the imaginary time interval t → t ∈ [0, (− i~2)β] with the initial

conditions

Sn(x′′, 0; x′) = −i~∂n

∂xn

[i~α(x− x′)2 − i~

2ln

π

)]∣∣∣∣x=x′′

; n = 0, ..., N, (4.26)

The second part is the application of the Schrodinger propagator by using the results

of the previous step {Sn[x′′, (− i~2)β; x′]}; n = 0, ..., N, as the initial conditions for the

propagation of eqs.(3.14)-(3.16) on the real time interval t ∈ [0, t′]. We use the tilde sign

over the to indicate the imaginary propagation (as in section 3.2). The final result of the

propagation yields

〈x′′|O(t)|x′〉 ≈ exp

[i

~S0(x

′′, t′; x′)]

. (4.27)

Apparently, from eq.(4.22) the ZEVCA formulation requires four propagations to

calculate the FS auto correlation function. Making an additional approximation allows

a reduction to two propagations. Using the set of {Sn(x′′, t′; x′)}; n = 0, ..., N, that are

the byproducts of eqs.(3.14)-(3.16) propagation, we can write a Taylor expansion around

x′′ (similarly to eq.(3.19))

S(x′′ + ∆x′′, t′; x′) ≈n=N∑n=0

Sn(x′′, t′; x′)n!

(∆x′′)n, (4.28)

26

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where ∆x′′ is a small deviation from x′′. Hence,

〈x′′ + ∆x′′|O(t′)|x′〉 ≈ exp

[i

~S0(x

′′ + ∆x′′, t′; x′)]

. (4.29)

Applying this result to eq.(4.22) we note that 〈∆x2|O(t)|0〉 and 〈∆x2|O(t)|∆x1〉 can be

approximated from the propagations for 〈0|O(t)|0〉 and 〈0|O(t)|∆x1〉 respectively. This

observation allows for the reduction of propagated trajectories from four to two.

In fig.4.7 we present the results for FS auto correlation function for temperatures of

T = 1000 and T = 500, where the plots are given as function of the order of truncation

N . For temperatures T . 450 we obtained large deviations from the exact result.

Approximation of solely the Schrodinger propagator

For T . 500 we can obtain results with the correct order of magnitude when applying

ZEVCA to approximate only the Schrodinger propagator. A split-operator propaga-

tor is used to apply the imaginary time operator and derive the thermal wavefunction

ψ[x, (− i~2)β; x′] from the initial Gaussian wavepacket (eq.(4.24)). A phase field corre-

sponding to the thermal wavefunction is derived from ansatz (1.1), S[x, (− i~2)β; x′] =

−i~ ln{ψ[x, (− i~2)β; x′]}. We match a set of phase derivatives {Sn[x′′, (− i~

2)β; x′]}; n =

0, ..., N to grid point x′′, where the derivatives (n ≥ 1) are obtained from a polynomial

expansion of S[x, (− i~2)β; x′] and N relates to the order of truncation. The set of phase

derivatives are the initial conditions for real time ZEVCA propagation. This process is

performed twice for deriving 〈0|O(t)|∆x1〉 and 〈0|O(t)|∆x1〉 out of which 〈∆x2|O(t)|0〉and 〈∆x2|O(t)|∆x1〉 are obtained using the Taylor expansion (eq.(4.28)).

In fig.4.8 we present the results for this approximation. Note that in this case the

results for T = 1000 and T = 500 are less accurate than when the approximation is

applied to the combined operator. The reason is a mutual exclusion of errors of both

parts of the combined operator.

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0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6 T=1000

t(fs)

CF

S(t

)

8 10 12 14 16 18 20 224.6

4.65

4.7

4.75

4.8x 10

−6

Exact QM

Asymptotic

N=2

N=4

N=6

N=8

(a)

ZOOM

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−8 T=500

t(fs)

CF

S(t

)

8 10 12 14 16 18 20 22

3.6

3.7

3.8

3.9x 10

−8

Exact QMAsymptoticN=2N=4N=6

(b)

ZOOM

Figure 4.7: Flux-side auto correlation function as function of truncation order N fortemperatures T = 1000 ((a)) and T=500 ((b)). The results were obtained by calculating

the action of the combined operator O(t) = e−iHt~ e−

βH2 using ZEVCA. The exact quantum

mechanical result (Exact QM) is obtained by using a split operator scheme. Note thevirtually perfect agreement with the exact results for T = 1000 at N = 8.28

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0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6 T=1000

t(fs)

CF

S(t

)

8 10 12 14 16 18 20

4.7

4.75

4.8

4.85x 10

−6

Exact QM

Asymptotic

N=2

N=4

N=5

(a)

ZOOM

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4x 10

−8 T=500

t(fs)

CF

S(t

)

Exact QMAsymptoticN=2N=4N=6

(b)

0 5 10 15 20 250

0.5

1

1.5

2

x 10−10 T=300

t(fs)

CF

S(t

)

Exact QMAsymptoticN=2N=4N=6

(c)

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−12 T=200

t(fs)

CF

S(t

)

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−12 T=200

t(fs)

CF

S(t

)

Exact QMAsymptoticN=2N=4N=6

(d)

Figure 4.8: Flux-side auto correlation function as function of temperature and truncationorder N . Plots (a), (b), (c) and (d) correspond to temperatures of 1000, 500, 300 and200 Kelvin respectively. The results correspond to the approximation of the Schrodinger

propagator e−βH2 by ZEVCA where the Boltzmann operator is applied by a split operator

scheme.

29

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4.3 Summary

We have presented ZEVCA, a new approximation for quantum dynamics calculations

that is a cross between a grid method and a semiclassical method. The formulation

was applied to the calculation of tunneling probabilities, low energy eigenvalues and

thermal rate constants with surprisingly good accuracy. The ZEVCA formulation has

several advantages: (1) The derivation of the formulation is straightforward and the

equations of motion are readily solvable by standard numerical software. (2) The ZEVCA

approximation yields the solution of the TDSE at a fixed and predetermined position.

This allows for easy application to quantum quantities that require only local information.

(3) The ZEVCA formulation requires the calculation of the potential and its derivatives

only at a fixed position. This is in contrast to most semiclassical methods that propagate

trajectories in configuration space and require the calculation of the potential (or its

derivatives) at each time step. (4) No root search is needed as in many other semiclassical

methods. (5) The extension to imaginary time propagation is easily attainable. (6) By

taking N →∞, ZEVCA formally gives the exact quantum result. Although we have not

conducted a rigorous comparison of timings, for the applications presented here ZEVCA

was found to be several orders of magnitude faster than the exact quantum calculation

using a Split Operator method.

Still, the numerical examples reveal a number of limitations. First, the convergence

to the exact result as a function of N seems to be asymptotic, in the sense that there

is an optimal choice of N . Second, the method has difficulty at nodal positions. Sur-

prisingly, this is not a result of the original ansatz (eq.(1.1)) but is a limitation imposed

by the condition of fixed trajectories. Since the trajectories are fixed, clearly they can-

not cross. We demonstrate in chapter 5 that making the ansatz (1.1) but allowing for

contributions from crossing trajectories can produce interference effects such as nodes

and oscillations[38]. Third, in section 5.3.1 present limitations on the relation between

the position of the initial wavefunction, the derivatives of the potential and the choice of

x(0). These limitations originate from the need to incorporate in the initial conditions of

ZEVCA “sufficient” data for a successful propagation.

An alternative to the procedure we have described is to discard the CQHJE and

construct a hierarchy of ODEs for the wavefunction ψn[x(0), t] itself, instead of for the

complex phase, Sn[x(0), t]. We have explored this direction but found that it produces

very poor results. In the case of an initial Gaussian wavepacket it is readily verified

that the truncation for N ≥ 2 does not entail any approximation to the complex phase

derivatives Sn[x(0), 0] (see eqs.(4.4)). This is not the case for the derivatives of the initial

wavefunction ψn[x(0), 0] itself, which go to infinity as a function of N for every choice

30

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of x(0). This observation provides additional justification for making the transformation

ψ → eiS/~ in the first place.

31

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Chapter 5

Method of Bohmian mechanics with

complex action (BOMCA)

5.1 Introduction

The appearance of the classical HJ equation for complex S(~x, t) on the LHS of the CQHJE

suggests a natural choice of the velocity field in eq.(3.10)

~v(~x, t) =~∇S(~x, t)

m. (5.1)

To appreciate more easily this choice (see also Apps. A-C) let us focus on the one-

dimensional instance where v(x, t) = S1

m. We write the four first equations of the set

(3.14)-(3.16) for some arbitrary N > 4

dx

dt=

S1

m, (5.2a)

dS0

dt=

S21

2m− V +

i~2m

S2, (5.2b)

dS1

dt= −V1 +

i~2m

S3, (5.2c)

dS2

dt= −V2 − S2

2

m+

i~2m

S4, (5.2d)

...

where we use the notation of eq.(3.17). The equation of motion for the phase S0

(eq.(5.2b)) has a familiar structure from classical mechanics: we recognize the classi-

cal Lagrangian L =S2

1

2m− V = 1

2mv2 − V and an addition quantum potential term i~

2mS2

(eq.(2.10)). Recognizing dS1

dtas the rate of change of the momentum, we identify in the

32

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RHS of eq.(5.2c) a classical force term, −V1, and a quantum force term, i~2m

S3. Had we

taken the truncation order at N = 2 (S3 = S4 = 0) then the trajectory equations of

motion, eqs.(5.2a) and (5.2c), would be precisely the classical equations of motion; the

only quantum addition would have been the quantum potential in eq.(5.2b). This is what

we meant when we stated in the Introduction that in the BOMCA formulation to the

lowest order QM can be thought of as classical mechanics with an additional geometrical

phase and normalization (coming from the quantum potential) but no additional forces.

In App. C we show that the N = 2 case is also unique in the sense that for an initial

Gaussian wavepacket propagating in up to a quadratic potential, the BOMCA equations

of motion are analytically solvable and yield the exact quantum result.

The simple structure of the equations of motion is a direct result of the complexifi-

cation of the phase in ansatz (1.1) (see section 8.2). However, there is a price to pay for

this complexification: the resulting trajectories propagate in the complex plane. Since

we are interested in the wavefunction on the real axis, we need to find trajectories that

at a given time tf reach the real axis. From these trajectories we can reconstruct the

wavefunction. This is actually not much different from the “root search” problem known

from the semiclassical literature. Though this is a nontrivial drawback we show that the

complexification is also a blessing in disguise. Firstly, complex trajectories can literally

go where real trajectories can not. We show in section 5.3.1 that complex trajectories

with no additional quantum force can “tunnel” through a potential barrier in the deep

tunneling regime and yield surprisingly accurate tunneling probabilities[40]. Secondly,

the crossing of complex trajectories allows us to obtain interference patterns which have

been a major stumbling block of Bohmian-related trajectory methods (section 5.3.3)[38].

The origin of the nodal problem in BM can be traced to the hydrodynamic equations,

eqs.(2.3) and (2.4), which are the first step in the derivation of the Bohmian formulation

(see App. B). The RHS of eq.(2.4) (the quantum potential) can be seen to diverge where

the wavefunction has nodes. Numerically the difficulty is even more severe — well before

a node is formed, when the amplitude of the wavefunction exhibits only nodeless ripples,

the quantum trajectories are highly unstable due to rapid oscillations in the quantum

potential[15].

Since nodes in QM arise from interfering amplitudes, it is only natural to attempt

to solve the nodal problem by applying the superposition principle — to decompose

the wavefunction into two nodeless parts and to propagate each part separately using

trajectories. Promising results based on this idea have been obtained using two different

approaches: the Counter Propagating Wave Method (CPWM)[12] and the Covering

Function Method (CFM)[13]. However, in both these methods the decomposition and

the trajectory propagation are separate steps.

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A somewhat more natural strategy is to let the trajectories themselves determine the

decomposition. Consider the rule, familiar from semiclassical mechanics[25], that if differ-

ent initial conditions lead to the same final position the amplitudes are superposed. Such

a procedure implicitly defines a decomposition of the wavefunction, each component of

which is expected to be nodeless. Hence, one might guess that the difficulties associated

with the quantum potential should disappear without the need for any explicit decom-

position. We show below that this is indeed the case for BOMCA — the trajectories

experience no particular difficulties with the quantum potential at nodes; moreover, su-

perposing the contributions of crossing trajectories yields accurate interference patterns

in close agreement with the exact QM.

There is a subtlety that we have glossed over: if quantum trajectories with different

initial conditions could lead to the same final position it would violate the no-crossing

rule, which formally applies to BOMCA as well as to conventional BM (see App. D).

Indeed, the no-crossing rule is circumvented in our numerical implementation, although it

is not clear if this is a result of the reformulation of the PDE for S in terms of a hierarchy

of ODEs or due to the truncation of this hierarchy. We have found that in practice the

no-crossing rule is also circumvented in conventional BM, leading to the possibility of

applying superposition there as well, but our preliminary tests show that straightforward

superposition of convectional Bohmian trajectories obtained by DPM bears no relation

to the exact QM.

We start in section 5.2 with an explanation of the root search process. Section 5.3

deals with the applications of the BOMCA formulation: tunneling probabilities in a one-

dimensional system (section 5.3.1), tunneling amplitudes in a two-dimensional coupled

potential system (section 5.3.2) and interference patterns via crossing trajectories (section

5.3.3). In section 5.4 we summarize.

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5.2 Root search

Since most of this chapter is devoted to one-dimensional systems we describe the root

search process in the one-dimensional case. To obtain the wavefunction on the real axis

at time tf , in principle we need to propagate eqs.(3.14)-(3.16) with a set of initial po-

sitions {x0j} such that {xj(tf ; x0j

)} ∈ R at a specified time tf . Note that the x0js are

parameters of the trajectories. Tracing back the initial positions from the final positions

resembles the computationally expensive “root search” problem familiar from the semi-

classical literature. In semiclassical formulations one usually has fixed initial positions

and fixed final positions with the initial momentum unspecified. In the BOMCA case the

initial momentum is specified by the initial position (eq.(3.18)), hence the freedom is the

complex initial position where the final position is specified and real. Suppose the initial

positions are restricted to a region where ∂∂x0

x(tf ; x0) 6= 0 and that the mapping of initial

positions x0 to final positions x(tf ; x0) is an analytic function. Then the inverse mapping

x 7→ x0(tf ; x) is also analytic. Consequently we can write

x0(tf , xb) = x0(tf ; xa) +

∫ xb

xa

∂x0(tf ; x′)

∂x′dx′, (5.3)

where we are free to choose the final positions xa, xb and the integration contour. For

simplicity, suppose we have found an initial condition x0 such that xa = x(tf ; x0) ∈ R.

Varying xb ∈ R and choosing the integration contour to be the real interval between xa

and xb, we obtain from eq.(5.3) a curve x0(tf ; xb) of initial conditions that reach the real

axis at tf . We refer to this curve as a branch. This can be translated to a numerical

scheme for generating initial positions that map to the vicinity of the real axis, by writing

an iterative discrete equation based on eq.(5.3)

x0j+1= x0j

+δx0j

δxj

∆x′; j ≥ 1 (5.4)

where we define δx0j≡ x0j

− x0j−1and δxj ≡ xj(tf ; x0j

)− xj−1(tf ; x0j−1). ∆x′ is a small

step along the real axis. Since the final positions are not exactly on the real axis an

extrapolation process is used to extract the complex phase along the real axis from the

action values at the set of final positions {xj(tf ; x0j)}.

35

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5.3 Applications

5.3.1 Tunneling probabilities in 1D

As a first numerical example we consider the one-dimensional scattering of an initial

Gaussian wavepacket (eq.(4.3) from an Eckart potential (eq.(4.2)). We take xc = −.7,

αc = 30π, D = 40, β = 4.32 and m = 30. In fig.5.1 we depict several complex quantum

trajectories for the case of translational energy E = p2c/2m = 0, for N = 2. We remind

the reader that the one-dimensional N = 2 BOMCA equations of motion are obtained

by inserting N = 2 into eqs.(3.14)-(3.16). Note that the complex values of x and p = S1

allow the trajectories to “tunnel” through the barrier centered at x = 0. In fig.5.2

(a) we compare the exact wavefunction at t = 0.85, E = 50 with the BOMCA results

for truncation at N = 2, ..., 5. Note that the transmitted part of the wavefunction is

nearly converged for N = 2, suggesting that BOMCA will be very efficient for calculating

tunneling probabilities. In fig.5.2(b) we consider the case of extremely deep tunneling —

E = 0 — and focus on the transmitted part of the wavefunction. The method converges

uniformly, and even for truncation at N = 2 where there is no quantum force, the

agreement with the exact results is excellent. For N = 3 there is already a quantum force

correction to the equations of motion of the trajectories.

The asymptotic tunneling probability T (E) is calculated by integrating the absolute

square of the wavefunction for x > 0 at a sufficiently long time, t = 1 (eq.(4.5)). In

figs. 5.3(a) and 5.3(b) we compare the exact tunneling probabilities as a function of E

with the results obtained from BOMCA and conventional BM. The exact results were

computed by a split operator wavepacket propagation. The BOMCA results were cal-

culated by propagating 50 complex quantum trajectories. The conventional BM results

were calculated using the numerical formulation developed by Lopreore and Wyatt[3].

The BOMCA formulation allowed the exploration of tunneling over the whole energy

range, while the conventional BM formulation proved unstable at low energies (E . 4).

Moreover, the BOMCA results are significantly more accurate than the BM results for

all energies below the barrier height (E < D), even using just the classical equations of

motion (N = 2). Note the improvement in the accuracy of the BOMCA results as N

increases, suggesting convergence to the exact quantum result.

36

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−1 −0.5 0 0.5 1 1.5 2−0.45

−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

Real(x)

Imag

(x)

Initial trajectoryConsequent trajectoriesInitial positionFinal position

Figure 5.1: Several complex quantum trajectories for the scattering of a Gaussian froman Eckart barrier centered around x = 0 where E = p2

c/m = 0, t = 1 and N = 2. Thetrajectories were obtained through the numerical scheme described in the text (eq.(5.4)).Note that the trajectories initiate at <(x0) ' −.7 = xc and reach <[xf (x0; tf )] > 0,=[xf (x0; tf )] ' 0. Hence, these trajectories “tunnel” through the barrier.

37

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−0.5 0 0.5 1 1.50

0.5

1

1.5

Real(x)

|ψ|2

N=2N=3N=4N=5Exact

(a)

0 0.2 0.4 0.6 0.8 1 1.20

1

2

3

4

5x 10−7

Real(x)

|ψ|2

N=2N=3N=4N=5N=6Exact

0 0.1 0.2 0.3 0.4

4.2

4.4

4.6

x 10−7

(b)

Zoom

Figure 5.2: Exact wavefunction vs. BOMCA reconstructed wavefunction for the scatter-ing of a Gaussian from an Eckart barrier. Plot (a) corresponds to t = 0.85 and E = 50.Plot (b) focuses on the transmitted part of the wavefunction for the case of extremelydeep tunneling — t = 1 and E = 0. Note the convergence to the exact wavefunction asN increases.

38

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0 10 20 305 15 25 35 40 4510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

E

T(E

)

ExactConventional BMN=2N=3N=4N=5N=5

4.16 4.17

10−3.5

10−3.47

10−3.44

10−3.41

Barrier Height

(a)

0 20 40 6010 30 50 70 9080

10−4

10−3

10−2

10−1

E

Log(

Rel

ativ

e E

rror

)

Conventional BMN=2N=3N=4N=5N=6

Barrier Height

(b)

Figure 5.3: (a) Comparison between the tunneling probabilities obtained by BOMCA,conventional BM and the exact results. The inset shows an enlargement of the results forE ≈ 4.17, the last point for which the BM formulation was stable; for E . 4 we couldnot obtain stable results from the conventional BM formulation. (b) Log of the relativedivergence from the exact results.

39

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5.3.2 Tunneling amplitudes in 2D

We examine a two-dimensional tunneling process. The initial wavefunction is a Gaussian

wavepacket

ψ(x, y, 0) =

(4αxαy

π2

)1/4

exp

[−αx(x− xc)

2 +i

~pc(x− xc)− αyy

2

](5.5)

where αx = αy = 94.25, xc = −.5, p2c/2m = 155, m = 30 and ~ = 1. The coupled

potential is given by

V (x, y) = (−1

2mω2y2 + b) exp(−βx2), (5.6)

where ω = 2π, b = 225, β = 15.7. In fig.5.4 we plot cross sections of the transmitted part

of the wavefunction (x > 0) at tf = 0.55. The two-dimensional BOMCA equations are

obtained by inserting eq.(5.1) into eq.(3.11) and solving eqs.(3.10)-(3.12), where the ~w’s

are two-dimensional vectors. We see that as in the one-dimensional case, a good approx-

imation is obtained even at truncation order N = 2 where the trajectories obey classical

equations of motion. The mere complexification of the trajectories is sufficient to yield

tunneling effects even though the average translational energy of the initial wavepacket

E = 155 is significantly lower than the barrier height b = 225 (The asymptotic time

tunneling probability is ∼ 3.5%). We have verified that the parameters we use in this

case and in the previous 1D example yield results that are attributed to tunneling effects

and not to classical transmission that originates from high energy components of the

Gaussian wavepacket.

40

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0.5 1 1.50

0.005

0.01

0.015

0.02

0.025

0.03

x

|ψ(x

,y)|

2

Exact QMN=2N=3

y=0

y=1

y=0.5

y=1.5

Figure 5.4: Four cross sections of a transmitted part (x > 0) of a two-dimensionalwavefunction. The system is an initial two-dimensional Gaussian wavepacket scatteredfrom a coupled barrier. The systems parameters are given in the text.

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5.3.3 Interference

To demonstrate the incorporation of interference effects using BOMCA we consider a

one-dimensional scattering of an initial Gaussian wavepacket from an Eckart potential.

The system is identical the one described in section 5.3.1. The average translational

energy of the initial Gaussian is E = p2c/m = 10 < D. We focus on trajectories that

end up at a final time tf = 0.995 with x(tf ) < 0, and thus contribute to the reflected

part of the wavefunction. tf is chosen as sufficiently long for the wavepacket to scatter

from the barrier and interference effects to appear. First we focus on the N = 2 BOMCA

approximation. As we elaborate in section 8.4 the N = 2 BOMCA equations of motion

are closely related to those of GGWPD[27, 28], which also uses complex trajectories (see

also reference [56]). As a result, many of the insights in reference [28] concerning multiple

root trajectories can be carried over to the case of complex BM, although we have found

that the structure and the number of the root branches generally depend on the value of

N , the order of truncation.

In fig.5.5 we plot three branches that contribute to the reflected wavefunction. A

branch is defined as the locus of initial positions of trajectories that end at time tf at real

xf with xf corresponding to a reflected segment of the wavefunction, xf ∈ [−1,−0.05].

Two sample trajectories are depicted emerging from each branch, one that ends at position

xf = −1 and one that ends at xf = −0.05. Thus, to each final position (for this segment of

the reflected wavefunction) there correspond three initial positions — one originating from

each of the three branches — and the wavefunction should therefore include contributions

from all three branches. As in reference [28], at short times only one branch contributes to

the final wavefunction. We refer to this branch as the “real branch” since it incorporates

a trajectory that stays on the real axis at all times (the trajectory that initiates from

xc). At longer times, secondary branches begin to make a significate contribution to the

final wavefunction. There is apparently no fundamental limitation on the number of the

secondary branches, although in the section of the complex plane depicted in fig.5.5 no

other branches were found. In fig.5.5, branch (2) is the real branch while branches (1)

and (3) are secondary branches.

In fig.5.6(a) we plot |ψj(x, tf )| = | exp[iSj(x, tf )/~]|, j = 1, 2, 3, where j corresponds

to each of the three branches. In fig.5.6(b) we present the result of adding pairs of

branches, |ψ| = |ψj + ψi|, i 6= j. We see that to a good approximation the rippled parts

of the wavefunction are described by a simple superposition with a proper choice of i

and j. Note that as we approach the region of the transmitted part of the wavefunction

(x & −0.2) a single branch (branch (1)) provides a good approximation whereas the

contributions of the other branches either diverge or go to zero. In fact, this single

branch is responsible for the transmitted wavefunction (xf > 0, not shown), and hence

42

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−1 −0.8 −0.6 −0.4 −0.2 0−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Real(x)

Imag

(x)

Branch (3)

Branch (2)

Branch (1)

Trajectory

Figure 5.5: Scattering of a Gaussian from an Eckart potential barrier within the N = 2BOMCA approximation (the parameters are given in the text). Three branches and sixsample complex trajectories are depicted. The branches are the locus of initial positions oftrajectories that end at time tf = 0.995 at real xf , −1 < xf < −0.05, and thus contributeto the reflected wavefunction. The figure shows two sample trajectories emerging fromeach branch, one that ends at xf = −1 and one that ends at xf = −0.05.

we will call this the “transmitted branch”. At energies on the order of magnitude of the

barrier height, the “transmitted branch” is the real branch, but at lower energies (and

longer time scales) there is a crossover and the transmitted branch is one of the secondary

branches. The existence of a single transmitted branch coincides well with the nodeless

character of the transmitted part of the wavefunction. By the same token, in section

5.3.1 we showed that the reflected part of the wavefunction can be well approximated by

a single branch if it has no ripples or nodes.

The issue of which branches should be included in the sum and when, is obviously

of central importance to the method; at present, we do not have a rigorous justification

for the neglect of certain branches. A partial discussion is given in reference [28] in the

context of GGWPD, but the BOMCA formulation, being more general, requires a more

comprehensive discussion.

We now turn to the next order of approximation, N = 3. The equations of motion

for N = 3 include a quantum force correction. At this level of truncation four branches

contribute to the reflected wavefunction at tf . Figure 5.7(a) depicts the contribution

of the four branches, |ψj|, j = 1, .., 4. The result of adding pairs of branches |ψ| =

43

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−0.8 −0.6 −0.4 −0.20

0.5

1

1.5

2

2.5

3

X

|ψ|

N=2

Exact QM|ψ

1|

|ψ2|

|ψ3|

(a)

−0.8 −0.6 −0.4 −0.20

0.5

1

1.5

2

2.5

3

3.5

X

|ψ|

N=2

Exact QM|ψ

1+ψ

2|

|ψ2+ψ

3|

(b)

Figure 5.6: (a) N = 2 BOMCA approximation corresponding to each of the threebranches j = 1, 2, 3 (see fig.5.5), |ψj(x, tf )| = | exp[iSj(x, tf )/~]|. In the region of thetransmitted wavefunction (starting from x ≈ −0.2), a good approximation to the exactresult is obtained by just a single branch (branch (1)). (b) The result of adding contri-butions from pairs of branches |ψ| = |ψj + ψi|, i 6= j. Different choices of i and j areuseful in different regions of the wavefunction.

44

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−0.8 −0.6 −0.4 −0.20

0.5

1

1.5

2

2.5

3

X

|ψ|

N=3

Exact QM|ψ

1|

|ψ2|

|ψ3|

|ψ4|

(a)

−0.8 −0.6 −0.4 −0.20

0.5

1

1.5

2

2.5

3

3.5

X

|ψ|

N=3

Exact QM|ψ

1+ψ

4|

|ψ2+ψ

3|

(b)

Figure 5.7: (a) N = 3 BOMCA approximation corresponding to four branches|ψj(x, tf )| = | exp[iSj(x, tf )/~]|, j = 1, ..., 4. (b) The result of adding contributions frompairs of branches |ψ| = |ψj + ψi|, i 6= j. Except for a slight decrease in accuracy leftto the maximum, the results for N = 3 where a quantum force term is added are betterthan for N = 2 (complex classical trajectories).

45

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|ψj + ψi|, i 6= j is given in fig.5.7(b). There is a significant increase in accuracy relative

to N = 2 in the vicinity of the maximum and to its right, although there is a slight

decrease in accuracy relative to N = 2 to the left of the maximum.

5.4 Summary

We presented BOMCA, a novel formulation of QM inspired by conventional BM. This

formulation yields simpler equations than conventional BM at the expense of complex

trajectories (see also section 8.2). Moreover, BOMCA allows a direct and simple deriva-

tion of uncoupled trajectories that may be used to reconstruct the wavefunction. The

tunneling amplitudes obtained by BOMCA in one and two dimensions were in excellent

agreement with the exact results even in the deep tunneling regime. We showed that even

classical equations of motion are sufficient to obtain very accurate results provided that

an extra, nonclassical term is added to the action integral. We also have demonstrated

that BOMCA accounts for interference and nodal structures of wavefunctions in a simple

and natural way.

To summarize, the BOMCA formulation presents four central contributions: 1) to

exploit the fact that the quantum force in the CQHJE is much smaller than in the

RQHJE (eq.(2.4)); 2) to solve the CQHJE using complex trajectories, 3) to provide ex-

plicit equations of motion for the complex trajectories, equations in which there is strictly

no communication between trajectories and 4) to demonstrate how to obtain interference

effects using crossing trajectories. The result is a remarkable new approach which in

principle is an exact formulation of quantum mechanics using complex trajectories that

do not communicate with each other. In chapter 8 we elaborate on the relationships and

benefits of BOMCA in comparison with other methods.

In spite of the conceptual difficulties that crossing trajectories may pose for BM as

an interpretational tool of QM, this notion introduces a powerful numerical tool and

might even enrich the orthodox interpretation of the Bohmian formulation (a discussion

on trajectory crossing appears in App. D). These results demonstrates great promise

for BOMCA as a versatile alternative to current semiclassical methods. However, several

issues still require a more comprehensive understanding. First, what are the convergence

properties of the method as higher order approximations are taken to the quantum force,

that is increasing the value of N? Can rigorous rules be derived for the summation of the

different branches? What is the relation between the exact phase that diverges at a node

and the approximate BOMCA formulations that can account for nodes via a bipolar or

multipolar expansion?

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Chapter 6

Interference with real trajectories

As mentioned in the Introduction, the nodal problem is currently one of the main obstacles

to performing numerical calculations using BM. In this chapter, we present preliminary

results showing that an oscillatory wavefunction in close agreement with the quantum

result can be obtained using eqs.(3.14)-(3.16) with real trajectories. We refer to the pro-

cess described below as real trajectory version of BOMCA. Consider an initial Gaussian

wavepacket, eq.(4.3), propagating in a Morse potential

V (x) = A{[1− exp(−βx)]2 − 1

}. (6.1)

The parameters of the initial Gaussian are αc = 0.5, xc = 9.342 and pc = 0 where we

take m = ~ = 1. As for the Morse potential parameters, A = 10.25 and β = 0.2209.

The final propagation time, tf = 5.93, is chosen so as to produce a strongly oscillating

pattern. Figure 6.1 depicts the potential and wavefunctions.

In BOMCA the velocity is taken to be a function of S1, v = S1

m. We now present an

alternative procedure; we take v to be completely independent of S1 and set a classical

equation of motion for v:dv

dt= −V1[x(t)]

m. (6.2)

Equation (6.2) is then supplemented to the set of eqs.(3.14)-(3.16) where we set N = 2.

Furthermore, we extend the freedom in the choice of the velocity field to include the

initial conditions for the velocity, which we take to be

v[x(0), 0] =={S1[x(0), 0]}

m=

2αc

m[x(0)− xc]; (6.3)

S1[x(0), 0] is obtained by inserting eq.(4.3) into eq.(3.18) and setting n = 1. Equation

(6.3) defines real initial conditions; taken together with the equations of motion this yields

real classical trajectories (note that if we had taken the initial velocity to be complex,

47

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−5 0 5 10 150

0.2

0.4

0.6

0.8

x

|ψ|

|ψ(x,0)||ψ(x,t

f)|

Potential

Figure 6.1: An initial Gaussian wavepacket propagating in a Morse potential. The pa-rameters of the system are given in the text. The final propagation time tf equals roughlyhalf of the oscillation period of a classical particle located at the center of the initial Gaus-sian, xc. The exact wavefunction was calculated using the split operator method with aFourier basis.

v[x(0), 0] = S1[x(0),0]m

, we would have obtained the N = 2 BOMCA equations. The next

step is to solve the equations of motion for a series of initial positions {x(0)} in the

vicinity of xc.

In fig.6.2 we plot the trajectories obtained. The trajectories can be divided into two

overlapping groups that we referred to as branches. The first (reflected) branch (marked

as solid lines) is the locus of trajectories that have reached their classical turning point

and were reflected by the exponential barrier of the potential. The second branch (marked

as dashed lines) is the locus of trajectories that by tf did not reach their classical turning

point. Thus, to an arbitrary final position x (x & −2.8) these correspond two initial

positions and two associated trajectories.

The question arises, if two trajectories and therefore two values of S0(x, tf ) correspond

to each final position x, how should we determine the wavefunction ψ(x, tf )? The clear

distinction between the two branches allows us to associate a wavefunction with each

branch, which we will call ψ1(x, tf ) and ψ2(x, tf ). We apply the superposition principle

to reconstruct the wavefunction at x, obtaining

ψ(x, tf ) = ψ1(x, tf ) + ψ2(x, tf ). (6.4)

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−5 0 5 10 15 200

1

2

3

4

5

6

x

t

t f

Figure 6.2: Real classical trajectories obtained by solving Newton’s second law of motion(eq.(6.2)) for a Morse potential (eq.(6.1)) with initial conditions (6.3). The parameters ofthe propagation are given in the text. The final propagation time tf is marked explicitly.The trajectories can be divided into two overlapping branches. The first branch (markedas solid lines) is the locus of trajectories that have reached their classical turning pointand were reflected by the exponential barrier of the potential. The second branch (markedas dashed lines) is the locus of trajectories that by tf did not reach their classical turningpoint.

In fig.6.3 we compare the exact wavefunction along with |ψ1(x, tf )|, |ψ2(x, tf )| and |ψ(x, tf )| =|ψ1(x, tf ) + ψ2(x, tf )|. For values of x larger than ≈ −2.8 (the position of the maximum

of |ψ(x, tf )|), the superposition yields a surprisingly accurate approximation of the os-

cillating wavefunction: even though the wavefunctions ψ1(x, tf ) and ψ2(x, tf ) exhibit no

oscillations whatsoever, their superposition yields strong oscillations and a node near the

maximum. Note that ψ2(x, tf ) provides the “main” contribution to the final wavefunc-

tion; ψ1(x, tf ), which originates from the reflected branch, contributes to ψ(x, tf ) only

where the wavefunction oscillates. As the classical turning point is approached, the am-

plitudes arising from the different branches diverge and the superposition of contributions

appears to have no physical significance.

As we elaborate in section 8.4, the BOMCA equations for N = 2 are identical to those

of Generalized Gaussian Wavepacket Dynamics (GGWPD)[27, 28]. In both cases, the

trajectories that are propagated obey classical equations of motion but are complex. In

this chapter we described a modification of BOMCA in which the classical trajectories are

taken to be real. It is interesting to speculate if there might be a connection with thawed

49

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−5 0 5 100

0.2

0.4

0.6

0.8

x

|ψ|

|ψExact

|

|ψ1|

|ψ2|

|ψ1+ψ

2|

Figure 6.3: A comparison between the exact wavefunction with the contribution ofthe first branch |ψ1(x, tf )|, the second branch |ψ2(x, tf )| and a superposition of both|ψ(x, tf )| = |ψ1(x, tf ) + ψ2(x, tf )|. For x & −2.8 (the position of the maximum of theexact wavefunction) the superposition procedure yields a surprisingly accurate approx-imation of the oscillating wavefunction: even though the wavefunctions |ψ1(x, tf )| and|ψ2(x, tf )| exhibit no oscillations whatsoever. The procedure is undefined in the classicallyforbidden region which the classical trajectories cannot access. The exact wavefunctionwas calculated using the split operator method with a Fourier basis.

Gaussian propagation[34], in which the equations of motion are the same as those for

GGWPD, but the trajectories are real. Consistent with this conjecture is the observation

that in thawed Gaussian propagation there is no need for a root search, as is true in the

real-trajectory version of BOMCA. However, the correspondence cannot be exact. In the

real-trajectory version of BOMCA presented here, in principle every point in coordinate

space is propagated, whereas in thawed Gaussian propagation the initial wavefunction is

decomposed into Gaussians and any decomposition that satisfies completeness is allowed.

Moreover, in real-trajectory BOMCA the trajectories have no width and no functional

form whereas in thawed Gaussian propagation there is always some residual signature

of the Gaussian decomposition. For example, Gaussians that reach the turning point

have part of their amplitude extending into the classically forbidden region, whereas in

real-trajectory BOMCA such penetration into the classically forbidden region is absent.

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Chapter 7

Method of complex time-dependent

WKB

7.1 Introduction

The WKB method[1] can be considered as the first of the semiclassical methods. Its date

of birth almost coincides with the publication of the Schrodinger equation in 1926, and

virtually every standard text book in QM has a description of the method. The basic

idea of the WKB method is to recast the wavefunction as the exponential of a function

and expand the function as a power series in ~. The WKB method is ordinarily applied

to the time-independent Schrodinger equation and provides for a good approximation

to the eigenstates as long as one is not too near a classical turning point. It is only

natural that as part of the effort to develop time-dependent semiclassical methods, a

time-dependent version of the WKB method would be explored. Surprisingly little work

has been done in this direction[20, 23, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. A decade

ago, Boiron and Lombardi[26] developed a complex trajectory version of time-dependent

WKB (TDWKB), which we refer to as CTDWKB. In conventional WKB the leading

order term in the phase of the wave function is taken to be O(~−1) and the leading order

term in the amplitude is taken to be O(~0). In contrast, the CTDWKB formulation

treats the amplitude and phase on an equal footing. The price to pay for this procedure

is that the resulting classical trajectories propagate in the complex space.

The CTDWKB equations of motion can be solved analytically and yield the exact

wavefunction for an initial Gaussian wavepacket in a potential with up to quadratic

terms[26]. The first-order method was tested numerically by Boiron and Lombardi for

scattering of a Gaussian wavepacket from a potential barrier. They showed that the

method produced very good results as long as the wavefunction did not exhibit interfer-

ence effects in the form of oscillations or nodes. Here we present a simple modification to

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CTDWKB that provides an accurate description of oscillations in the wavefunction. We

show that complex classical trajectories, similar to real classical trajectories, can cross

in configuration space. By superposing the contributions from two or more crossing tra-

jectories, interference effects are obtained. We take CTDWKB a step further in another

direction by showing that the approximation generally improves when incorporating ad-

ditional terms in the series expansion. Since the WKB expansion is an asymptotic series,

this observation is non-trivial.

In section 7.2 we formulate the CTDWKB method. Our derivation is more compact

than the Boiron-Lombardi derivation and demonstrates how to obtain the equations of

motion for higher orders of the expansion in a simple manner. In Section 7.3 we apply

the formulation to a Gaussian initial wavepacket propagating in a quartic double-well

potential. We demonstrate that superposing the contributions of crossing trajectories

leads to interference effects and that incorporating higher order terms in the expansion

improves the approximation. Section 7.4 contains a summary and concluding remarks.

Following Boiron and Lombardi we will refer to the CTDWKB method in the body of

the chapter as the Complex Trajectory Method (CTM) for short.

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7.2 Formulation

7.2.1 Time-independent vs. Time-dependent WKB

For simplicity we present the one-dimensional version of the CTM derivation. The gen-

eralization to multi-dimensions can be performed in a straightforward manner. The

conventional WKB derivation begins by inserting the ansatz

ψ(x) = exp

[i

~S(x)

], (7.1)

into the time-independent Schrodinger equation. The result is

1

2m

(dS

dx

)2

+ V (x)− i~2m

d2S

dx2= E, (7.2)

where E is an energy eigenvalue. If we assume that S(x) can be expanded asymptotically

as a power series in ~

S(x) = S0(x) + ~S1(x) + ~2S2(x) + ... =∞∑

j=0

~jSj(x), (7.3)

then, by substituting the last equation into eq.(7.2) and equating powers of ~, a series of

coupled ODEs are obtained for the Sj’s.

The derivation of the TDWKB approximation starts with the CQHJE which we write

as

∂tS +1

2m(∂xS)2 + V =

i~2m

∂xxS, (7.4)

where we use in this chapter the notation ∂t ≡ ∂∂t

and ∂x ≡ ∂∂x

. Inserting into eq.(7.4) a

time-dependent version of eq.(7.3)

S(x, t) =∞∑

j=0

~jSj(x, t), (7.5)

yields∞∑

j=0

~j∂tSj +1

2m

∞∑j1,j2=0

~j1+j2∂xSj1∂xSj2 + V =i

2m

∞∑j=0

~j+1∂xxSj. (7.6)

Note that the Sjs do not stand for partial derivative of the phase as was the case in the

previous chapters when we used the notation of eq.(3.17). By equating terms having the

53

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same powers of ~ we obtain the classical HJ equation for S0(x, t)

∂tS0 +1

2m(∂xS0)

2 + V = 0, (7.7)

and equations of motion for Sn(x, t), n ≥ 1

∂tSn +∂xS0

m∂xSn =

i

2m∂xxSn−1 − 1

2m

n−1∑j=1

∂xSj · ∂xSn−j. (7.8)

Conveniently, each equation depends only on lower order terms (in contrast to the uni-

fied derivation equations of motion). The next step in TDWKB is to convert eqs.(7.7)

and (7.8) into a set of ODEs by looking at the evolution of S0, S1, . . . along classical

trajectories.

7.2.2 Integrating along classical trajectories

As we mentioned earlier, the first term in the ~ power expansion, S0, obeys the classical HJ

equation (eq.(7.7)). The solution of the classical HJ equation along classical trajectories

is presented in App. A. Here we present the final result

dx

dt=

∂xS0

m, (7.9)

d(∂xS0)

dt= −∂xV, (7.10)

dS0

dt=

1

2m(∂xS0)

2 − V. (7.11)

Noting that ∂xS0 is the momentum one easily recognizes Newton’s second law of motion as

eqs.(7.9) and (7.10), and the classical equation of motion for the action, S0 (eq.(7.11)). In

summary, equations (7.9)-(7.11) are a closed set of equations that yield S0 along classical

trajectories.

We turn to the higher order terms in the series Sn, n ≥ 1. Our aim is to obtain

equations of motion for these terms along trajectories defined by eq.(7.9). Recognizing

the LHS of eq.(7.8) as the Lagrangian time derivative of Sn, we can write

dSn

dt=

i

2m∂xxSn−1 − 1

2m

n−1∑j=1

∂xSj · ∂xSn−j. (7.12)

These equations are not a closed set of ODEs since they depend on partial derivatives

such as ∂xxSn−1. We close the set of equations by deriving equations of motion for the

partial derivatives on the RHS of eq.(7.12) (∂xxSn−1 and ∂xSj, j = 1, ..., n − 1). We

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demonstrate the process by deriving equations of motion for S1 and S2. Inserting n = 1

into eq.(7.12) yieldsdS1

dt=

i

2m∂xxS0. (7.13)

An equation of motion for ∂xxS0 is obtained by taking a second spatial partial derivative

of eq.(7.7),

∂xxtS0 +1

m

[∂xS0 · ∂xxxS0 + (∂xxS0)

2]+ ∂xxV = 0, (7.14)

and rewriting it asd(∂xxS0)

dt= − 1

m(∂xxS0)

2 − ∂xxV. (7.15)

This equation is derived in reference [26] by a cumbersome finite difference scheme. Note

that an equation of motion for any order of spatial derivatives of S0 can be derived in a

similar fashion by taking consecutive spatial derivatives of eq.(7.14) and then grouping

together the Lagrangian time derivative terms. Equations (7.13) and (7.15) provide a

closed set of equations of motion for S1.

Inserting n = 2 into eq.(7.12) yields

dS2

dt=

i

2m∂xxS1 − 1

2m(∂xS1)

2. (7.16)

The equations of motion for ∂xS1 and ∂xxS1 are obtained by first inserting n = 1 in

eq.(7.8). We then derive two equations by taking a first and a second spatial partial

derivative of the result. By grouping the Lagrangian time derivatives of ∂xS1 and ∂xxS1

in each of the two equations separately we obtain

d(∂xS1)

dt=

i

2m∂xxxS0 − 1

m∂xS1 · ∂xxS0, (7.17)

d(∂xxS1)

dt=

i

2m∂xxxxS0 − 1

m∂xS1 · ∂xxxS0 − 2

m∂xxS1 · ∂xxS0. (7.18)

The last equations depend in turn on ∂xxxS0 and ∂xxxxS0. As mentioned earlier, the

equation of motion for these terms can be obtained by additional spatial derivatives of

eq.(7.14), a process that yields

d(∂xxxS0)

dt= − 3

m∂xxS0 · ∂xxxS0 − ∂xxxV, (7.19)

d(∂xxxxS0)

dt= − 1

m

[4∂xxS0 · ∂xxxxS0 − 3(∂xxxS0)

2]− ∂xxxxV. (7.20)

Equations (7.15) and (7.16)-(7.20) provide a closed set of equations of motion needed for

S2. The scheme we described for S1 and S2 can be extended to any of the higher order

terms in the expansion. Note that incorporating higher order terms Sn into the TDWKB

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approximation does not affect the classical trajectories associated with S0, defined by

eqs.(7.9) and (7.10). We now turn to the source of the distinction between conventional

TDWKB and CTM.

7.2.3 Initial conditions and complex classical trajectories

In conventional TDWKB the initial wavefunction is “divided” between S0(x, 0) and

S1(x, 0)

ψ(x, 0) = A(x) exp[iφ(x)] = exp

[i

~S0(x, 0) + S1(x, 0)

], (7.21)

where A(x) and φ(x) are the initial amplitude and phase respectively, both taken to be

real. The phase is related to the zero-order term S0 and the amplitude to the first-order

correction term S1 according to

S0(x, 0) = ~φ(x), S1(x, 0) = −i ln[A(x)], (7.22)

and Sn(x, 0) = 0 for n ≥ 2. Note that the initial conditions specified by eqs.(7.22) yield

classical trajectories that propagate on the real axis since S0 and its spatial derivatives are

real quantities (see eqs.(7.9) and (7.10)). In contrast, in CTM the amplitude and phase

are treated on an equal footing with far-reaching consequences. The initial wavefunction

is specified solely by S0(x, 0)

S0(x, 0) = −i~ ln[ψ(x, 0)], Sn(x, 0) = 0, n ≥ 1. (7.23)

Since S0 is generally complex and since the initial velocity v(x, 0) ≡ ∂xS0(x, 0)/m, the

trajectories propagate in the complex plane even if the initial positions are on the real

axis (=[x(0)] = 0). This observation requires us to look at the analytic continuation of

the wavefunction in the complex plane and find ways to extract the wavefunction on the

real axis. In section 8.3 we argue that the N = 2 BOMCA equations of motion yield the

same complex classical trajectories as CTM (with the same initial conditions). Hence,

we use the root search process of section 5.2 in the CTM formulation as well.

7.2.4 Complex root search and superposition

One of the benefits of conventional TDWKB and CTM compared with BOMCA, is that

the trajectories obey the classical equations of motion and are independent of the order of

the phase expansion we incorporate in the final wavefunction. But the fact still remains

that for an arbitrary initial position x(0) ∈ C and an arbitrary final propagation time tf

the final position x(tf ) is complex and yields an “analytically continued” wavefunction

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at x(tf )

ψ[x(tf ), tf ] ≈ exp

{i

~

N∑j=0

~jSj[x(tf ), tf ]

}, (7.24)

where the non-negative integer N is the order of the approximation. Section 5.2 and

references [26, 28, 40] include discussions of root search algorithms for the derivation

of initial positions that reach the real axis at a given time. We will just remind the

reader that the complex root search we employ exploits the assumption that the mapping

x(0) 7→ x(tf ) is analytic. This property allows for an iterative process that detects the

initial positions that correspond to real final positions. However, as demonstrated in

references [27, 28, 38, 41] and in section 7.3.1, for an arbitrary potential and final time,

the mapping is not one-to-one. Generally, more than one initial position ends at a final

position (whether real or complex). This makes the search for trajectories that end on

the real axis more complicated but it has an important advantage in terms of interference

effects.

Our main observation is that the contribution of multiple trajectories in CTM can

accumulate to an interference pattern (Similarly the the observation in section 5.3.3). For

simplicity we make the following assumption. Suppose that L trajectories end at final

time tf on real position x(tf ) and that the final wavefunction can be approximated by a

superposition of contributions

ψ[x(tf ), tf ] ≈L∑

l=1

exp

{i

~Sl [x(tf ), tf ]

}, (7.25)

where each trajectory (denoted by the index l) is associated with a phase Sl[x(tf ), tf ]

Sl[x(tf ), tf ] =N∑

j=0

~jSlj[x(tf ), tf ], (7.26)

that is calculated by the CTM equations of motion. In section 7.3 we show that this

assumption is too simplified and does not hold at all times and all positions. For example,

for positions associated with a tunneling part of the wavefunction, only one of the multiple

trajectories should be taken into account. A partial discussion on the superposition of

contributions from complex trajectories appears in reference [28] in the GGWPD context.

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7.3 Numerical Results

In this section we examine numerically the CTM formulation when allowing for the

superposition of complex trajectories. For ready comparison the physical system we

choose is identical to the one studied by Boiron and Lombardi (reference [26] section

IVB). The potential considered is a quartic double-well

V (x) = 1.25× 10−4(x4 − 400x2). (7.27)

The initial wavefunction is a Gaussian wavepacket (eq.(4.3)) where αc = 1, xc = 0, pc = 5,

and we take m = ~ = 1. The initial conditions for the terms in the ~ power-expansion of

the phase are

S0(x, 0) = iαc~(x− xc)2 + pc(x− xc) + γc = ix2 + 5x + γc, (7.28)

∂xS0(x, 0) = 2iαc~(x− xc) + pc = 2ix + 5, (7.29)

∂xxS0(x, 0) = 2iαc~ = 2i, (7.30)

∂jxS0(x, 0) = 0, j ≥ 3, (7.31)

∂jxSk(x, 0) = 0, j ≥ 0, k ≥ 1, (7.32)

where ∂jxSk ≡ ∂jSk

∂xj .

In section 7.3.1 we analyze the first order approximation of CTM (N = 1, S =

S0 + ~S1) and the properties of the trajectories. Section 7.3.2 is dedicated to the next

order of the approximation (N = 2, S = S0 +~S1 +~2S2). We omit an analysis of N = 0

since it is well presented in reference [26] and yields poor results.

7.3.1 First Order approximation, N = 1

The first order approximation of CTM requires the solution of eqs.(7.9)-(7.11), (7.13)

and (7.15). The first two equations define the complex classical trajectories and the next

three equations yield S0 and S1. We start by analyzing the complex classical trajectories.

As mentioned above, the mapping x(0) 7→ x(tf ) is not one-to-one. In section 8.3 we show

that the N = 1 CTM equations and N = 2 BOMCA equations are identical. Hence, it

is not a surprise that in this case we also found more than one branch, that is, locus of

initial positions that end up at the same final (real) position. In figures 7.1(a) and 7.1(b)

we plot complex classical trajectories for tf = 3 and tf = 6 respectively. The initial

positions of the trajectories can be divided into three branches, the real branch and two

secondary branches. The real branch is characterized by the property that it includes

the initial position of a trajectory that propagates solely on the real axis. We refer to

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this trajectory as the real trajectory. It can be readily verified that for a Gaussian initial

wavefunction there is only a single real trajectory that initiates at x(0) = xc (see eqs.(7.9)

and (7.10) together with eq.(7.29)). In fig.7.1(b) we depict the real trajectory explicitly.

The secondary branches are defined simply as the groups of initial positions that do not

belong to the real branch. Generally, the branches might be infinitely long curves in the

complex plane. We use the term branches to refer to the locus of initial positions that

leads to final positions where the wavefunction is significantly different from zero. Hence,

the branches are curves of finite length in the complex plane, although clearly there is

some arbitrariness to their length.

In fig.7.1(a) we see that at short time scales the secondary branches are centered far

from the neighborhood of the real axis. We can show analytically that for small times

tf the initial positions that comprise the real branch obey |x(0)| = O(tf ) whereas the

secondary branches obey |x(0)| = O( 1tf

). Note that the linear dependence of the initial

momentum on position (eq.(7.29)) allows trajectories with initial positions far from the

real axis to reach a real final position in a short time. Unlike the secondary branches, the

real branch is centered in the vicinity of the real axis at all times. The initial position

x(0) = xc is a fixed point of the real branch and prevents the real branch (recall that this

is the locus of initial positions) from “straying” from the neighborhood of the real axis

as the final time tf is increased. At intermediate times (time scales comparable to the

time of the collision of the wavefunction with the barrier, 4 . tf . 7) secondary branch

(1) reaches the vicinity of the real axis (fig.7.1(b)) and at longer time scales it continues

in the direction of the positive imaginary axis. As we demonstrate below, the proximity

of secondary branch (1) to the real axis is closely related to the size of its contribution to

the final form of the wavefunction and its role in interference effects. Secondary branch

(2) does not reach the vicinity of the real axis for any of the time scales specified below.

The contribution of this branch to the absolute value of the final wavefunction (eq.(7.25))

is negligible (of the order of 10−35). Hence, from here on we ignore secondary branch (2)

and refer to secondary branch (1) as the secondary branch.

As we mentioned in section 7.2.4, the existence of more than one branch motivates

the attempt to superpose the contributions of the real branch and secondary branch in

the final wavefunction

ψ[x(tf ), tf ] = ψR + ψS; ψR = exp

(i

~SReal

), ψS = exp

(i

~SSec

), (7.33)

where SReal and ψR are the phase and wavefunction associated with the real branch, and

SSec and ψS correspond to the secondary branch. In figures 7.2(a), 7.2(b) and 7.2(c) we

compare the exact wavefunction with the numerical results obtained by applying CTM

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−60 −40 −20 0 20 40 60−30

−20

−10

0

10

20

30

Real(x)

Imag

(x)

tf=3

Real Branch

Secondary Branch (2)

(a)

Secondary Branch (1)

−5 0 5 10 15 20 25 30−2

−1

0

1

2

Real(x)

Imag

(x)

tf=6

Secondary Branch (1)

Real Branch

(b)

Real Trajectory

Figure 7.1: Complex classical trajectories with initial positions (marked as circles) andreal final positions (marked as pluses) at (a) tf = 3 and (b) tf = 6. The trajectoriesarise from an initial Gaussian wavepacket propagating in a quartic double-well potential.The Gaussian is centered at x = 0 and has positive initial momentum (the physicalparameters are given in the text). In plot (a) we demonstrate that each final positionarises from three initial positions. The initial positions are divided into a real branch andtwo secondary branches. The real branch is defined as incorporating a trajectory thatremains on the real axis at all times. The real trajectory is specifically indicated in plot(b).

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using a two-branch superposition. The figures indicate that when the wavefunction does

not exhibit oscillations, the contribution of the real branch is sufficient to obtain a good

approximation to the wavefunction. But at intermediate times, when the wavefunction

exhibits interference effects, the contribution of both branches must be included. This

last observation applies in the spatial range up to the classical turning point (x ' 24),

beyond which the combined contribution diverges from the exact result.

We turn to a closer inspection of this divergence. In fig.7.3 we plot the contribution

at tf = 6 of each individual branch and their superposition. Starting from the vicinity of

x ' 22, we observe an exponential increase of ψS. For x & 23 we have a discontinuity of

the approximation, as we discard the contribution of the secondary branch and include

just the real branch. A description of this divergence appears in reference [28] in the

context of the GGWPD formulation.

It is interesting to compare the time-dependence of the real and secondary branch

contributions to the final approximation. A qualitative measure of the contribution of

each branch is given by the imaginary part of the phase since

|ψR| =∣∣∣∣exp

(i

~SReal

)∣∣∣∣ = exp

[−=(SReal)

~

], (7.34)

and a similar relation applies for ψS and =(SSec). In figures 7.4(a) and 7.4(b) we plot

=(SReal) and =(SSec) respectively for a series of final times tf . We see that the secondary

branch has a significant magnitude only at intermediate times. This observation coincides

well with the need to include the contribution of the secondary branch to the final wave-

function only at these times. The exponential growth of ψS that is observed in fig.7.3 is

also apparent in fig.7.4(b), in the negative parts of the graphs for tf = 5 and tf = 6. The

divergent magnitude of ψS is in contrast to the finite magnitude of ψR that is observed

in fig.7.4(a). A discontinuity in the derivative of =(SR) at tf = 5 and tf = 6 is also

observed. This discontinuity appears slightly prior to the points where the contribution

of the secondary branch begins to diverge.

A close inspection of the complex trajectories at tf = 5 and tf = 6, reveals an

interesting property of the real trajectory: the real trajectory acts as a boundary between

two “regimes” of complex trajectories arising from the real branch. This can be seen in

fig.7.5, where the trajectories that initiate from =[x(0)] > 0 are seen to reach the real

x-axis at values lower than the real trajectory while trajectories with =[x(0)] < 0 seem

to go past the barrier and reach the real x-axis at values higher than the real trajectory.

These two regimes correspond to the two legs of the “v”-shaped graph of =(SReal) in

fig.7.4(a): the trajectories arising from initial positions with =[x(0)] > 0 correspond to

the left leg of the “v” while trajectories with =[x(0)] < 0 correspond to the right leg of

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0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

x

|ψ|2

Exact QM

|ψR|2

|ψR+ψ

S|2

(1)

(2)(3) (4)

(5)

(a)

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

x

|ψ|2

(9) (8)(7)

(6)

(b)

16 18 20 22 240

0.05

0.1

0.15

0.2

0.25

x

|ψ|2

(8)

(7)

(6)

(c)

Figure 7.2: A comparison between the exact quantum wavefunction and CTM (N = 1)with a two-branch superposition. The comparison is at a series of final times tf specifiedby the numbers in the parentheses. The plots arise from an initial Gaussian wavepacketcentered at x = 0 with a positive average momentum, propagating in a quartic double-well potential (the parameters are given in the text). (a) Initially right-propagatingwavefunction; (b) the reflected wavefunction; (c) a zoom on a section of (b). For tf = 5in (a) and tf = 6 in (b) and (c) we plot the results both for just the real branch |ψR|2and for the combination of branches |ψR + ψS|2. The interference pattern obtained bysuperposing the contributions is clearly observed.

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14 16 18 20 22 240

0.2

0.4

0.6

0.8

1

x

|ψ|

tf=6, N=1

Exact QM|ψ

R|

|ψS|

|ψR+ψ

S|

Figure 7.3: CTM approximation at tf = 6 for N = 1. The contributions of each branch tothe wavefunction are depicted by plotting |ψR|, |ψS| and |ψR + ψS|. Note the exponentialincrease of ψS begins around x ' 22. For x ' 23; we discard the contribution of thesecondary branch and include just the real branch, leading to a discontinuity of the CTMapproximation.

the “v”.

7.3.2 Second Order approximation, N = 2

In this section we analyze the effect of incorporating S2 into the CTM approximation.

In addition to the five equations that are needed for obtaining the complex trajectories,

S0 and S1, we need to solve eqs.(7.16), (7.17) and (7.20). In fig.7.6(a) we depict the

approximate wavefunction for N = 2 at tf = 6. Comparing the N = 2 result with

the N = 1 result plotted in fig.7.3, we conclude that other than an interval in the

neighborhood of x ' 22.5, the N = 2 result (dashed line) lies on top of the exact result

(solid line) and is significantly better then the N = 1 result. For x & 23, where we

incorporate solely the real branch contribution, the improvement in the approximation is

graphically evident from the plots. For x ≤ 22 we calculated the relative error between

the absolute value of the approximations and the exact wavefunction using all the data

points depicted in figs.7.3 and 7.6(a). The results are presented in fig.7.6(b). For N = 1

the mean relative error is 0.34% while for N = 2 the mean relative error is 0.11%. We see

that the approximation worsens in the vicinity of the discontinuity of ψR. In the vicinity

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0 10 20 300

1

2

3

4

5ℑ

(SR

eal)

x

(9)

(5)

(6)(4)

(7)(8)

(a)

0 10 20 30−5

0

5

10

15

ℑ(S

Sec

)

x

(9)(8)

(6)

(5)

(4)

(7)

(b)

Figure 7.4: (a) =(SReal) and (b) =(SSec) are depicted at a series of final times tf (givenin the parentheses). The results are limited to the spatial interval for which the absolutevalue of the exact wavefunction is significantly larger than zero. The imaginary partof the phase allows for a qualitative estimate of the contribution of each branch to theprobability |ψR + ψS|2, see eq.(7.34). Plot (b) shows that =(SSec) drops below ∼ 2 onlyfor a finite interval of intermediate times. Therefore only for this time range does thesecondary branch makes a significant contribution to the wavefunction.

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−5 0 5 10 15 20 25 30−5

−4

−3

−2

−1

0

1

Real(x)

Imag

(x)

Real Branch, tf=5

Initial positionsFinal positionsTrajectoriesReal Trajectory

Figure 7.5: Complex classical trajectories that correspond to the real branch at tf = 5.The real trajectory acts as a boundary between two “regimes” of the complex trajectories.Initial positions with =[x(0)] < 0 seem to go past the potential wall x ' 22. Thetwo “regimes” can be related to the singular behavior of the derivative of =(SReal) atintermediate times (fig.7.4(a)).

of x ' 22.5, the N = 2 results are worse than the N = 1 results; moreover, in the N = 2

case ψS as well as ψR exhibits a discontinuity.

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14 16 18 20 22 240

0.2

0.4

0.6

0.8

1

x

|ψ|

tf=6, N=2

Exaxt QM|ψ

R|

|ψS|

|ψR+ψ

S|

(b)

14 16 18 20 2210

−4

10−3

10−2

10−1

100

101

x

Rel

ativ

e E

rror

(%

)

N=1N=2

(b)

Figure 7.6: (a) The second order (N = 2) CTM approximation is depicted for tf = 6.A discontinuity appears at x ' 22.5 for both ψR and ψS. (b) The relative error betweenthe absolute value of the exact quantum wavefunction and the CTM approximation forN = 1 and N = 2, based on the data in fig.7.3 and fig.7.6(a). A comparison of therelative errors indicates a clear improvement when taking an additional order in theCTM approximation.

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7.4 Summary

We have presented a formulation of complex time-dependent WKB (CTDWKB) that

allows the incorporation of interfering contributions to the wavefunction. The central idea

in CTDWKB presented by Boiron and Lombardi[26] is to include both the amplitude and

the phase in the lowest order term of the conventional time-dependent WKB method.

The rationale behind this substitution is to treat the phase and the amplitude on equal

footings in the limit ~→ 0. The benefits of the method are twofold. Firstly, CTDWKB

exhibits accuracy superior to the conventional TDWKB[26]. Secondly, no singularities

appear in the integration of the equations of motion of the trajectories (singularities may

appear for the phase). The method has two main drawbacks. First, the trajectories that

emerge obey the classical equations of motion but propagate in the complex plane (due to

complex initial conditions), requiring analytic continuation of the quantum wavefunction.

Second, the reconstruction of the wavefunction on the real axis requires a root search

process. This process can be eased by exploiting the analytic mapping between initial

and final position.

We have incorporated into the CTDWKB method the possibility of contributions from

multiple crossing trajectories. Boiron and Lombardi claim (section V in reference[26])

that they use the root search procedure “excluding de facto such double contributions”,

although they appreciate the benefit that double contributions have in the GGWPD

formulation. As we have demonstrated here, considering double contributions allows de-

scription of interference effects that are missing in the Boiron-Lombardi formulation of

CTDWKB. Moreover, we have showed how to derive higher orders terms of the approx-

imation in a straightforward manner. This process was applied for the derivation of a

second order term in the CTDWKB approximation. The results for N = 2 were better

than for N = 1 except for a small interval in the vicinity of the classical turning point. It

was also observed that even though there are no singularities in the integration of the CT-

DWKB equations of motion, a singularity appears in the real branch ψR at intermediate

times. For N = 2 an irregularity also appears in the part of the wavefunction associated

with the secondary branch ψS. We demonstrated that when a singularity appears in ψR

(at intermediate times), the real trajectory acts as the boundary between two groups

of trajectories associated with the real branch. Each of these groups contributes to a

different side of the singularity.

The CTDWKB formulation has several issues that require more comprehensive study.

The most critical issue is to give an analytic explanation of the need to include the

contributions from multiple classical trajectories (with zero relative phase) and why in

some cases these contributions diverge. Some insight into the analytic structure of the

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complex classical trajectories was given in reference [28] in the context of GGWPD;

however, we believe that a more general understanding of this structure is yet to be

developed. This structure presumably is relevant to the question of when the CTDWKB

formulation converges to the exact quantum mechanical result. We saw that in most

parts of configuration space N = 2 performed better than N = 1, but in other parts

of configuration space, where there were singularities, N = 2 performed worse. What

determines the position and time-dependence of these singularities in ψR at intermediate

times? What is the relation between the singularities in CTDWKB vs. conventional

time-dependent WKB? Is there any fundamental limitation on the time scale for which

the method is accurate? Since WKB plays such a central role in QM in general and in

semiclassical mechanics in particular, we believe that these questions are of great general

interest. The developments described here together with the answers to some of the above

questions could make the time-dependent WKB formulation a competitive alternative to

current time-dependent semiclassical methods.

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Chapter 8

Remarks on the relationship between

BOMCA and other methods

As we mentioned in the Introduction, the unified derivation presented in chapter 3 and

more specifically the BOMCA formulation (chapter 5) has many connections with other

formulations such as the CTM of Boiron and Lombardi[26], GGWPD of Huber and

Heller[27, 28] and others. In this chapter we give a survey of these relations.

8.1 BOMCA and the modified deBroglie-Bohm ap-

proach to quantum mechanics

The idea of exploring the CQHJE using complex trajectories was discussed in the context

of BM in 2002 by John[22]. John’s work is perhaps the most closely related antecedent

of BOMCA. John starts with the one-dimensional version of the CQHJE and interprets

the wavefunction as a guiding wave for complex trajectories, analogous to the de Broglie-

Bohm interpretation in the case of real trajectories. The projection of the complex

trajectories onto the real axis is argued to be related to the actual particle trajectory (see

App. B).

However, John applied his ideas only to analytical examples, for example the Gaussian

in the HO potential and the step potential. Because all of John’s examples are analyt-

ically solvable it is easy to overlook the fact that John’s formulation assumes that the

full wavefunction is known from a separate calculation. In mathematical terms, John’s

equation (7) (in reference [22]) for the velocity

mx =∂S

∂x=~i

1

ψ

∂ψ

∂x(8.1)

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is written in the form of a mixed ODE and a PDE, while a bona fide trajectory method

expresses the equations of motion in terms of ODEs alone. Thus John’s work does not pro-

vide a general prescription to replace quantum calculations with trajectory calculations.

John provides a trajectory interpretation of QM, not a trajectory-based formulation of

the quantum theory in which solutions of ODEs are used to solve the TDSE. One should

also notice that there is no separation in eq.(8.1) into a classical force contribution and

a quantum force contribution as appears in the BOMCA formulation (see eq.(5.2c)).

8.2 Unified derivation, BOMCA and the derivative

propagation method

As indicated in section 3.1, the idea of using a hierarchy of equations to evaluate the

quantum potential term in the CQHJE has been used in the context conventional (real)

Bohmian methods by Burghardt et al.[6], Wyatt et al.[7] and Makri et al.[8]. The method

developed by Wyatt et al. is referred to as the Derivative Propagation Method (DPM).

In this method, the spatial partial derivatives of the real amplitudes and real phases in

the hydrodynamic equations, eqs.(2.3) and (2.4), are obtained by writing a hierarchy

of ODE’s that are propagated along real quantum trajectories. By using the unified

derivation presented in section 3.1 we obtain here a simplified derivation of the DPM

method that yields an equivalent but more compact set of equations than has appeared

previously in the literature[53]. Moreover, this derivation allows for an easily accessible

comparison of DPM with BOMCA. For simplicity we demonstrate this derivation for

truncation order N = 2. Setting a velocity field

v =<(S1)

m(8.2)

and inserting it into eqs.(3.14)-(3.16) with N = 2 yields

dx

dt=<(S1)

m, (8.3a)

dS0

dt=

[<(S1)]2

2m− V +

i~2m

S2 +[=(S1)]

2

2m, (8.3b)

dS1

dt= −V1 − iS2

m=(S1), (8.3c)

dS2

dt= −V2 − S2

2

m. (8.3d)

Equations (8.3a-d) are equivalent to those of the N = 2 DPM as they appear in the

literature[4, 15], as we now show. In the DPM, the wavefunction is represented by the

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ansatz

ψ(x, t) = exp

[c(x, t) +

i

~s(x, t)

], (8.4)

where s(x, t) and c(x, t) are real functions. Ansatz (8.4) is essentially identical to the

conventional Bohmian ansatz ψ(x, t) = A(x, t) exp[

i~s(x, t)

]if one identifies A(x, t) =

exp[c(x, t)]). Equating eq.(8.4) with eq.(1.1) yields

S(x, t) = s(x, t)− i~c(x, t). (8.5)

Inserting eq.(8.5) into eqs.(8.3a-d) and dividing the results into their real and imaginary

parts yields

dx

dt=

s1

m, (8.6a)

ds0

dt=

s21

2m− V +

~2

2m(c2 + c2

1), (8.6b)

ds1

dt= −V1 +

~2

mc1c2, (8.6c)

ds2

dt= −s2

2

m+~2

mc22 − V2, (8.6d)

dc0

dt= − s2

2m, (8.6e)

dc1

dt= −c1s2

m, (8.6f)

dc2

dt= −2c2s2

m. (8.6g)

These equations are equivalent to eqs.(8.3a-d), and are readily seen to be equal to

eqs.(10.10) of reference [15] by taking N = 2 and identifying x(t) = s1

m

The trajectories in the DPM propagate on the real axis, removing the need to ex-

trapolate the wavefunction to the real axis, as is required in BOMCA. However, as we

demonstrate, the classical structure that exists in BOMCA is no longer present in DPM

(see also section 5.1). The N = 2 BOMCA equations of motion are obtained by inserting

N = 2 into eqs.(3.14)-(3.16) with the velocity field v = S1

m, hence

dx

dt=

S1

m, (8.7a)

dS0

dt=

S21

2m− V +

i~2m

S2, (8.7b)

dS1

dt= −V1, (8.7c)

dS2

dt= −V2 − S2

2

m. (8.7d)

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Comparing eqs.(8.3a-d) and (8.7a-d), we see that a quantum force term appears in the

equation of motion for the momentum S1 (eq.(8.3c)), while the equation of motion for the

phase S0 (eq.(8.3b)) has an “extra” quantum potential term [=(S1)]2

2m. When transformed

to the conventional DPM equations (eqs.(8.6a-g)) we see there is a quantum correction

to the trajectories. At the same order of truncation, N = 2, the quantum force in

BOMCA vanishes. This is the basis for our claim that BOMCA has a higher degree

of localization than conventional Bohmian formulations. In a forthcoming publication

by Wyatt et al.[54] further numerical evidence is presented indicating that for the same

order of truncation, the BOMCA formulation yields more accurate results than DPM.

8.3 BOMCA and complex time-dependent WKB

BOMCA and the complex time-dependent WKB (CTDWKB) method use different series

expansions of the complex phase of ansatz (1.1). While BOMCA applies a Taylor series

of the phase, the CTDWKB method uses a power series in ~. Nevertheless, the equations

of motion for N = 2 BOMCA and N = 1 CTDWKB can be shown to be identical as we

now show. The N = 1 CTDWKB equations are (7.9)-(7.11), (7.13) and (7.15) with the

initial conditions given by eq.(7.23). We remind the reader that the subscripts in these

equations stands for the order of the ~ series expansion term (see eq.(7.5)). We see that

eqs.(7.9) and (7.10) are identical to eqs.(8.7a) and (8.7c) and yield classical equations of

motion. By defining S = S0 + ~S1 and using eqs.(7.11) and (7.13) we can write

dS

dt=

dS0

dt+ ~

dS1

dt=

1

2m(∂xS0)

2 − V +i~2m

∂xxS0. (8.8)

The equivalence with eq.(8.7b) is evident. We conclude the comparison by noting that

the equation of motion for ∂xxS0, eq.(7.15), is identical to eq.(8.7d).

For N > 2 the equivalence between the BOMCA and CTDWKB equations breaks

down. In this context one should notice two differences between the methods: (1) incor-

porating higher order terms in the CTDWKB approximation does not effect the results

for lower order terms; since each equation of motion of Sn depends only on lower terms

of the expansion (Sj, j = 0, ..., n − 1). This is not the case with BOMCA, where each

equation of motion depends on both lower and higher terms, giving rise to feedback. (2) a

consequence of (1) is that in CTDWKB, the equations of motion of the trajectories remain

classical whereas in BOMCA, the inclusion of higher orders of the approximation affect

the complex trajectories by adding a quantum force that yields quantum trajectories.

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8.4 BOMCA and generalized Gaussian wavepacket

dynamics

The Generalized Gaussian Wavepacket Dynamics (GGWPD) developed by Huber, Heller

and Littlejohn [27, 28] is closely related to the BOMCA formulation. We show that for

an initial Gaussian wavepacket the equations of motion of GGWPD are de facto identical

to the N = 2 BOMCA equations. The GGWPD equations of motion, eqs.(2.9a-d) in

reference [28] are

dx

dt=

p

m, (8.9a)

dp

dt= −dV

dx, (8.9b)

dc

dt= iL− i~α

m(8.9c)

dt=

i

2

(d2V

dx2− 4α2

m

), (8.9d)

where L = 12mx2 − V (x). By making the transformations x → x, p → S1, c → S0 and

α → S2 one can show the equivalence between eqs.(8.9a-d) and eqs.(8.7a-d).

Equation (2.13) in reference [28] yields the final wavefunction of the GGWPD method

at position xf

〈xf |e− iHt~ |g; q, p〉 =

branches

exp

[c(t)

~

], (8.10)

where |g; q, p〉 represents a Gaussian wavepacket with position and momentum expecta-

tion values of q and p respectively. The final wavefunction in GGWPD is given by a

sum of complex phase contributions analogous to our treatment of interference effects in

BOMCA formulation in section 5.3.3. However, two main differences separate BOMCA

and the GGWPD method: (1) The GGWPD has no generalization to an arbitrary initial

wavefunction, whereas in BOMCA the only restriction on the initial wavefunction is its

analyticity, and (2) The GGWPD method presents no systematic way to increase the ac-

curacy of the approximation, that is, the method presents no higher order approximations

whereas BOMCA does. From this perspective, BOMCA can be seen as a generalization

of GGWPD. Since the N = 2 BOMCA equations are equivalent to the N = 1 CTDWKB

equations, as shown in section 8.3, CTDWKB can also be viewed as a generalization of

GGWPD that allows the incorporation of any (analytic) initial wavefunction and includes

higher order approximations.

Strongly related to the work of Huber and Heller, but developed from a completely

different angle, is the extensive work by de Aguiar and coworkers on semiclassical approx-

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imations to the coherent state propagator, both in one [55, 56, 57] and multiple [58, 59]

dimensions. Once again, this work is restricted to specific initial and final wavefunc-

tions, but the formalism does in principle allow calculations of increasing accuracy, and

a detailed study of the need for multiple trajectories has been undertaken [57].

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Part II

Miller-Good transformation for the

mapped Fourier method

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Chapter 9

Miller-Good transformation for the

mapped Fourier method

9.1 Introduction

The prevalent methods for obtaining numerical solutions of the Schrodinger equation use

grid representations: a continuous wavefunction is represented in terms of a discrete and

finite set of time-evolving complex amplitudes at a set of grid points [65, 66, 67, 68, 69,

70, 71, 72, 73, 74, 75]. Finding such an accurate representation with the minimal number

of grid points is one of the main problems in numerical calculations in QM. This problem

becomes crucial in multidimensional problems since the number of grid points required

increases exponentially with the dimension. The amplitudes at the grid points can be

interpreted as the coefficients of localized basis functions. These are known as pseudospec-

tral basis sets as apposed to the conventional orthogonal non-localized basis sets known

as a spectral basis. The representation of the wavefunction on a finite discrete basis can

be viewed as acting with a projection operator on the wavefunction. This projection

operator needs to be applied to the operators which map the wavefunction as well. When

evaluating the Hamiltonian matrix operator it proves very useful to apply a pair of pseu-

dospectral and spectral bases that expand the same space since it is usually convenient

to evaluate the kinetic energy in the spectral and the potential energy in the pseudospec-

tral basis. Two important examples of such pairs of spectral and pseudospectral basis

sets are the Discrete Variable Representation (DVR) [65] and the Fourier method (FM)

[66, 67, 68, 69, 70, 71, 72, 73, 74]. The application of the Fourier method for obtaining

the Hamiltonian matrix is called the Fourier grid Hamiltonian (FGH) [75].

The Fourier method is based on the Fast Fourier Transform algorithm which is easily

implemented numerically. In the Fourier method the wavefunction and the operator which

map it are represented on equally spaced grid points in coordinate and momentum space

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while the phase space representation is a finite rectangular grid. As a result, it is most

suited to problems where the energy shell in phase space is close to a rectangular shape

and the local de Broglie wavelength does not change significantly in coordinate space.

Since in many examples this is not the case, the standard Fourier method is wasteful

in the number of grid points required to describe the wavefunction. This problem was

tackled successfully for one-dimensional problems through a mapping procedure of the

coordinate system known as the mapped Fourier method [72, 76, 77] and is still an active

area of research [78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92].

The underlying idea of the mapping procedure is to reshape the wavefunction to fit

into the rectangular grid shape of the Fourier method so the wasted phase space area

is minimal; in this way the numerical efficiency can be increased through a reduction

in the number of grid points. The most successful of the mapping procedures utilizes

the local de Broglie wavelength as a local grid spacing [71, 74]. The mapping function

which is derived is proportional to the classical action function and as such is closely

related to the extrema and nodal positions of the wavefunction. As a consequence of the

coordinate mapping, the kinetic and the potential energy operators become correlated

and the Hamiltonian is rendered non-Hermitian. A derivative term and extra potential

energy terms may also appear. An additional drawback of current mapping procedures is

the difficulty of extending them to multidimensional problems. This is in fact the main

motivation for trying to find new mapping procedures.

Here we present a new mapping procedure which we refer to as the Miller-Good

mapped Fourier method (MIGMAF). The inspiration for the MIGMAF originated from

two sources. First is the idea that the amplitude of a wavefunction might possess relevant

information for the mapping procedure and not only the extrema and nodal positions as

in current procedures. Second is the partial success by Pechukas [63, 64] to extend the

Miller-Good (MG) method to multidimensional problems. The underlying idea of the

MG method is to find a solution of the time independent Schrodinger equation (TISE) by

a transformation from a known solution of the TISE with a given potential to an unknown

solution of a different potential. The MG method consists of a coordinate mapping and

an amplitude rescaling

ψ(q) = A(q)ψ[Q(q)], (9.1)

where q, Q are the old and new coordinates respectively, ψ(q), ψ(Q) are the unknown

and known solutions and A(q) is the amplitude pre-factor. If the extrema and nodal

positions of the mapped wavefunction ψ(Q) are at constant spacings then they are a

natural choice for the grid points in the implementation of the Fourier method. Moreover,

the MG transformation can be utilized to produce a simpler mapped Hamiltonian than

that obtained solely by a coordinate mapping.

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This chapter is organized as follows: Section 9.2 reviews the FGH method and the

mapping procedures in the general framework of numerical methods in QM. In section 9.3

we describe the Miller-Good method. Section 9.4 is dedicated to a detailed description

of the MIGMAF method. In this section (9.4.1) we present a new formulation and

extension of the MG method, referred to as the exact MG method. In section 9.5.1 we do

a numerical analysis of the exact MG vs. the conventional MG method and in 9.5.2 we

compare the exact eigenvalues of the HO to those obtained through the MIGMAF and

current coordinate mapping schemes. Section 9.5.3 is dedicated to preliminary results

we obtained for a two-dimensional coupled potential. In section 9.6 we summarizes the

chapter.

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9.2 Review of the Mapped Fourier Grid Hamiltonian

9.2.1 The Fourier grid Hamiltonian (FGH)

The FGH method refers to the evaluation of the matrix elements of a Hamiltonian using

the Fourier method. This is a specific application of a pseudospectral method on a grid

of evenly spaced points. Normally this matrix is diagonalized to obtain the eigenvalues

and eigenvectors of the Hamiltonian. In this section we review the FGH method and its

position in the general framework of the numerical methods for solving the Schrodinger

equation. We follow the guide lines of the derivation as presented in reference [19].

Any numerical calculation of a quantum mechanical system requires the truncation

of the infinite dimensional Hilbert space on which the system is defined. This truncation

can be seen as a projection onto an N -dimensional reduced Hilbert space. The truncated

Hamiltonian can be write as

HN = PNHPN = PNTPN + PNV PN = TN + VN , (9.2)

where PN is the projection operator, H is the Hamiltonian, T = − ~22m

d2

dq2 is the kinetic

energy term and V is the potential energy term. The subscript N refers to a projection

on to the truncated N -dimensional subspace. The projection operator can be expressed

by utilizing a set of N orthogonal functions {φn}Nn=1 where

PN =N∑

n=1

|φn〉〈φn|. (9.3)

The representation of a wavefunction and its operators by applying such a basis is known

as a spectral representation. The kinetic energy term can be calculated relatively easy if

the functions {φn}Nn=1 are non-localized

Tnm = 〈φn|TN |φm〉 = 〈φn|PNTPN |φm〉

=N∑

i,j=1

〈φn|φi〉〈φi|T |φj〉〈φj|φm〉 =N∑

i,j=1

δni〈φi|T |φj〉δjm = 〈φn|T |φm〉. (9.4)

It is very useful, if it is possible, to express the same projection operator by a set of

spatially localized orthogonal basis functions {θn}Nn=1, in which case

PN =N∑

n=1

|φn〉〈φn| =N∑

n=1

|θn〉〈θn|. (9.5)

The potential operator matrix element can be approximated as being diagonal in the

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localized basis set

Vnm = 〈θn|VN |θm〉 ' V (qn)δnm, (9.6)

where qn is the effective spatial center around which θn is localized. Such basis sets are

called pseudospectral. Ultimately, a single representation has to be chosen to obtain the

matrix element of the complete Hamiltonian, but this is easily accomplished provided

the unitary transformation between the two representations is known. The equivalence

of the projection operator in the two representations guarantees that the reduced Hilbert

space on which the Hamiltonian is represented is defined consistently. Two important

examples of such relations between a spectral and pseudospectral representation are the

DVR method, which will not be described here, and the Fourier method. The FGH is an

application of the latter.

The unique feature of the Fourier method is that the grid points around which the

pseudospectral basis set are centered, are equally spaced. If the number of grid points is

N and the extent in coordinate space is L then the spectral basis in the Fourier method

is chosen as

φn(q) =exp (in∆kq)√

L=

exp (iknq)√L

; −N

2+ 1 ≤ n ≤ N

2; 0 ≤ q ≤ L, (9.7)

where ∆k = 2πL

is the spacing in k-space and kn = n∆k. The spacing between the grid

points in the spatial coordinate q is ∆q = LN

. The range of the kn’s is [−K,K] where

K = N2∆k = π

∆q. It can be shown that the corresponding pseudospectral basis are sinc

like functions which take the form

θn(q) =

N2∑

j=−N2

+1

exp (iq 2πjL

)√L

exp (−iqn2πjL

)√N

; qn = n∆q; n = 1, ...N, (9.8)

where qn are the equally spaced grid points. The width of each function is approximately

π/K. The spectral and pseudospectral bases which are related to the Fourier method

fulfill the relation

〈φl|θn〉 = φl(qn), (9.9)

which we use below. In the FGH method these two basis sets are used to obtain the

Hamiltonian matrix element

Hnm = 〈θn|HN |θm〉 = 〈θn|TN |θm〉+ 〈θn|VN |θm〉 = Tnm + Vnm. (9.10)

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The potential energy term Vnm is given by eq.(9.6), while the kinetic term is given by

Tnm = 〈θn|T |θm〉 =∑

l1,l2

〈θn|φl1〉〈φl1|T |φl2〉〈φl2|θm〉

=∑

l1,l2

φl1(qn)〈φl1|T |φl2〉φl2(qm)

=∑

l1,l2

φl1(qn)~2k2

l1

2Mδl1l2φl2(qm)

=~2

2M

l

exp (iklqn)√L

k2l

exp (iklqm)√L

' ~2

2M

∫ K

−K

exp (ikqn)√2K

k2 exp (ikqm)√2K

dk

=~2

2M

{K2

3n = m

2K2

π2

(−1)m−n

(m−n)2n 6= m

, (9.11)

where M is the mass of the particle. In the first stage we applied eq.9.4. In the second

stage we inserted the spectral basis as a complete set of the reduced Hilbert space,

next we used the result from eq.(9.9). In the 4th stage we used the fact that in the

spectral basis the kinetic matrix element is diagonal. The 5th stage consists of applying

the delta function and writing the spectral basis explicitly. In the 6th stage the sum is

approximated by an integration over the span in k-space. After plugging eqs.(9.6) and

(9.11) into eq.(9.10) we can diagonalize the complete Hamiltonian matrix and obtain its

eigenvalues and eigenfunctions.

9.2.2 Phase space analysis, or why mapping ?

The phase space area covered when applying the Fourier method in one dimension is a

rectangle with dimensions

qrange = L =2π

∆k(9.12)

prange = ~(kmax − kmin) = 2~K =h

∆q. (9.13)

Note that the grid spacing in coordinate space corresponds to the grid range in momentum

space, and vice versa. The area covered in phase space is qrange×prange = Lh∆q

= Nh. The

area is proportional to the number of grid points and the area per grid point is Planck’s

constant. The FGH method can represent eigenfunctions of a quantum system if their

representation in phase space is fully covered by the rectangle defined by (qrange, prange).

For quantum mechanical bound states this condition can be closely met since such states

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decay exponentially in the classically forbidden regions of phase space. One source of

numerical inefficiency of the FGH method originates from the fact that the energy shell

which depicts the boundary of the forbidden region is generally non-rectangular in shape.

For example, the energy shell of the HO is a circle. This implies a maximum numerical

efficiency of (π4)d, where d is the dimensionality of the system. Other potentials might

have a more complicated energy shell which can further reduce the numerical efficiency.

The FGH method has an additional and more important source of inefficiency, namely

different regions of space corresponds to extremely different dynamics. For instance, if

there are regions where the motion is slow and other regions where the motion is fast,

the constant grid spacing leads to an unnecessarily large number of grid points.

The numerical efficiency can be enhanced significantly by mapping the coordinates.

The mapping procedure,known as the mapped Fourier method, deforms the rectangular

shape in phase space so that it matches the shape of the classical energy shell and hence

the dynamics of the system. This procedure allows a reduction in the number of grid

points required to obtain a good representation of the quantum system.

9.2.3 The mapping procedure

The mapped Fourier method is a three stage procedure [77]. The first stage is the

mapping process of the space coordinate. The second stage is the transformation of the

Hamiltonian to match the new Hilbert space. Finally, the FGH method is implemented

on the new working grid using the transformed Hamiltonian.

Space coordinate mapping process

Two mapping strategies have been applied for the mapped Fourier method. The first

strategy is based on an intelligent guess of the mapping function. For example, the

function

q = Q− A arctan(βQ), (9.14)

where q and Q are the original and new coordinates respectively and A, β are parameters,

was used as a mapping function for the Coulomb potential [77]. The mapping allows a

more efficient use of the Fourier method since a larger part of the energy shell is captured

by rectangular boundaries.

A more systematic strategy exploits the semiclassical local de-Broglie wavelength to

optimize the grid spacing [71, 72, 74, 76]. The first step in this procedure is to define a

transformation

s(q) =π

p(q)=

1√2m[Vmax − V (q)]

, (9.15)

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where p(q) is the momentum, V (q) is the potential of the system and Vmax is the energy

of the system. Note that s(q) is small near regions with large momentum. The next step

is to choose a constant grid step

∆Q = 1 = s(q)dQ

dq; (9.16)

hence, the mapped coordinate is

Q(q) =1

π

∫ q

q0

√2m[Vmax − V (q)]dq =

1

π

∫ q

q0

p(q)dq. (9.17)

It is convenient to choose q0 as the classical turning point (V (q0) = Vmax). Note that Q(q)

is the classical action function divided by π,hence, this mapping is a point transformation

and as such it is a canonical transformation[93].

Mapping of the Hamiltonian

The one-dimensional TISE is of the form

Hψ(q) = Eψ(q); H = − ~2

2m

d2

dq2+ V (q), (9.18)

where ψ(q) is an eigenfunction and E is the corresponding eigenvalue. The mapping of

the coordinate system transforms the Hamiltonian operator. After defining a mapping

between old (q) and new (Q) coordinates the transformed Hamiltonian will satisfy the

equation

Hψ(Q) = Eψ(Q), (9.19)

where ψ(Q) ≡ ψ[q(Q)]. H can be obtained by noting that

d

dq=

dQ

dq

d

dQ=

1

J

d

dQ

d2

dq2=

1

J

d

dQ

(1

J

d

dQ

)= −JQ

J3

d

dQ+

1

J2

d2

dQ2, (9.20)

where we have defined J(Q) ≡ dqdQ

and JQ = dJdQ

. Inserting eq.(9.20) into the TISE yields

[− ~

2

2m

(1

J2

d2

dQ2− JQ

J3

d

dQ

)+ V (Q)

]ψ(Q) = Eψ(Q), (9.21)

where V (Q) ≡ V [q(Q)]. The operator in the square brackets is H and the transformed

kinetic energy operator is T = H − V .

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The FGH method on the mapped Hamiltonian

In section 9.2.1 we reviewed the FGH method and in eqs. (9.6), (9.10) and (9.11) we

showed how the Hamiltonian matrix elements are obtained. We reexamine this derivation

for the mapped Hamiltonian. By analogy to eq.(9.10)

Hnm = 〈θn|HN |θm〉 = 〈θn|TN |θm〉+ 〈θn|VN |θm〉 = Tnm + Vnm, (9.22)

where {θn}Nn=1 are the pseudospectral basis set in the mapped coordinate Q. The mapped

potential term can be approximated as in eq.(9.6)

Vnm = 〈θn|VN |θm〉 ' V (Qn)δnm, (9.23)

where Qn is the effective spatial center of θn(Q). As for the mapped kinetic term

Tnm = 〈θn|TN |θm〉 = 〈θn|T |θm〉= − ~

2

2m

[〈θn| 1

J2

d2

dQ2|θm〉 − 〈θn|JQ

J3

d

dQ|θm〉

]

' − ~2

2m

[1

J2(Qn)〈θn| d2

dQ2|θm〉 − JQ(Qn)

J3(Qn)〈θn| d

dQ|θm〉

], (9.24)

where in the second stage we applied the result from eq.(9.4), in the third stage we

inserted the definition of T from eq.(9.21). In the last stage we utilized the locality of

the pseudospectral basis. The first term in the square brackets can be calculated in an

identical fashion to the derivation in eq.(9.11)

〈θn| d2

dQ2|θm〉 =

{K2

3n = m

2K2

π2

(−1)m−n

(m−n)2n 6= m

. (9.25)

As for the second term, we need only to change k2 by k inside the integration of the last

stage of eq.(9.11)

〈θn| d

dQ|θm〉 '

∫ K

−K

exp (ikqn)√2K

kexp (ikqm)√

2Kdk =

{0 n = mKπ

(−1)m−n

m−nn 6= m

. (9.26)

Inserting eqs.(9.26) and (9.25) into eq.(9.24) completes the expression of the mapped

kinetic energy term. At this stage we still have not determined the number and values

of the Qn’s and the corresponding qn’s . This determination will define the dimensions

of the Hamiltonian matrix and, since K = π∆Q

also the value of K. These choices are

important since the central idea of the mapped FGH is to reduce the number of grid

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points required for constructing the Hamiltonian matrix[72, 76].

If a single energy level is known or can be approximated, then the following algorithm

proves very useful for calculating the mapped FGH. Say that the N th energy level EN is

given for a bounded potential V (q). The two classical turning points can be calculated

by solving V (q) = EN ; let qc be one of the turning points. In eq.(9.16) we chose ∆Q = 1,

hence we take Qn = n; n = 1, ...N . The corresponding set of qn’s is obtained by solving

the collection of equations

1

π

∫ qn

qc

√2m[EN − V (q)]dq = Qn = n, (9.27)

which derive from eq.(9.17). As can be seen from eq.(9.32) below, the qn’s obtained by

this algorithm are the WKB approximation for the position of the extrema of the N th

eigenfunction. Since the size of the Hamiltonian matrix obtained by this procedure is

N×N we can expect N energy levels. In fig.9.1 we depict this coordinate transformation.

−5 −4 −3 −2 −1 0 1 2 3 4 5

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

qn

Qn(q

n)

Figure 9.1: This figure presents the solutions of the coordinate transforation (eq.(9.27))for the HO potential V = 1

2mω2q2, where ω is the angular frequency. Here ω = m = ~ = 1

and N = 15. Note the higher density of grid points in areas with higher momentum(around qn = 0) and the equal grid spacings ∆Q = 1 for the Qn’s.

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9.3 Review of the Miller-Good method

In this section we review the MG method [60, 61] as presented by Child [62]. The MG

method is an approximation scheme for obtaining solutions of the TISE. The method

is composed of the MG transformation and an additional semiclassical approximation

which is based on the WKB approximation. However, the MG method usually does not

suffer from the classical turning point divergence that the WKB approximation is known

for. As discussed in the introduction, the MG method plays an important role in the

mapping procedure we present in section 9.5.

The underlying idea of the MG method is to utilize a mapping from a known solution

of the TISE to obtain an unknown solution for a different potential. The rationale of the

mapping is that the qualitative shape of any bounded wavefunction is determined by the

number of nodes and the exponential decay beyond the classical turning points. We seek

a bounded solution of the TISE

[d2

dq2+ k2(q)

]ψ(q) = 0; k2(q) =

2m

~2[E − V (q)], (9.28)

where the solution of the system

[d2

dQ2+ k2(Q)

]ψ(Q) = 0; k2(Q) =

2m

~2[E − V (Q)], (9.29)

is known. The MG method consorts of a scheme for finding a function A(q) and a mapping

Q(q) such that a MG transformation relates ψ(Q) to an approximation of ψ(q), ψapp(q):

ψapp(q) = A(q)ψ[Q(q)]. (9.30)

A(q) and Q(q) are obtained by inserting the WKB approximations of ψ(q) and ψ(Q) into

a MG transformation

ψWKB(q) = A(q)ψWKB(Q). (9.31)

The WKB approximations are given by

ψWKB(q) = k−12 (q) exp

(i

∫ q

qc

k(q′)dq′)

, (9.32)

ψWKB(Q) = k−12 (Q) exp

(i

∫ Q

Qc

k(Q′)dQ′)

, (9.33)

where qc and Qc are the corresponding classical turning points. Inserting eqs.(9.32) and

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(9.33) into eq.(9.31) suggests a solution

A(q) =

{k[Q(q)]

k(q)

} 12

, (9.34)

∫ Q

Qc

k(Q′)dQ′ =

∫ q

qc

k(q′)dq′. (9.35)

Note that by differentiating eq.(9.35) with respect to q and using the chain role we obtain

dQ

dq=

k(q)

k[Q(q)]= A(q)−2. (9.36)

The mapping Q(q) is derived from eq.(9.35) by correlating the upper limits of the in-

tegration intervals. Plugging in Q(q) and A(q) into eq.(9.30) yields ψapp(q). It can be

shown that ψapp(q) satisfies the equation

[d2

dq2+ k2(q) + γ(q)

]ψapp(q) = 0, (9.37)

where

γ(q) = −{

k(q)

k[Q(q)]

} 12 d2

dq2

{k(q)

k[Q(q)]

}− 12

= −A′′(q)A(q)

. (9.38)

Eq.(9.34) was utilized in the last stage. The validity of the approximation is obtained by

comparing eqs.(9.28) and (9.37)

γ(q) ¿ k2(q) =2m

~2[E − V (q)]. (9.39)

In the limit of ~ → 0 this extreme inequality is satisfied, hence, the MG method is an

additional semiclassical approximation. This is expected since the WKB approximation

was utilized.

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9.4 The Miller-Good transformation for the mapped

Fourier method (MIGMAF)

In this section we propose a novel mapping procedure which is derived from the MG

method. Unlike previous procedures (as described in section 9.2.3) which apply only

a coordinate mapping, the new procedure utilizes an amplitude mapping as well. A

byproduct of the new procedure is a new formulation of the MG method. This new

formulation, which does not utilize the WKB approximation, yields in addition to the

conventional MG method a new method for obtaining an exact MG transformation. In

order to distinguish the two methods we refer to the former as the approximate MG

method and the latter as the exact MG method. In section 9.4.1 we present the new

formulation and derive the exact MG method. Section 9.4.2 is dedicated to the mapped

Hamiltonian which derives from the new procedure and in section 9.4.3 we apply the

FGH on the mew mapped Hamiltonian.

9.4.1 The exact Miller-Good method

We seek a solution of the TISE ψ(q) (eq.(9.28)) through a MG transformation to a known

solution ψ(Q) (eq.(9.29)). More specifically we require a function A(q) and a coordinate

mapping Q(q) such that

ψ(q) = A(q)ψ[Q(q)], A(q) 6= 0; ∀q, (9.40)

The first step in obtaining A(q) and Q(q) is to insert ψ[Q(q)] from eq.(9.40) into eq.(9.29),

hence [d2

dQ2+ k2(Q)

]ψ(q)

A(q)= 0. (9.41)

The operator in the square brackets can be mapped to the q coordinate system by writing

d2

dQ2= −JQ

d

dq+ J2 d2

dq2(9.42)

in a similar manner to eqs.(9.20). We insert the last result into eq.(9.41) and after several

steps we obtain

[d2

dq2+

1

J2

(JQ − 2A′J

2

A

)d

dq+

1

A

(2A′2

A− A′′

)− JQA′

J2A+

k2

J2

]ψ(q) = 0, (9.43)

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where A′ = dAdq

. By comparing eqs.(9.43) and (9.28) we require that the prefactor of the

first derivative be zero and the mapped potential energy term be equal to k2(q)

JQ − 2A′J2

A= 0, (9.44)

1

A

(2A′2

A− A′′

)− JQA′

J2A+

k2

J2= k2. (9.45)

By dividing eq.(9.44) by J and integrating over dQ we obtain

J(Q) = DA2(q)

(=

dq

dQ

), (9.46)

where D is an integration constant which we take as 1 henceforth and we remind ourselves

the definition of J in the round brackets. Note that eq.(9.46) is equivalent to eq.(9.36).

Inserting JQ from eq.(9.44) into eq.(9.45) yields

k2(q) =k2(Q)

A4(q)− A′′(q)

A(q)⇒ A′′(q) = −A(q)k2(q) +

k2(Q)

A3, (9.47)

where we obtained a second order ODE for A(q). Equations (9.46) and (9.47) are a set of

coupled ODE’s from which A(q) and Q(q) can be obtained. The MG transformation which

is obtained through A(q) and Q(q) yields an exact solution ψ(q) since no approximation

was utilized in the derivation. This result is the exact MG method which is an extension

of the approximate MG method (section 9.3) as presented in the literature[61, 62].

The equations for the approximate MG (AMG) method can be readily obtained by

noting that the term −A′′/A in eq.(9.47) is γ(q) from eqs.(9.37) and (9.38). Hence, in

the limit ~→ 0 eq.(9.47) reduces to

k2(q) =k2(Q)

A4AMG(q)

⇒ AAMG(q) =

{k[Q(q)]

k(q)

} 12

=

{E − ˜V [Q(q)]

E − V (q)

} 14

, (9.48)

which is identical to eq.(9.34).

9.4.2 The mapped MIGMAF Hamiltonian

In section 9.2.3 we outlined the procedure of obtaining the mapped Hamiltonian after a

coordinate transformation (eq.(9.21)). Here we follow the procedure for the MIGMAF

Hamiltonian. Inserting ψ(q) from eq.(9.40) into eq.(9.28) yields

[d2

dq2+ k2(q)

]A(q)ψ(Q) = 0. (9.49)

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By inserting the second result of eqs.(9.20) into the last equation we transform the oper-

ator in the square brackets to the new Q-space coordinates, after several steps we obtain

[1

J2

d2

dQ2+

1

JA

(2A′ − AJQ

J2

)d

dQ+ k2 +

A′′

A

]ψ(Q) = 0. (9.50)

The prefactor of the first derivative is nullified by inserting JQ from eq.(9.44), hence, we

are left with [1

A4

d2

dQ2+

2m

~2(E − V ) +

A′′

A

]ψ(Q) = 0, (9.51)

where k2 is expressed explicitly and eq.(9.46) is used to express J2. We multiply by − ~22m

and add Eψ, hence

[− ~2

2mA4

d2

dQ2+ V [q(Q)]− ~2

2m

A′′

A

]ψ(Q) = Eψ(Q). (9.52)

The operator in the square brackets is the MIGMAF mapped Hamiltonian. This operator

is simpler in form, and its matrix elements are more easily calculated than the coordinate

mapped Hamiltonian of eq.(9.21), that possesses a first derivative term. It is interesting

to notice that the additional potential term in eq.(9.52) is identical to the “quantum po-

tential” which is obtained in the Bohmian formulation of QM (eq.(2.5)). This connection

can be traced back to the ansatz that is applied in the formulation of BM, eq.(2.1). The

amplitude A plays a similar scaling role in both the MG transformation (eq.(9.1)) and in

ansatz 2.1.

9.4.3 FGH method applied on MIGMAF mapped Hamiltonian

The MIGMAF mapped Hamiltonian of eq.(9.52) can be divided into a mapped kinetic

term

T = − ~2

2mA4

d2

dQ2, (9.53)

and a mapped potential term

V = V [q(Q)]− ~2

2m

A′′

A. (9.54)

The matrix elements of the MIGMAF mapped Hamiltonian are given as in eq.(9.22)

Hnm = Tnm + Vnm, (9.55)

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where

Tnm = 〈θn|T |θm〉 = 〈θn|[− ~2

2mA4

d2

dQ2

]|θm〉

' − ~2

2mA4[qn(Qn)]〈θn| d2

dQ2|θm〉 ' − ~2

2mA4[qn(Qn)]

{K2

3n = m

2K2

π2

(−1)m−n

(m−n)2n 6= m

,(9.56)

and

Vnm = 〈θn|V |θm〉 = 〈θn|[V [q(Q)]− ~2

2m

A′′

A

]|θm〉 (9.57)

'{

V [qn(Qn)]− ~2

2m

A′′[qn(Qn)]

A[qn(Qn)]

}δnm.

We used eq.(9.25) for the kinetic energy term and the locality of the pseudospectral basis

in both the kinetic and potential energy term. For the application of the MIGMAF

method we need to specify the known solution of the TISE (ψ(Q)) on which to map the

unknown solution (ψ(q)). Clearly these choices are directly related to the values of the

potentials V (q) and V (Q). The set of qn(Qn)’s is also required. These specific choices

specify A[qn(Qn)], A′′[qn(Qn)] and K in eqs.(9.56) and (9.57). In the next section we

analyze a mapping from the eigenfunctions of the infinite square well to the eigenfunctions

of the HO.

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9.5 Numerical examples

In this section we apply the exact MG method and the MIGMAF method on the one-

dimensional HO. The mapping is carried out on sine eigenfunctions that are the solutions

of the infinite square well potential. This specific choice has several benefits for MIGMAF.

First, these eigenfunctions and the corresponding eigenvalues are given analytically. Sec-

ondly, since k2(Q) = 2m~2 E is a constant, eq.(9.47) can be solved independently from

eq.(9.46). Thirdly, a natural choice for the Qn’s are the equally spaced positions of the

extremum of the sine functions. The corresponding set of qn(Qn)’s are approximately

the extrema positions of the HO eigenfunction. This relation can be seen by combining

AAMG(q) from the approximate MG method (eq.(9.48)) with eq.(9.46)

dQ

dq= A(q)−2 =

√E − V (q)

E⇒ Q =

1√2mE

∫ q

p(q)dq, (9.58)

which is the action transformation up to a constant. As we know from the WKB ap-

proximation (eq.(9.32)) this transformation maps appropriate equal spacings to extrema

positions. The main drawback of this mapping is that the eigenfunctions of the infinite

square well potential are of finite support, hence the mapping is limited in range.

In section 9.5.1 we compare the results of the approximate MG method and the exact

MG method with the exact wavefunction. In section 9.5.2 we compare eigenvalues of

MIGMAF Hamiltonian matrices where A(q), A′′(q) and the Qn(qn)’s are calculated by

the approximate MG method and the exact MG method.

9.5.1 Comparison of the approximate MG method and the ex-

act MG method

The mapping q(Q) and the function A(q) that solves the set of eqs.(9.46) and (9.48), when

inserted in the MG transformation (eq.(9.1)), are the standard, approximate MG method.

The exact MG method is obtained by the solution of eqs.(9.46) and (9.47). We examine

these solutions for a mapping between the N th eigenfunction of the one-dimensional HO

and the N th eigenfunction of the infinite square well potential. The infinite square well

potential, eigenfunctions and eigenvalues are given by

V (Q) =

{0 0 ≤ Q ≤ L

∞ Q ≤ 0; Q ≥ L; ψ(Q) =

√2

Lsin

(nπQ

L

); En =

n2π2~2mL2

, (9.59)

where L is the width of the well and n = 1, ...,∞ is an integer. The HO potential is given

by V (q) = 12mω2q2, where ω is the angular frequency.

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In the vicinity of the well, eqs.(9.46) and (9.47), that correspond to the exact MG

method, take the form

dQ

dq= A−2(q), (9.60)

A′′(q) =2m

~2

{[EN − V (q)]A(q) +

EN

A3(q)

}, (9.61)

where EN is the eigenvalue of the N th HO eigenfunction (Note that knowledge of EN is

also a necessity in the standard action transformation procedure (eq.(9.27)) and in the

WKB approximation (eq.(9.32))). Eq.(9.61) can be solved independently of eq.(9.60) but

it depends on initial conditions A(q), A′(q) for a certain q. For an even n we can exploit

the expected symmetry of ψ(q) and take the initial condition A′(q = 0) = 0. A(q = 0)

can be approximated by utilizing AAMG(q) from eq.(9.48)

A(q = 0) ' AAMG(q = 0) =

(EN

EN

) 14

. (9.62)

For an odd n we can use the WKB approximation to find the approximate position of

an extremum and apply similar considerations for the initial conditions. The numerical

solution of eq.(9.61) is inserted in eq.(9.60), and after an integration over q the coordinate

mapping is obtained.

The approximate MG method requires the solution of eqs.(9.46) and (9.48) which

take the form

dQ

dq= A−2

AMG(q), (9.63)

AAMG(q) =

[EN

EN − V (q)

] 14

. (9.64)

Since AAMG(q) is readily given, only an integration of eq.(9.63) is needed for the coordi-

nate mapping. As we showed in eq.(9.58) this integration results in the action function,

which for the HO potential can be obtained analytically. Hence the solution of eq.(9.63)

is

Q(q) = ω

√m

2EN

[q

2

√q2c − q2 +

qc

2arcsin

(q

qc

)], (9.65)

where the integration is carried over [0, q]. qc is a classical turning point that is obtained

through the relation EN = 12mω2q2

c .

In fig.9.2 we compare the predictions of the approximate MG method and the exact

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MG method for the 11th HO wavefunction. The latter is examined for two variations on

the initial conditions. The plots denoted as exact MG (1) corresponds to A′(q = 0) = 0

and A(q = 0) = AAMG(q = 0) as in eq.(9.62). Exact MG (2) corresponds to the exact

initial conditions where A′(q = 0) = 0 and A(q = 0) = ψ(q=0)

ψ(Q=0). The latter requires prior

knowledge of ψ(q = 0). For the approximate MG method the specific mapping to sine

functions does not eliminate the classical turning points divergence as is expected from

this method [62]. This is since the mapping to sine functions is no different from the

WKB approximation that expresses the wavefunction in terms of complex exponents.

The exact MG method plots, on the other hand, are stable both through and beyond the

turning point. In particular, exact MG (2) exhibits excellent agreement with the exact

wavefunction.

In fig.9.3 A(q) is examined for the approximate MG method and exact MG (1) and

(2). The plots for the approximate MG method and exact MG (2) behave in a similar

smooth manner; on the other hand, A(q) for the exact MG method (1) exhibits a wavy

behavior. Apparently, the small change in initial condition of A(q = 0) is the source

of this change. Note that the local maxima of A(q) are approximately localized in the

positions of the extrema of ψ(q) whereas the local minima of A(q) are approximately

localized in the positions of the nodes of ψ(q).

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0 1 2 3 4 5 6

−0.4

−0.2

0

0.2

0.4

0.6

q

ψ(q

)

exact HO WFexact MG (1)approx’ MGexact MG (2)

(a)

3.8 4 4.2 4.4 4.6 4.8 5 5.20.4

0.45

0.5

0.55

0.6

0.65

0.7

q

ψ(q

)

exact HO WFexact MG (1)approx’ MGexact MG (2)

(b)

Figure 9.2: The predictions of the approximate MG method and the exact MG methodfor the 11th HO wavefunction for q > 0. The exact MG method is examined for twochoices of the initial conditions. Exact MG (1) refers to approximate initial conditions(A′(q = 0) = 0, A(q = 0) = AAMG(q = 0)) and exact MG (2) refers to exact initial

conditions (A′(q = 0) = 0, A(q = 0) = ψ(q=0)

ψ(Q=0)). Plot (b) is a zoom of plot (a) in

the vicinity of qc. Note the accuracy of exact MG (2) which corresponds to the exactinitial conditions, and the divergence of the approximate MG method around the classicalturning point, located at qc ' 4.58

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0 1 2 3 4 5 6

0.9

0.95

1

1.05

1.1

1.15

q

A(q

)

exact MG (1)approx’ MGexact MG (2)

Figure 9.3: A(q) as obtained by the approximate MG method and the exact MG methodwith its two variations on the initial conditions. Note how the minor change in the initialconditions at A(q = 0) results in the wavy behavior of exact MG (2) which correspondsto the exact initial conditions.

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9.5.2 MIGMAF vs. Coordinate Mapping

In eqs.(9.55), (9.56) and (9.57) we presented the expressions for the matrix elements

of the MIGMAF mapped Hamiltonian. This matrix can be diagonalized to obtain the

eigenvalues of Hamiltonian. For eqs.(9.56) and (9.57) to be fully defined we need to

specify the values of qn(Qn) and A(qn), A′′(qn). In the previous section we solved the set

of eqs.(9.60), (9.61) and eqs. (9.63), (9.64) for the purpose of expressing the wavefunction

of the approximate MG method and the exact MG method. The mapping Q(q) and the

function A(q) of either set of equations can be utilized to obtain qn(Qn), A(qn) and A′′(qn)

through the following procedure. As mentioned earlier, we choose the set of Qn’s as the

equally spaced extrema of the sine wavefunctions. The next step is to choose the mapping

equations of either the approximate MG method or exact MG method and do the reverse

mapping to obtain the set of qn’s. For the mapping of the approximate MG method,

the A(qn)’s are given by inserting the qn’s in eq.(9.64), the second derivative of A(q) is

obtained by differentiation of this equation. As for the mapping of the exact MG method,

both A(qn) and A′′(qn) are obtained numerically from the solution of eq.(9.61).

In figures 9.4 and 9.5 we depict the eigenvalues that derive from the MIGMAF method

vs. the coordinate mapping method and the unmapped FGH. This is done for the map-

ping of the 15th and 41th wavefunctions of the HO, hence, the Hamiltonian matrices are

of dimensions 15 × 15, 41 × 41 respectively. For the analysis of the unmapped FGH

we obtained ∆q by dividing the spacing between the turning by N . We conclude from

fig.9.4 that the unmapped FGH Hamiltonian (eqs.(9.6), (9.10) and (9.11)) eigenvalues

and the eigenvalues which correspond to exact MG (2) differ significantly from the exact

eigenvalues. The eigenvalues obtained by the unmapped FGH method exhibit for n & N2

an increasing divergence from the exact eigenvalue. This result is the main motivation of

the mapping schemes, to try to optimize the number of grid points required to describe

the quantum system. Exact MG (2) corresponds to a scheme that utilizes the exact MG

method with exact initial conditions to obtain qn(Qn) and A(qn), A′′(qn). The reason this

scheme does not yield good approximations to the eigenvalues can be understood from

fig.9.6 where A′′A

is plotted for the approximate MG method and the exact MG (1) and

(2). The values of A′′(qn)A(qn)

are part of the mapped potential energy term (eq.(9.57)). Since

the set of A′′(qn)A(qn)

which corresponds to exact MG (2) are localized in the minima of the

plot, the data inserted in the potential term is misleading and results in a distortion of

the eigenvalues. The eigenvalues which derive from the approximate MG method actually

give the best results for the MIGMAF Hamiltonian. As we see from fig.9.6 the values ofA′′(qn)A(qn)

for this scheme seem to average the plots of the the exact MG method schemes.

Apparently the divergence in the vicinity of the turning points is also of importance. The

MIGMAF method, despite the simpler form of its Hamiltonian, is no better, and in fact

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somewhat inferior to the coordinate mapping scheme.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

2

4

6

8

10

12

14

16

18

n

En

unmapped FGHcoordinate mappingexact eigenvaluesapprox’ MGexact MG (1)exact MG (2)

(a)

20 25 30 35 40

20

25

30

35

40

45

50

55

En

unmapped FGHcoordinate mappingexact eigenvaluesapprox’ MGexact MG (1)exact MG (2)

(b)

Figure 9.4: The eigenvalues of five Hamiltonian matrices are compared with the exacteigenvalues of the HO potential. Plot (a) corresponds to N = 15 and plot (b) to N = 41.The unmapped FGH Hamiltonian is constructed using eqs.(9.10), (9.6) and (9.11). TheHamiltonian which relate to the coordinate mapping applies eqs.(9.22), (9.23) and (9.24),whereas the MIGMAF Hamiltonians which are related to the MG methods apply eqs.(9.55), (9.56) and (9.57). The exact MG and approximate MG methods differ by theprocess of obtaining A(q) and the mapping Q(q). The approximate MG method utilizeseqs.(9.63), (9.64) whereas the exact MG method applies eqs.(9.60), (9.61). Exact MG(2) uses exact initial conditions and exact MG (1) uses approximate initial conditions.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

1

2

3

4

5

6

7

8

n

%

coordinate mappingapprox’ MGexact MG (1)

(a)

5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

3

n

%

coordinate mappingapprox’ MGexact MG (1)

(b)

Figure 9.5: The relative error of the eigenvalues depicted in fig.9.4. Plot (a) correspondsto N = 15 and plot (b) to N = 41. The analysis is for the coordinate mapped Hamilto-nian and two MIGMAF Hamiltonians, the approximate MG method and exact MG (1)(approximate initial conditions).

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−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

q

A’’/

A

approx’ MG exact MG (1) exact MG (2)

Figure 9.6: The value of A′′(q)A(q)

as obtained for the approximate MG method (eq.(9.64))

and the exact MG method (eq.(9.61)) with the approximate initial conditions (exactMG (1)) and the exact initial conditions (exact MG (2)). The dot, circle and plus sign

correspond to values of A′′(q)A(q)

at the qn’s. Note that the qn’s are in close proximity in

all three schemes. The term A′′(q)A(q)

is part of the mapped potential energy term in the

MIGMAF procedure (eq.(9.54)).

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9.5.3 Two dimensional mapped Fourier method

Here present preliminary attempts to extend the mapped Fourier method to multi-

dimensional problems. Currently, we have obtained moderate success in a two-dimensional

(2D) coordinate mapping scheme. The difficulty to extend the successful one-dimensional

coordinate mapping lays in the inability to generalize the coordinate mapping of eq.(9.17)

to multi-dimensions. We tackle this problem by using the nodal structure of a known

high-quantum-numbers eigenstate as a basis for new grid points. Such a grid needs to be

dense enough to describe any wavefunction corresponding to a lower quantum numbers.

The motivation for this idea comes from the one-dimensional (1D) coordinate mapping

which is closely related to the extrema locations of the wavefunction (as can be seen from

comparing eq.(9.17) with the phase term of the WKB approximation).

We start from the 2D TISE

[− ~

2

2m

2∑j=1

∂2

∂q2j

+ V (~q)

]ψ(~q) = Eψ(~q), (9.66)

where ~q = (q1, q2). For a general coordinate mapping ~Q = ~Q(~q) it can be shown that

eq.(9.66) transforms to

{− ~

2

2m

2∑i,j=1

[g2

ij

∂2

∂Q2i

+∂gij

∂qj

∂Qi

]+ 2

2∑j=1

g1jg2j∂2

∂Q1∂Q2

+ V ( ~Q)

}ψ( ~Q) = Eψ( ~Q),

(9.67)

where

gij =∂Qi

∂qj

, V ( ~Q) = V [~q( ~Q)], ψ( ~Q) = ψ[~q( ~Q)]. (9.68)

The term in the squiggly braces in eq.(9.67) is the new mapped Hamiltonian H. The next

step is to apply the 2D FGH on H. The difficulty with this application can be drastically

reduced by making two choices. First, we choose the new coordinate system ~Q = (Q1, Q2)

to be orthogonal. Second, we take the Fourier basis set to be a multiplication of two 1D

Fourier basis sets, the first as a function of Q1 coordinate and the second as a function

of Q2. For each basis set we determine Nj equally spaced grid points Qj,k where j = 1, 2

corresponds to each axis and k = 1, ...Nj. Hence, in the corresponding 2D pseudospectral

basis representation, a basis element is given by |θm, θn〉 = |θm(Q1)〉|θn(Q2)〉, where

|θm(Qj)〉 denotes a Qj=1,2 dependent basis element with an effective spatial center around

the grid point Qj,m. These choices render all the derivative operators in H separable to

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a 1D Fourier basis sets, for example

〈θn′ , θm′|g211

∂2

∂Q21

|θn, θm〉 ' g11(q1,n′ , q1,m′)δmm′〈θn′| ∂2

∂Q21

|θn〉,

〈θn′ , θm′ |g11g21∂

∂Q1∂Q2

|θn, θm〉 ' g11(q1,n′ , q1,m′)g21(q1,n′ , q1,m′)〈θn′| ∂

∂Q1

|θn〉〈θm′| ∂

∂Q2

|θm〉,

where qi,n is a grid point in the ~q-coordinate system that relates to the corresponding

Qi,n grid point and δmm′ is Kronecker’s delta function. The innerproducts are readily

obtained by the 1D FGH method[66, 67, 68, 69, 70, 71, 72, 73, 74].

At this point we are left with the key determination of the coordinate mapping func-

tion. As mentioned above, we use the nodal structure of a known high-quantum-numbers

eigenstate for the mapping function. For a specific example we look at a system with a

potential of the form

V (q1, q2) =1

2mω2

1q21 +

1

2mω2

2q22 + λq2

1q2, (9.69)

where m = 1, ω1 = 1.1, ω2 = 1 and λ = −0.042, all quantities are in atomic units. The

mapping process is done in the following manner: each maximum position corresponds

to coordinates [q1,n, q2,m] where n, m = 1, ..., Nj. This constitutes the grid system in

the “old” coordinate system. We set a 2D cartesian grid with an arbitrary distance

∆Q between neighboring points. Each grid point is located at [Q1,n, Q2,m] where Qj,l =

l∆Q; l = 1, ..., Nj; j = 1, 2. This defines the grid system in the “new” coordinate system.

The correlation function [Q1,n, Q2,m] → [q1,n, q2,m] is the required 2D mapping function.

We use numerical tools provided by Matlab to calculate gij and∂gij

∂qjneeded for H matrix

elements.

In fig.9.7 (a) we plot the approximately 640th eigenstate of the quantum system. In

fig.9.7 (b) we depict the “old” and “new” coordinate grid points. In fig.9.8 we compare

the relative error of the eigenvalues we obtain by diagonalizing H and the eigenvalues

which correspond to the harmonic oscillator approximation of eq.(9.69). Till roughly the

40th eigenvalue the mapping process yield good agreement with the exact eigenvalues. As

the quantum numbers increase we lose numerical accuracy.

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−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

q1

q 2

(a)

−8 −6 −4 −2 0 2 4 6 8−8

−6

−4

−2

0

2

4

6

8

q1, Q

1

q 2, Q2

(b)

Figure 9.7: Plot (a) depicts the approximately 640th eigenstate of a quantum system withthe potential given by eq.(9.69). This eigenstate was obtained by the conventional 2DFourier method. In plot (b) we depict the grid points that correspond to ~q-coordinate

system (circles) and the new ~Q-coordinate system (crosses).

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0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

n

%

Mapped HamiltonianHO approx’

Figure 9.8: Relative error between the first exact 100 eigenvalues of a quantum systemwith a potential given by eq.(9.69) and: 1) the eigenvalues obtained by diagonalizing H.2) the eigenvalues obtained by the harmonic oscillator approximation of eq.(9.69).

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9.6 summary

We presented a new family of mapped Fourier methods. The advantage of the MIGMAF

is the simplicity of the mapped Hamiltonian which is obtained. This Hamiltonian in-

cludes a correction to the potential energy term instead of a first derivative term as in

existing mapping procedures. The result is a reduction in the computational effort in

obtaining the Hamiltonian matrix elements. Since the best results for the MIGMAF

were obtained through the mapping and A(q) obtained by the approximate MG method,

the correction to the potential term is readily obtained analytically. As a byproduct

of the procedure we reformulated the standard MG method in a manner that does not

involve the WKB approximation. This allowed an exact formulation of the MG trans-

formation. The drawback of the MIGMAF is that it did not produce better results than

the coordinate mapping schemes.

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Appendix A

Review of the classical

Hamilton-Jacobi equation

In this Appendix we review the derivation of the classical Hamilton-Jacobi (HJ) equation

and its connection with Newton’s second law of motion. This derivation is given in most

standard text books on analytical classical mechanics[93]; here we follow the derivation

given in reference [33], specializing to one-dimension for simplicity.

Hamilton’s principle and the classical HJ equation

The HJ equation can be derived from Hamilton’s principle. Consider two points in real

configuration space, initial point (q0, t0) and final point (q, t). From among all possible

paths {q(t′)}|(q,t)(q0,t0) (where t′ goes from t0 to t) that join these initial and final end points,

the classical path is the path that extremizes the action integral

I(q, t; q0, t) =

∫ (q,t)

(q0,t0)

L(q(t′), q(t′), t′)dt′; (A.1)

that is δI = 0, where L(q, q, t) is the Lagrangian of the system. If we denote the action

integral associated with the extremal path as S(q, t; q0, t0) (the action field) one may show

that the differential as a function of the arguments of S is

dS = pdq −Hdt− p0dq0, (A.2)

where p ≡ ∂L(q,q,t)∂q

and p0 = ∂L(q0,q0,0)∂q0

. H is the Hamiltonian which is given in phase

space by a Legendre transformation, H(q, p, t) ≡ pq − L. Note that q is the velocity at

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final position q. Since we can also write

dS =∂S

∂qdq +

∂S

∂tdt +

∂S

∂q0

dq0, (A.3)

where the initial time t0 is not varied, we deduce from comparing eqs.(A.2) and (A.3)

that

p =∂S

∂q, H = −∂S

∂t, p0 = − ∂S

∂q0

. (A.4)

The second equation in (A.4) is the classical HJ equation (or HJ equation, in short)

∂S

∂t+ H(q,

∂S

∂q, t) = 0, (A.5)

where we have used the first equation in (A.4) as well. The more familiar form of this

equation (eq.(7.7)) is derived if we focus on a single particle of mass m in a scalar potential

V . The Lagrangian in Cartesian coordinates is given by

L(x, x, t) =1

2mx2 − V (x, t), (A.6)

hence, the momentum is given by

p =∂L

∂x= mx, (A.7)

and the Hamiltonian takes the more familiar form

H(x, p, t) = px− L =p2

2m+ V (x, t). (A.8)

Inserting the last result into eq.(A.5) and using the first equation in (A.4)

p =∂S(x, t)

∂x, (A.9)

yields the HJ equation for a single particle

∂S

∂t+

1

2m

(∂S

∂x

)2

+ V = 0. (A.10)

Classical HJ equation and Newton’s second law of motion

The equivalence of the action field governed by the classical HJ equation with Newton’s

second law of motion can be seen by solving eq.(A.10) along its characteristic curves

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defined by a velocity field

v =dx

dt=

1

m

∂S

∂x. (A.11)

Note that the RHS of eq.(A.11) is identified as the momentum divided by mass (see

eq.(A.9)). Taking the spatial partial derivative of eq.(A.10) yields

∂2S

∂x∂t+

1

m

(∂S

∂x

)(∂2S

∂x2

)+

∂V

∂x= 0. (A.12)

This equation can be written in terms of the velocity field v as

∂v

∂t+ v

∂v

∂x= − 1

m

∂V

∂x. (A.13)

Recognizing the LHS of the last equation as the Lagrangian time derivative of v, dvdt

=∂v∂t

+ v ∂v∂x

, results in Newton’s second law of motion

dv

dt= − 1

m

∂V

∂x

∣∣∣∣x=x(t)

. (A.14)

Equation (A.11) and eq.(A.14) are a closed set of ODE’s that describes a classical trajec-

tory. Note that the classical trajectories in and of themselves do not yield the solution of

the classical HJ equation. To this end one needs to apply the Lagrangian time derivative

on the phasedS

dt=

∂S

∂t+ v

∂S

∂x=

∂S

∂t+ mv2 =

1

2mv2 − V. (A.15)

We recognize on the RHS of eq.(A.15) the classical Lagrangian introduced in eq.(A.6).

Hence, the solution of the HJ equation along classical trajectories is given by integrating

eq.(A.15) along the set of equations (A.11), (A.14). Note that we started with a PDE

and ended up with a solution given by a closed set of ODE’s. This is an application of

the method of characteristics in the theory of PDE’s. Notice that the solution of the

action by eq.(A.15) is not the global solution of eq.(A.10) but a solution along a classical

trajectory which is a particular curve in configuration space.

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Appendix B

Review of Bohmian mechanics

Since the hydrodynamic formulation of QM (eqs.(2.3) and (2.4)) and conventional (real)

BM provide the background for this work we give here a short review of BM as both an

interpretational tool and as a numerical tool. For the former we follow reference [33]. For

a comprehensive description of the development of BM as a numerical tool the reader is

referred to reference [15].

The basic postulates

BM is an attempt to recover the notion of causality in QM and reintroduce into QM a

particle that follows a definite track in space and time. The postulates of BM are:

1. An individual physical system consists of a wave propagating in space and time

together with a point particle which moves continuously under the guidance of the

wave.

2. The wave is mathematically described by ψ(x, t) that is a solution of the Schrodinger

equation.

3. The particle motion is obtained as the solution x(t) to the equation

dx

dt=

1

m

∂S(x, t)

∂x

∣∣∣∣x=x(t)

,

where S(x, t) is the phase of ψ(x, t) = A(x, t) exp[

i~S(x, t)

]. An ensemble of possi-

ble motions associate with the same wavefunction is generated by varying x(0).

4. The probability that a particle in the ensemble lies between the points x and x+dx

at time t is

A2(x, t)dx,

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where A2(x, t) = |ψ(x, t)|2.

The first three postulates are a consistent theory of motion, while the last postulate

is added to ensure the compatibility of the motion of the ensemble with the results of

conventional QM. It has the effect of selecting from all the possible motions implied

by dxdt

= 1m

∂S∂x

∣∣x=x(t)

those that are compatible with an initial position distribution of

ρ(x, 0) = |ψ(x, 0)|2. In the Bohmian formulation the primary function of the wavefunc-

tion is its action on the point particle and not its statistical meaning. BM differs from the

Copenhagen approach by stating that |ψ(x, t)|2dx yields the probability of the particle

to be in the interval [x, x + dx] at time t and not the probability of finding or measuring

the particle. This distinction is at the heart of a century old debate between the Copen-

hagen interpretation and its rivals in the form of the EPR experiment, the many-world

interpretation of quantum mechanics, the Bohmian formulation and consequent hidden

variable interpretations.

Wave and particle equations of motion

The introduction of the particle concept into QM is motivated by decomposing the TDSE

into two real equations by expressing the wavefunction in a polar form

ψ(x, t) = A(x, t) exp

[i

~S(x, t)

], (B.1)

where A(x, t) ≥ 0 and S(x, t) are real functions. As can be seen from this ansatz, S has

the dimensions of action and A2 yields the probability density. Inserting eq.(B.1) into

the TDSE and dividing the result into its real and imaginary parts yields

∂S

∂t+

1

2m

(∂S

∂x

)2

+ V =~2

2m

1

A

∂2A

∂x2, (B.2)

∂A

∂t+

1

m

∂S

∂x

∂A

∂x+

A

2m

∂2S

∂x2= 0. (B.3)

These equations are called the hydrodynamic formulation of QM due to their similarity

to classical hydrodynamic field equations. We recognize the (classical) HJ equations on

the LHS of eq.(B.2) with an additional quantum potential term on the RHS.

Q = − ~2

2m

Axx

A. (B.4)

By analogy with classical mechanics (see App. A) a velocity field is defined as

v ≡ 1

m

∂S

∂x, (B.5)

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in accordance with postulate number 3. Using this velocity field and defining a probability

density ρ ≡ A2 we can rewrite eq.(B.3) as

∂ρ

∂t+

∂x

m

∂S

∂x

)= 0. (B.6)

Noting that ρm

∂S∂x

is a probability current we identify eq.(B.3) as a continuity equations

that ensures the conservation of probability.

In App. A we showed the equivalence between the classical HJ equation and Newton’s

second law of motion. We can follow the same process and obtain an equation of motion

for the velocity from eq.(B.2)

mdv

dt=

[−∂V

∂x+~2

2

∂x

(1

A

∂2A

∂x2

)]∣∣∣∣x=x(t)

, (B.7)

where the first term in the square brackets is the classical force and the second term is

referred as the quantum force.

The explicit coupling of the hydrodynamic equations with eq.(B.7) yield trajectories

that depend on the probability density via the quantum potential and quantum force.

In practice this complication is never faced by the community that studies BM from

its interpretational perspective. This community does not solve eqs.(B.2) and (B.3) but

uses the solution of the TDSE, ψ(x, t), to obtain S(x, t) and A(x, t) from ansatz B.1.

The hydrodynamic equations are a tool of the Bohmian formulation that facilitates the

concept of the propagating particle by the similarity to the classical equations. From

these arguments we can also deduce that the Bohmian formulation has no computational

benefits over conventional QM since the TDSE should be solved in any case.

Bohmian mechanics as a numerical tool

As mentioned in the Introduction, in 1999 two groups working independently, Mayor,

Askar and Rabitz[2], and Lopreore and Wyatt[3], developed a new approach for solving

the hydrodynamic equations and then reconstructing the wavefunction by eq.(B.1). Both

groups carried out their calculations in the Lagrangian picture where grid points move

along trajectories with velocities given by eq.(B.5). The equations of motion in this

approach are obtained by applying the Lagrangian time derivative ddt

= ∂∂t

+ v ∂∂x

on

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A(x, t) and S(x, t). The results are

dS

dt=

1

2m

(∂S

∂x

)2

− V [x(t)] +~2

2m

1

A

∂2A

∂x2, (B.8)

dA

dt= − A

2m

∂2S

∂x2, (B.9)

where we recognize the quantum potential as the rightmost term on the RHS of eq.(B.8).

Equations (B.7), (B.8), (B.9) and dxdt

= v are not a closed set since on the RHS of

these equations, the quantum potential (∝ 1A

∂2A∂x2 ), quantum force (∝ ∂

∂x( 1

A∂2A∂x2 )) and

∂2S∂x2 are unknown. All the non-locality of QM is contained in these quantities, which

are evaluated by finite difference computational methods in references [2] and [3]. In

both these references an ensemble of trajectories is propagated and information from the

neighborhood of each trajectory is used to evaluate the partial derivatives that are needed.

As one might expect, most of the effort in using the Lagrangian version of the hydro-

dynamic equations goes into evaluating the quantum potential and quantum force[4, 5,

6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. This calculation becomes acute in the vicinity of nodes.

The quantum potential diverges when A → 0, leading to numerical instability as soon as

interference effect starts to appear in the wavefunction.

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Appendix C

BOMCA - Analytically solvable

examples

In this Appendix we solve analytically the one-dimensional BOMCA equations of motion

for two cases where the initial wavefunction is a Gaussian wavepacket. In the first case

we examine a free particle propagation and in the second case we study the propagation

in a HO potential. We also show that in both instances the solution we obtain at order

N = 2 is equivalent to the exact quantum mechanical solution. The N = 2 BOMCA

equations of motion are given by taking N = 2 in eqs.(3.14)-(3.16) yielding

dx

dt=

S1

m, (C.1a)

dS0

dt=

S21

2m− V +

i~2m

S2, (C.1b)

dS1

dt= −V1, (C.1c)

dS2

dt= −V2 − S2

2

m, (C.1d)

where we use the notation On ≡ ∂nO∂xn |x(t).

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Free particle Gaussian dynamics

For a free particle, V (x) = 0, eqs.(C.1a-d) become

dx

dt=

S1

m, (C.2a)

dS0

dt=

S21

2m+

i~2m

S2, (C.2b)

dS1

dt= 0, (C.2c)

dS2

dt= −S2

2

m. (C.2d)

The initial Gaussian wavepacket is

ψ(x, 0) = exp

[−αc(x− xc)

2 +i

~pc(x− xc) +

i

~γc

], (C.3)

where γc = − i~4

ln(

2αc

π

)takes care of the normalization condition and αc is a given

constant. xc and pc are the expectation values of the position and momentum operators

respectively. For a trajectory that starts from an arbitrary x(0) ∈ C, the initial conditions

of eqs.(C.2) are given by inserting eq.(C.3) into eq.(3.18)

S0[x(0), 0] = S0(0) = i~αc[x(0)− xc]2 + pc[x(0)− xc] + γc, (C.4a)

S1[x(0), 0] = S1(0) = 2i~αc[x(0)− xc] + pc, (C.4b)

S2[x(0), 0] = S2(0) = 2i~αc. (C.4c)

We note that the solutions of eqs.(C.2a-d) will be expressed as a function of time with

x(0) a parameter. Equations (C.2a) and (C.2c) are a closed set whose solution is straight-

forward: S1(t) = S1(0) and

x(t) = x(0) +1

mS1(0)t. (C.5)

Hence, the complex quantum trajectories that correspond to the free particle Gaussian

are straight lines of constant velocity (v = S1[x(0),0]m

). A locus of such trajectories is

depicted in fig.C.1.

The solution of eq.(C.2d) is

S2(t) =mS2(0)

m + S2(0)t. (C.6)

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−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Real(x)

Imag

(x)

Figure C.1: A group of complex quantum trajectories that correspond to a freely prop-agating Gaussian wavepacket. The trajectories are given by eq.(C.5). The initial posi-tions of the trajectories are located in the vicinity of the center of the initial Gaussianwavepacket xc = 0. Both the initial and final positions of the trajectories are denoted bythe circles. The initial Gaussian parameters (eqs.(C.4)) are pc =

√2000, αc = 30π and

m = 30. The final propagation time is tf = 1.

Inserting this result into the RHS of eq.(C.2b) yields after a temporal integration

S0(t) = S0(0) +S2

1(0)

2mt +

i~2

ln

[1 +

S2(0)

mt

]. (C.7)

The wavefunction along the trajectory x(t) is given by

ψ[x(t), t] = exp

[i

~S0(t)

]. (C.8)

The exact solution of the free propagating Gaussian is given by[34]

ψ(x, t) = exp

[−αt(x− xt)

2 +i

~pt(x− xt) +

i

~γt

], (C.9)

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where

pt = pc, xt = xc +pc

mt, (C.10a)

αt =αc

1 + 2i~αc

mt, (C.10b)

γt = γc +p2

c

2mt +

i~2

ln

[1 +

2i~αc

mt

]. (C.10c)

To see the equivalence between eq.(C.8) and eq.(C.9) one needs to insert x(t) from

eq.(C.5) into eq.(C.9) (x → x(t)). We then write eq.(C.8) in terms of x(0), xc, pc,

αc, γc and t. After this somewhat cumbersome process solutions (C.8) and (C.9) are seen

to be identical.

Harmonic oscillator Gaussian dynamics

Inserting the HO potential, V (x) = 12mω2x2, into eqs.(C.1a-d) yields

dx

dt=

S1

m, (C.11a)

dS0

dt=

S21

2m− 1

2mω2x2 +

i~2m

S2, (C.11b)

dS1

dt= −mω2x, (C.11c)

dS2

dt= −mω2 − S2

2

m. (C.11d)

The initial wavefunction is given by eq.(C.3) where eqs.(C.4) are the initial conditions of

eqs.(C.11a-d) and for simplicity we chose a coherent state, αc = mω2~ . Equations (C.11a)

and (C.11c) are a closed set whose solution is given by the HO equation of motion

x(t) = x(0) cos(ωt) +S1(0)

mωsin(ωt), (C.12)

S1(t) = S1(0) cos(ωt)−mωx(0) sin(ωt). (C.13)

In fig.C.2 we plot a group of complex quantum trajectories for the HO example (eq.(C.12)).

It can easily be verified that the solution of eq.(C.11d) for S2(0) = 2i~αc = imω

is S2(t) = S2(0). Inserting x(t), S1(t) and S2(t) into eq.(C.11b) and performing the

temporal integration one can show that

S0(t) =1

2[x(t)S1(t)− x(0)S1(0)]− 1

2~ωt. (C.14)

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−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Real(x)

Imag

(x)

Figure C.2: A group of complex quantum trajectories corresponding to the propagation ofa coherent state Gaussian wavepacket in a one-dimensional HO potential. The trajectoriesare given by eq.(C.12). Since the propagation is for one period of the HO (τ = 2π

ω= 1), the

circles correspond to the initial positions (x0’s) and final positions. The initial positions ofthe trajectories are located in the vicinity of the center of the initial wavepacket, xc = 0.5.The initial parameters of the Gaussian are pc = 0, ω = 2π and m = 30. Note that theonly trajectory to stay on the real axis corresponds to x0 = xc.

The wavefunction along the trajectory is given by eq.(C.8) where S0(t) is given by

eq.(C.14).

The exact quantum solution is given by eq.(C.9) where[34]

αt =mω

2~, (C.15a)

xt = xc cos (ωt) +pc

mωsin (ωt), (C.15b)

pt = pc cos (ωt)−mωxc sin (ωt), (C.15c)

γt = γc +1

2(ptxt − pcxc)− ~ω

2t. (C.15d)

The equivalence between the exact solution and the N = 2 BOMCA solution is obtained

in an identical way as in the free particle example. We insert eq.(C.12) into eq.(C.9)

(with eqs.(C.15)) and then we write eq.(C.8) (with eq.(C.14)) as a function of x(0), xc,

pc, αc, γc and t.

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Note that in both the free particle example and the HO example, the trajectory that

initiates at x(0) = xc remains on the real axis. The reason is that for N = 2 there is no

complex quantum force (the equations of motion are classical) and the initial conditions

of the x(0) = xc trajectory are real. This is a general observation valid for x(0) = xc and

N = 2 BOMCA equations with any potential.

Gaussian wavepacket dynamics is the quintessential example of classical-quantum cor-

respondence. Our observation that the N = 2 BOMCA equations for Gaussian dynamics

yield the exact QM results coincide well with our intuition that for Gaussian dynamics

there should be no quantum force correction. As we stated above, in conventional BM

this is not the case and a quantum force of magnitude equal to the classical force appears.

This difference is one of the justifications to explore the complex quantum HJ equation

as a starting point for studying classical-quantum correspondence.

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Appendix D

Trajectory crossing and the

uniqueness of the action field

The superposition of phase contributions from crossing trajectories allows for the incor-

poration of interference effects in BOMCA (section 5.3.3) and CTDWKB (chapter 7).

Hence, the two related issues of trajectory crossing and the multi-valuedness of the phase

field that the crossing implies are of fundamental importance to the formulations we pre-

sented here. In this Appendix we review our understanding of these two issues in classical

mechanics, Bohmian mechanics and BOMCA. For simplicity, the arguments presented

here are given for one-dimensional systems.

Trajectory crossing in classical mechanics

Classical trajectories can cross in configuration space. A straightforward example would

be the crossing of two trajectories propagating under a one-dimensional harmonic oscilla-

tor potential with zero initial velocity and initial positions of x(0) and −x(0) respectively.

After 1/4 of a period the trajectories will cross at x = 0. However, classical trajectories

in phase space do not cross, that is, two trajectories with initial coordinates (x1(0), p1(0))

and (x2(0), p2(0)) respectively will not reach the same coordinate (x, p) at the same time.

Note that in bound potentials classical phase space trajectories create orbits that do not

cross. The no-crossing rule can be derived by elimination. The classical equations of

motion can be written as the set of ODEs

dx

dt=

p

m, (D.1)

dp

dt= −∂V (x, t)

∂x

∣∣∣∣x=x(t)

. (D.2)

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The Picard-Lindelof theorem guarantees that if ∂V (x,t)∂x

is Lipschitz continuous then the

classical equations of motion have a solution and it is unique. Suppose that two trajec-

tories that initiate from (x1(0), p1(0)) and (x2(0), p2(0)) cross at (x, p) after time t. If we

do a backward propagation from (x, p) the uniqueness of the solution (that still applies

for t → −t) will yield either (x1(0), p1(0)) or (x2(0), p2(0)), hence, trajectories cannot

cross in phase space.

In App. A we showed that the characteristics of the classical HJ equation (eq.(A.10))

fulfill Newton’s second law of motion. However, the solution of the classical HJ equation

being a non-linear PDE is not equivalent to the solution obtained by integrating Newton’s

equations (expressed by the set of ODEs, (D.1), (D.2) and (A.15)). First, note that the

initial condition of the HJ equation, S(x, 0), specifies an initial momentum for every

initial position x0 by the relation p(x0, 0) = ∂S(x,0)∂x

|x=x0 . The global solution of the HJ

equation corresponds to a whole ensemble of trajectories of identical particles initiating

from different initial states, (x0, p(x0, 0)), and propagating in a velocity field given by

eq.(B.5).

Second, for a given problem there can be more then one solution of the action field;

as such, the action field is not unique (unlike classical trajectories). Still, the different

action fields that correspond to the same system produce identical trajectories. These

observations can be demonstrated for free particle propagation. The action field

S(x, t) = − P 2

2mt + Px, (D.3)

where P is a constant, solves the HJ equation for V = 0. Using eq. (A.11) we readily

obtain the classical trajectory of a free particle

x(t) = x0 +P

mt, (D.4)

where x0 is an integration constant and the velocity is given by v = P/m. The ensemble

described by this action field corresponds to parallel straight lines of particles with the

same momentum P and a varying initial position x0.

The action field

S(x, t) =m

2t(x− x0)

2, (D.5)

also solves the HJ equation for a free particle. Taking the spatial partial derivative of

this equation and using eq. (A.11) we derive x = x0 +vt. This action field corresponds to

an ensemble of particles with a span of possible velocities but a fixed initial position x0.

Choosing v = P/m produces the same classical trajectory as obtained by using eq.(D.3)

or solving eqs.(D.1) and (D.2) for V = 0. Note that eqs.(D.3) and (D.5) correspond to

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different ensembles of free propagating particles.

Third, being a non-linear PDE, the HJ equation produces shocks for a general initial

action field even for V = 0, a fortiori for a general potential. By shocks we refer to areas

in space-time where the global solution exhibits singularities. Actually, inserting V = 0

into eq.(A.13) (which derives directly from the HJ equation) yields the inviscid Burgers’

equation∂v

∂t+ v

∂v

∂x= 0, (D.6)

which is a quintessential example of an equation displaying shocks. An illustration of

such a singularity can be demonstrated for the free particle propagation with the initial

field S(x, 0) = −αm2

x2, where α is a constant. From the velocity field v(x, 0) = 1m

∂S(x,0)∂x

=

−αx, we can deduce the classical trajectory for a particle that initiates from x0, x(t) =

x0 − αx0t. Hence, all trajectories cross at t′ = 1/α at position x(t′) = 0. By integrating

eq.(A.9) we can express the action along a trajectory

S[x(t), t] = S(x0, 0) + m

∫ x(t)

x0

vdx = −αm

2x2

0 − αx0m[x(t)− x0]. (D.7)

At (0, t′) we have S(0, t′) = αm2

x20 which depends on x0! Since all trajectories end up at

this position we have a multi-valued action field at (0, t′). Notice that this singularity in

the solution of the HJ equation does not contradict the well defined solution obtained by

integrating Newton’s equations (eqs.(D.1), (D.2) and (A.15)). Since the solution of the

HJ equation represents a simultaneous solution of an ensemble of particles. Such a global

solution is generally not continuous in all of configuration space since shocks are created.

This observation is not inconsistent with the fact that Newton’s laws of motion generally

produce a smooth, unique and continuous solution for each trajectory of the ensemble

where the local phase along a trajectory is given by integrating eq.(A.15).

Conventional Bohmian trajectories

When referring to conventional BM one should first distinguish between two approaches:

first, BM as an interpretational tool and second, the use of BM as a numerical tool to

calculate the wavefunction. In the former, the wavefunction—the amplitude and phase—

are assumed to be known from solving the TDSE and inserting the result into ansatz

(B.1). The trajectories are defined by the velocity field given by eq.(B.5). The classical

counterpart of this process is using the global solution of the classical HJ equation to

reconstruct classical trajectories. When using BM as a numerical tool, eqs. (B.2) and

(B.3) are solved along quantum trajectories that fulfill eq.(B.5) (see eqs.(B.8) and (B.9)).

The classical counterpart of this process is obtaining the phase by first solving the clas-

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sical equations of motion (eqs.(D.1) and (D.2)) and then integrating eq.(A.15) along the

classical trajectories.

In the previous section which dealt with classical trajectories, we saw that the global

solution of the phase (solution of the classical HJ equation) and the local solution which is

obtained by integration along classical trajectories are not equivalent. This suggests that

the question of whether the two approaches to BM yield identical trajectories, amplitudes

and phases is not trivial. The common wisdom is that the two are equvalent[15]; however

we believe that the classical analog suggests that the picture maybe more complicated

and that more study is needed to resolve this point.

We present the properties of the amplitude, the phase and trajectory crossing from

the Bohmian interpretational perspective. Assuming that the wavefunction is known, one

can use ansatz (B.1) to obtain the real amplitude and the real phase

A(x, t) = |ψ(x, t)|, (D.8)

S(x, t) = −i~ ln

[ψ(x, t)

|ψ(x, t)|]

. (D.9)

Equations (D.8) and (D.9) imply that A(x, t) is uniquely defined whereas S(x, t) is de-

fined where A(x, t) 6= 0 and up to 2πn~ where n is an integer. Although the phase is

multivalued, the velocity field that it yields v = 1m

∂S∂x

is single valued. Hence, the trajec-

tories that propagate in accordance with this velocity field cannot cross or even touch.

No such limitation exists in phase space, and indeed, Bohmian trajectories do cross in

phase space. (Mathematically, one can think of the addition of the quantum potential on

the RHS of eq.(B.8) as preventing the implementation of the Picard-Lindelof theorem).

To summarize, the crossing rules in Bohmian mechanics are opposite to those of classical

mechanics: trajectories do not cross in configuration space but are allowed to cross in

phase space.

Trajectories in Bohmian mechanics with complex action

BOMCA has several unique properties in comparison with classical mechanics and Bohmian

mechanics. First, BOMCA trajectories propagate in the complex plane. Second, the im-

plementation of BOMCA as an interpretational tool requires constructing complex quan-

tum trajectories from the solution of the TDSE (see eq.(8.1)). This is a difficult process

since it requires obtaining the analytical continuation of the wavefunction. A wavepacket

tends to diverge in the pure imaginary directions which makes the numerical integration

of the complex trajectories unstable. Still, this approach can use analytically solvable

examples of the TDSE to reconstruct complex quantum trajectories and analyze their

properties[22, 24] (see App. C). Third, in the implementation of BOMCA as a numer-

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ical tool, the complex quantum force is obtained by solving a hierarchy of ODEs. The

question of how these properties affect the crossing (or no-crossing) of BOMCA complex

quantum trajectories still requires a comprehensive study. Here we present our current

understanding of this issue.

Following the same rationale that is applied in the conventional Bohmian case, the

complex phase is given on the real axis (x ∈ R) by

S(x, t) = −i~ ln[ψ(x, t)] + 2πn~, (D.10)

if |ψ(x, t)| 6= 0 and where n is an integer. If the wavefunction is analytic then we can look

at S(x, t) for x ∈ C. Taking the spatial partial derivative of eq.(D.10) we see that the

velocity field of the complex trajectories, eq.(B.5), is single valued, hence, no crossing is

allowed in configuration space.

However we find in practice that numerical solution of BOMCA circumvents the no-

crossing rule. For example, let us look at the N = 2 BOMCA equations of motion,

eqs.(C.1a-d). The (complex) trajectories fulfill classical equations of motion that allow

crossing in configuration space and indeed we observe crossing in this instance, see fig.5.5.

It should be noted that in the DPM[7, 15] (see also section 8.2) trajectories also cross

although this is not allowed by the exact Bohmian equations of motion. Two questions

arise: (1) Is the trajectory crossing a product of the approximation induced by the trun-

cation or is the mere casting of the CQHJE in form of a hierarchy of ODEs sufficient to

yield crossing? (2) What is the justification for the superposition of phase contributions,

in other words, why does the sum of two (or more) “bad” approximations yield a good

approximation? (see for example fig.5.6a). These questions require further clarification,

and are currently being studied by us.

124

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Appendix E

List of publications

• Y. Goldfarb, D. J. Tannor, Bohmian mechanics with complex action, Abstracts of

Papers ACS 231, (2006).

• Y. Goldfarb, I. Degani, D. J. Tannor, Bohmian Mechanics with Complex Action: A

New Trajectory-Based Formulation of Quantum Mechanics, J. Chem. Phys. 125,

231103 (2006).

• Y. Goldfarb, I. Degani, D. J. Tannor, Semiclassical approximation with zero velocity

trajectories, Chem. Phys. 338, 106 (2007).

• Y. Goldfarb, J. Schiff, D. J. Tannor, Unified derivation of Bohmian methods and

the incorporation of interference effects, J. Phys. Chem. A 111, 10416 (2006).

• Y. Goldfarb, D. J. Tannor, Interference in Bohmian Mechanics with Complex Ac-

tion, J. Chem. Phys. 127, 161102 (2007).

• Y. Goldfarb, J. Schiff, D. J. Tannor, Complex trajectory method in time-dependent

WKB, J. Chem. Phys., submitted.

• Y. Goldfarb, I. Degani, D. J. Tannor, Response to Comment on “Bohmian mechan-

ics with complex action: A new trajectory-based formulation of quantum mechanics

[J. Chem. Phys. 125, 231103 (2006)]” by A. S. Sanz and S. Miret-Artes, J. Chem.

Phys. 127, 197101 (2007).

125

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צירקת

של התורה ת מדויקהצגהניקה קוונטית החל חיפוש אחר גילוי המכמאז בשנות החמישים . על חישוב לאורך מסלוליםתהיה מבוססתהקוונטית אשר

בהתבסס על ) מכניקה בוהמית (את שכזהצגה )David Bohm( דיויד בוהםפיתח מסלולים ה ב)de-Broglie (ברוליי- ודה)Madelung (ם קודמים של מאדלונגפיתוחי

בשנים האחרונות .נעים בנוכחות כוח קוונטי בנוסף לכוח הניוטוני הרגילשהוא התחדש העניין במכניקה בוהמית ככלי חישובי לאור הדינמיקה המקומית

פשר יתרונות חישוביים לא היתהעשויהתכונה זאת . לכאורה מאפשרבחינה לעומק של , אבל. עבור מערכות קוונטיות מורכבותמשמעותיים

ונטית של מכניקה קו)non-locality( מקומיות- מראה כי האיהמכניקה הבוהמיתבחלקה . מתחת לשטיח אל תוך הכוח הקוונטיהטאטו כי אם מהלא נעל, בהשראת מכניקה בוהמיתות חדשהצגות של התזה אנו מציגים מספר הראשון

המוצא שלנו הינהנקודת. הינה גודל מרוכב, S, ם הפעולה הקוונטיתבה complex quantum Hamilton-Jacobi(מרוכבת -יאקובי הקוונטית-משוואת המילטון

equation .(של להצגות אחרותמשוואה זאת זכתה לתשומת לב מועטה יחסית . ןשוואת שרדינגר התלויה בזמהתורה הקוונטית למרות היותה שקולה למ

מאפשר קבלת זה דבר , ל מסלול מקודם באופן עצמאיבהצגות השונות כל "יש לציין כי הצגות הנ. פונקצית הגל הקוונטית עבורמקומייםקירובים ר מסלולים תיאור של תבניות התאבכות כסכום של תרומות ממספמאפשרות

לחישוב פיזור ממחסום פוטנציאל במימד ההצגות החדשות משמשות. חוצים .קבועי תגובה תרמיים וחישוב ערכים עצמיים, אחד ושני מימדים

עבור שיטת בחלקה השני של התזה אנו בוחנים תהליך מיפוי חדש השיטה המוכרת מנצלת את ). Mapped Fourier method( פורייה הממופת

כך שנוצר תאורדינאטופונקצית הפעולה הקלאסית בכדי ליצר מיפוי של הקוהשיטה החדשה . מרווח שווה בין נקודות הקיצון והאפסים של פונקצית הגל

בכדי לבנות ) Miller-Good transformation(גוד - מילרתנעזרת בטרנספורמצייההמילטוניאן המתקבל . את המנעד של פונקצית הגל גםמיפוי המשלב

מטריצי שלו קל יותר לחישוב הגבתהליך המיפוי החדש הינו פשוט יותר והייצואנו מציגים תוצאות ראשוניות עבור ההכללה , בנוסף. ימותימאשר בשיטות הק

ים כאשר הפוטנציאל אינו בר חלוקהיטות מיפוי קימות לשני מימדישל ש)non-separable .(מיפוי ך למצוא תהליכיחלק ממאמץ מתמש םהינ פיתוחים אילו

.עיות שאינן ברות חלוקה במספר מימדיםברי חלוקה אשר ניתנים להכללה לב

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בור לשם קבלת התואריח

דוקטור לפילוסופיה

פתרון משוואת שרדינגר התלויה בזמן באמצעות

מסלולים קוונטיים מרוכבים

יאיר גולדפרב בהנחיית

טנור 'ד ידו' פרופ

ז"באלול התשסח "כ

מוגש למועצה המדעית של מכון ויצמן למדע

ישראל, רחובות