Chattaraj P.K. (Ed.) Quantum Trajectories

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Foreword xi

of the trajectory. To demonstrate the legitimacy of the idea we must envisage going

beyond the conventional formalism. Fortunately, and this is its second beneﬁt, the

trajectory potentially provides a way to do this by allowing the precise formulation

of questions concerning particle-like concepts such as speed and separation that are

ambiguous in a pure-wave context (examples include tunneling times and chaos).

Whether these ideas can be made empirical is not yet known.Advances in this direction

will presumably entail restrictions on the range of valid trajectory laws.

The third beneﬁt we shall mention, and arguably the most signiﬁcant recent devel-

opment in this ﬁeld, is the observation that the trajectory is a constructive aid in a very

practical sense: it can be made the basis of the quantum description in that the evolution

of a physical system may be deduced from the dynamics of an ensemble of trajec-

tories. This gives the trajectory theory a status somewhat analogous to Feynman’s

path integral technique but with a more comprehensible model. This insight has been

propelled not by the foundations of the physics community (who are largely unaware

of it) but by Robert Wyatt and the quantum chemists. In particular, it highlights the

power of a particular analogy between quantum mechanics and hydrodynamics. Not

only does the trajectory picture provide an alternative method of solving the wave

equation, it may even enable one to arrive at novel facts about quantum dynamics,

such as its symmetries, which otherwise would have been difﬁcult to obtain.

The advantages of the trajectory outlook we have mentioned, and others, are strik-

ingly exhibited in the contributions to this book. The breadth of topics attests to the

vibrancy of the ﬁeld. In view of the trajectory’s proven value, a reasonable question

to ask of conventional quantum treatments is: why has it been dropped?

Peter Holland

Green Templeton College

University of Oxford

Oxford, England, United Kingdom

Preface

The quantum dynamics of many-particle systems is traditionally understood by solv-

ing the time-dependent Schrödinger equation (TDSE) which involves the time evolu-

tion of the wavefunction, a function of (3N +1) variables. Attempts have been made

since the birth of quantum mechanics to express the dynamics in three-dimensional

space using quantities like charge- and current densities. In time-dependent density

functional theory (TDDFT) the map between an arbitrary time-dependent (TD) poten-

tial and the density is uniquely invertible up to an additive TD function in the potential.

This TDDFT allows us to formulate quantum ﬂuid dynamics (QFD) wherein the quan-

tum dynamics is described in terms of the ﬂow of a probability ﬂuid associated with the

charge- and current densities by solving an equation of continuity and an Euler-type

equation of motion.A quantum ﬂuid density functional theory may also be envisaged

in this context.

In order to understand the epistemological signiﬁcance, one needs to know the

quantum theory of motion (QTM) as well, in the sense of the classical interpretation

of quantum mechanics as developed by de Broglie and Bohm. In QTM, the complete

description of a physical system needs the simultaneous presence of the “wave”

and the “particle” as opposed to conventional quantum mechanics involving “wave–

particle duality” (“wave” or “particle” picture). The wave motion is governed by the

solution of the TDSE, and the motion of a point particle guided by that wave for a

given initial position is characterized by a velocity deﬁned as the gradient of the phase

of the wavefunction. An assembly of initial positions will constitute an ensemble of

particle motions (so-called quantum trajectories or Bohmian trajectories) guided by

the same wave, and the probability of having the particle in a given region of space

at a given time is provided by the quantum mechanical TD probability density.

Acrucial link between QTM and QFD is the quantum potential. In QTM, the parti-

cle experiences forces originating from both classical and quantum potentials, and in

QFD, the ﬂuid motion takes place under the inﬂuence of the external classical potential

augmented by the quantum potential.

Apart from providing conceptual insights into many-body quantum dynamics, the

quantum trajectory method provides an efﬁcient on-the-ﬂy computational method-

ology for solving both stationary and time-evolving states and hence encompassing

a large area of quantum mechanics. Both Lagrangian and Eulerian techniques may

be developed. Owing to the strong classical ﬂavor in the de Broglie–Bohm repre-

sentation of quantum mechanics, various mixed quantum–classical techniques have

been tried. It also provides an easy way to analyze the quantum domain behavior of

classically chaotic systems.

Several experts who have either developed the subject from inception or have

improved upon the subject to take it to the state-of-the-art level have written chapters

for this volume. The Foreword has been written by Professor Peter Holland, an author-

ity in this subject. This book will be useful to graduate students and other research

xiii

xiv Preface

workers in the related ﬁelds of chemistry, physics, mathematics, and computer sci-

ence.

I am grateful to all the authors who cooperated with me so that the book could be

published on time. I would like to thank Professor Peter Holland, Lance Wobus and

David Fausel, Santanab Giri and Soma Duley for their help in various ways. Finally,

I must express my gratitude toward my wife Samhita and my daughter Saparya for

their whole-hearted support.

Pratim Kumar Chattaraj

IIT Kharagpur, India

Editor

Pratim Kumar Chattaraj joined the faculty of the Indian Institute of Technology

(Kharagpur) after obtaining his BSc and MSc degrees from Burdwan University and

his PhD degree from the Indian Institute of Technology (Bombay). He is now a profes-

sor and the head of the Department of Chemistry and also the convener of the Center

for Theoretical Studies there. He has visited the University of North Carolina (Chapel

Hill) as a postdoctoral research associate and several other universities throughout the

world as a visiting professor. Apart from teaching, Professor Chattaraj is involved in

research on density functional theory, the theory of chemical reactivity, aromaticity in

metal clusters, ab initio calculations, quantum trajectories, and nonlinear dynamics.

He has been invited to deliver special lectures at several international conferences and

to contribute chapters to many edited volumes. Professor Chattaraj is a member of the

editorial board of the Journal of Molecular Structure (Theochem) and the Journal of

Chemical Science, among others. He is a council member of the Chemical Research

Society of India and a fellow of the Indian Academy of Sciences (Bangalore), the

Indian National Science Academy (New Delhi), the National Academy of Sciences,

India (Allahabad), and the West Bengal Academy of Science and Technology. He is

a J.C. Bose National Fellow.

List of Contributors

J. Alberto Beswick

Laboratoire Collisions Agrégats

Réactivité

IRSAMC, Université Paul Sabatier

Toulouse, France

Eric R. Bittner

Department of Chemistry

University of Houston

Houston, Texas

F. Borondo

Departamento de Química

Instituto Mixto de Ciencias

Matemáticas

CSIC–UAM–UC3M–UCM

Universidad Autónoma de Madrid

Cantoblanco, Madrid, Spain

Gary E. Bowman

Department of Physics and Astronomy

Northern Arizona University

Flagstaff, Arizona

Irene Burghardt

Département de Chimie

Ecole Normale Supérieure

Paris, France

Pratim K. Chattaraj

Department of Chemistry and Center

for Theoretical Studies

Indian Institute of Technology

Kharagpur, India

Chia-Chun Chou

Institute for Theoretical Chemistry and

Department of Chemistry and

Biochemistry

The University of Texas at Austin

Austin, Texas

D.-A. Deckert

Mathematisches Institut LMU

München, Germany

Arnaldo Donoso

Laboratorio de Física Estadística de

Sistemas Desordenados

Centro de Física, Instituto Venezolano

de Investigaciones Cientíﬁcas

Caracas, Venezuela

Soma Duley

Department of Chemistry and Center for

Theoretical Studies

Indian Institute of Technology

Kharagpur, India

D. Dürr

Mathematisches Institut LMU

München, Germany

Alon E. Faraggi

Department of Mathematical Sciences

University of Liverpool

Liverpool, United Kingdom

Edward R. Floyd

Coronado, California

Sophya Garashchuk

Department of Chemistry and

Biochemistry

University of South Carolina

Columbia, South Carolina

Swapan K. Ghosh

Theoretical Chemistry Section

Bhabha Atomic Research Centre

Mumbai, India

xvii

xviii List of Contributors

Santanab Giri

Department of Chemistry and Center

for Theoretical Studies

Indian Institute of Technology

Kharagpur, India

Sheldon Goldstein

Departments of Mathematics, Physics

and Philosophy

Rutgers University

Piscataway, New Jersey

Gebhard Grübl

Theoretical Physics Institute

Universität Innsbruck

Innsbruck, Austria

Peter Holland

Green Templeton College

University of Oxford

Oxford, United Kingdom

Dipankar Home

Department of Physics

Center for Astroparticle Physics and

Space Science

Bose Institute

Calcutta, India

Keith H. Hughes

School of Chemistry

Bangor University

Bangor, United Kingdom

Moncy V. John

Department of Physics

St. Thomas College

Kozhencherry, Kerala, India

Brian K. Kendrick

Theoretical Division

Los Alamos National Laboratory

Los Alamos, New Mexico

Munmun Khatua

Department of Chemistry and Center for

Theoretical Studies

Indian Institute of Technology

Kharagpur, India

Donald J. Kouri

Department of Chemistry

University of Houston

Houston, Texas

Craig C. Martens

Department of Chemistry

University of California, Irvine

Irvine, California

Marco Matone

Department of Physics “G. Galilei”—

Istituto

Nazionale di Fisica Nucleare

University of Padova

Via Marzolo, Padova, Italy

Christoph Meier

Laboratoire Collisions Agrégats

Réactivité

IRSAMC, Université Paul Sabatier

Toulouse, France

Salvador Miret-Artés

Instituto de Física Fundamental

Consejo Superior de Investigaciones

Cientíﬁcas

Madrid, Spain

Alok Kumar Pan

Department of Physics

Center for Astroparticle Physics and

Space Science

Bose Institute, Sector-V

Calcutta, India

Gérard Parlant

Institut Charles Gerhardt,

Université Montpellier 2

Montpellier, France

List of Contributors xix

Markus Penz

Theoretical Physics Institute

Universität Innsbruck

Innsbruck, Austria

P. Pickl

Institut für Theoretische Physik

ETH Zürich, Switzerland

Bill Poirier

Department of Chemistry and

Biochemistry, and Department of

Physics

Texas Tech University

Lubbock, Texas

Vitaly Rassolov

Department of Chemistry and

Biochemistry

University of South Carolina

Columbia, South Carolina

Àngel S. Sanz

Instituto de Física Fundamental

Consejo Superior de Investigaciones

Cientíﬁcas

Madrid, Spain

Utpal Sarkar

Department of Physics

Assam University

Silchar, India

Dmitrii V. Shalashilin

School of Chemistry

University of Leeds

Leeds, United Kingdom

Roderich Tumulka

Department of Mathematics

Rutgers University

Piscataway, New Jersey

Robert E. Wyatt

Department of Chemistry and

Biochemistry

Institute for Theoretical Chemistry

The University of Texas at Austin

Austin, Texas

Tarik Yefsah

Laboratoire Kastler Brossel

Département de Physique de l’Ecole

Normale Supérieure

Paris, France

Nino Zanghì

Dipartimento di Fisica

Università di Genova and INFN sezione

di Genova

Genova, Italy

Yujun Zheng

School of Physics

Shandong University

Jinan, People’s Republic of China

Bohmian Trajectories

as the Foundation

of Quantum Mechanics

Sheldon Goldstein, Roderich Tumulka,

and Nino Zanghì

CONTENTS

1.1 Bohmian Trajectories......................................................................................1

1.2 Bohmian Mechanics........................................................................................3

1.3 Equivariance ...................................................................................................4

1.4 The Quantum Potential ...................................................................................4

1.5 Connection with Numerical Methods .............................................................5

1.6 The Quantum Potential Again.........................................................................7

1.7 Wave–Particle Duality ....................................................................................7

1.8 The Phase Function S(q, t) .............................................................................8

1.9 Predictions and the Quantum Formalism........................................................9

1.10 The Generalized Quantum Formalism..........................................................10

1.11 What is Unsatisfactory About Standard Quantum Mechanics?....................11

1.12 Spin ...............................................................................................................12

1.13 The Symmetrization Postulate ......................................................................13

1.14 Further Reading ............................................................................................14

Bibliography ............................................................................................................15

Bohmian trajectories have been used for various purposes, including the numerical

simulation of the time-dependent Schrödinger equation and the visualization of time-

dependent wave functions. We review the purpose they were invented for: to serve as

the foundation of quantum mechanics, i.e., to explain quantum mechanics in terms of

a theory that is free of paradoxes and allows an understanding that is as clear as that

of classical mechanics. Indeed, they succeed in serving that purpose in the context of

a theory known as Bohmian mechanics, to which this article is an introduction.

1.1 BOHMIAN TRAJECTORIES

Let us consider a wave function ψ

(q) of non-relativistic quantum mechanics, deﬁned

on the conﬁguration space R

of N particles, taking values in the set C of complex

2 Quantum Trajectories

numbers, and evolving with time t according to the non-relativistic Schrödinger

equation,

i

∂ψ

∂t

=−



k=1



∇

+ V ψ

, (1.1)

where m

is the mass of the k-th particle, ∇

=∇



∂

∂x

∂

∂y

∂

∂z



is the derivative

with respect to the coordinates of the k-th particle, and V : R

→ R is a potential

function, for example the Coulomb potential

V (q

, ...,q

) =



1≤j<k≤N

− q

(1.2)

with e

the charge of the k-th particle.

With this wave function there is associated a family of trajectories in conﬁguration

space R

, the Bohmian trajectories, which are deﬁned to be those trajectories t →

Q(t) = (Q

(t), ...,Q

(t)) satisfying the equation

(t)



∇

(Q(t)). (1.3)

Put differently, to every wave function ψ : R

→ C there is associated a vector

ﬁeld v

on conﬁguration space according to

= (v

, ...,v

), v



∇

, (1.4)

and Equation 1.3 amounts to

dQ(t)

= v

(Q(t)). (1.5)

Since Equation 1.5 is an ordinary differential equation (ODE) of ﬁrst order (or rather,

a system of 3N coupled ODEs of ﬁrst order), it has, leaving aside the exceptions, a

unique solution for every choice of initial conﬁguration Q(0).

Another way of writing Equation 1.5 is

dQ(t)

|ψ

(Q(t)), (1.6)

where j

is the vector ﬁeld on R

usually called the probability current associated

with the wave function ψ,

= (j

, ...,j

), j



Im(ψ

∗

∇

ψ). (1.7)

Bohmian Trajectories as the Foundation of Quantum Mechanics 3

1.2 BOHMIAN MECHANICS

The theory that uses Bohmian trajectories as the foundation of quantum mechanics

is known as Bohmian mechanics; it arises if we take a Bohmian trajectory seriously.

Namely, Bohmian mechanics claims that in our world, electrons and other elementary

particles have precise positions Q

(t) ∈ R

at every time t that move according to

Equation 1.3. That is, for a certain Bohmian trajectory t → Q(t) in conﬁguration

space, it claims that Q(t) = (Q

(t), ...,Q

(t)) is the conﬁguration of particle

positions in our world at time t.

This picture is in contrast with the orthodox view of quantum mechanics, according

to which quantum particles do not have precise positions, but are regarded as “delo-

calized” to the extent to which the wave function ψ

is spread out. It is also in contrast

with another picture of the Bohmian trajectories that one often has in mind when using

Bohmian trajectories for numerical purposes: the hydrodynamic picture. According

to the latter, all the Bohmian trajectories associated with a given wave function (but

corresponding to different Q(0)) are on an equal footing, none is more real than the

others, they are all regarded as ﬂow lines in analogy to the ﬂow lines of a classical ﬂuid.

In Bohmian mechanics, however, only one of the Bohmian trajectories corresponds

to reality, and all the other ones are no more than mathematical curves, representing

possible alternative histories that could have occurred if the initial conﬁguration of

our world had been different, but did not occur.

As a consequence, talk of probability makes immediate sense in Bohmian mechan-

ics but not in the hydrodynamic picture: In Bohmian mechanics, with only one tra-

jectory realized, that trajectory may be random. In the hydrodynamic picture, with

all trajectories equally real, it is not clear what a probability distribution over the

trajectories could be the probability of, and what it could mean to say that a trajectory

is random.

Bohmian mechanics was ﬁrst proposed by Louis de Broglie (1892–1987) in the

1920s [5]; it is named after David Bohm (1917–1992), who was the ﬁrst to realize

that this theory provides a foundation for quantum mechanics [4]: The inhabitants

of a typical Bohmian world would, as a consequence of the equations of Bohmian

mechanics, observe exactly the probabilities predicted by the quantum formalism.

To understand how this comes about requires a rather subtle “quantum equilib-

rium” analysis [8] that is beyond the scope of this paper. An important element of

the analysis is however rather simple. It is the property of equivariance, expressing

the compatibility between the evolution of the wave function given by Schrödinger’s

equation and the evolution of the actual conﬁguration given by the guiding equa-

tion 1.6. This property will be discussed in the next section.

The upshot of the quantum equilibrium analysis is the justiﬁcation of the probabil-

ity postulate for Bohmian mechanics, that the conﬁguration Q of a system with wave

function ψ = ψ(q) is random with probability density |ψ(q)|

. Bohmian mechan-

ics, with the probability postulate, is empirically equivalent to standard quantum

mechanics. We will return to this point later and explain how this follows from the

equations.