Chattaraj P.K. (Ed.) Quantum Trajectories
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Quantum
Trajectories
ATOMS, MOLECULES, AND CLUSTERS
Structure, Reactivity, and Dynamics
Series Editor: Pratim Kumar Chattaraj
Aromaticity and Metal Clusters
Pratim Kumar Chattaraj
Quantum Trajectories
Pratim Kumar Chattaraj
ATOMS, MOLECULES, AND CLUSTERS
Structure, Reactivity, and Dynamics
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
Boca Raton London New York
Quantum
Trajectories
EDITED BY
Pratim Kumar Chattaraj
CRC Press
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Library of Congress Cataloging-in-Publication Data
Quantum trajectories / editor, Pratim Kumar Chattaraj.
p. cm.
Summary: “Applying quantum mechanics to many-particle systems is an active area
of research. The popularity of using quantum trajectories as a computational tool has
exploded over the last decade, finally bringing this methodology to the level of practical
application. This book explores this powerful tool to efficiently solve both static and
time-dependent systems across a large area of quantum mechanics. Many of the pioneers
who have either developed the subject from its inception or have improved on the subject
contribute to this volume. They offer their insights on Lagrangian and Eulerian methods
as well as various mixed quantum-classical techniques, among other topics”-- Provided
by publisher.
Includes bibliographical references and index.
ISBN 978-1-4398-2561-7 (hardback)
1. Quantum trajectories. I. Chattaraj, Pratim Kumar.
QC174.17.Q385Q36 2010
530.12--dc22 2010035467
Contents
Foreword...................................................................................................................ix
Preface.....................................................................................................................xiii
Editor........................................................................................................................xv
List of Contributors................................................................................................xvii
Chapter 1 Bohmian Trajectories as the Foundation of Quantum Mechanics .......1
Sheldon Goldstein, Roderich Tumulka, and Nino Zanghì
Chapter 2 The Equivalence Postulate of Quantum Mechanics:
Main Theorems...................................................................................17
Alon E. Faraggi and Marco Matone
Chapter 3 Quantum Trajectories and Entanglement ...........................................41
Edward R. Floyd
Chapter 4 Quantum Dynamics and Supersymmetric Quantum Mechanics........53
Eric R. Bittner and Donald J. Kouri
Chapter 5 Quantum Field Dynamics from Trajectories......................................73
Peter Holland
Chapter 6 The Utility of Quantum Forces ..........................................................87
Gary E. Bowman
Chapter 7 Quantum Trajectories in Phase Space ................................................95
Craig C. Martens, Arnaldo Donoso, and Yujun Zheng
Chapter 8 On the Possibility of Empirically Probing the Bohmian Model
in Terms of the Testability of Quantum Arrival/Transit
Time Distribution .............................................................................113
Dipankar Home and Alok Kumar Pan
v
vi Contents
Chapter 9 Semiclassical Implementation of Bohmian Dynamics.....................135
Vitaly Rassolov and Sophya Garashchuk
Chapter 10 Mixed Quantum/Classical Dynamics: Bohmian and
DVR Stochastic Trajectories ............................................................149
Christoph Meier, J. Alberto Beswick, and Tarik Yefsah
Chapter 11 A Hybrid Hydrodynamic–Liouvillian Approach
to Non-Markovian Dynamics...........................................................163
Keith H. Hughes and Irene Burghardt
Chapter 12 Quantum Fluid Dynamics within the Framework
of Density Functional Theory...........................................................183
Swapan K. Ghosh
Chapter 13 An Account of Quantum Interference from
a Hydrodynamical Perspective.........................................................197
Ángel S. Sanz and Salvador Miret-Artés
Chapter 14 Quantum Fluid Density Functional Theory and Chemical
Reactivity Dynamics ........................................................................223
Santanab Giri, Soma Duley, Munmun Khatua, Utpal Sarkar,
and Pratim K. Chattaraj
Chapter 15 Bipolar Quantum Trajectory Methods..............................................235
Bill Poirier
Chapter 16 Nondifferentiable Bohmian Trajectories..........................................251
Gebhard Grübl and Markus Penz
Chapter 17 Nonadiabatic Dynamics with Quantum Trajectories .......................263
Gérard Parlant
Chapter 18 Recent Analytical Studies of Complex Quantum Trajectories.........283
Chia-Chun Chou and Robert E. Wyatt
Contents vii
Chapter 19 Modified de Broglian Mechanics and the Dynamical Origin
of Quantum Probability....................................................................301
Moncy V. John
Chapter 20 Types of Trajectory Guided Grids of Coherent States
for Quantum Propagation.................................................................313
Dmitrii V. Shalashilin
Chapter 21 Direct Numerical Solution of the Quantum Hydrodynamic
Equations of Motion.........................................................................325
Brian K. Kendrick
Chapter 22 Bohmian Grids and the Numerics of Schrödinger Evolutions.........345
D.-A. Deckert, D. Dürr, and P. Pickl
Chapter 23 Approximate Quantum Trajectory Dynamics in Imaginary
and Real Time; Calculation of Reaction Rate Constants .................361
Sophya Garashchuk
Chapter 24 Dynamical Systems Approach to Bohmian Mechanics ...................381
F. Borondo
Index......................................................................................................................391
Foreword
It is a characteristic of contemporary culture that the aim of much public discourse in
quantum theory is not to educate but to mystify. One encounters a relentless message
that “nobody understands quantum mechanics,” that “an electron can be at two places
at once,”and that theworld is ruledby “quantum weirdness.”In this context the special
claim of the scientist is just that she comprehends more deeply than the populace why
she does not comprehend. It’s a lucrative market and it is rarely mentioned that a
rational explanation exists.
A currently popular approach that is often presented as if it is the official quan-
tum worldview is the “many-worlds” interpretation. This interpretation asserts that
everything that can happen according to quantum mechanics does happen, some-
where. The suggestion that an individual can have multiple versions may chime with
the public but should such a view be afforded special status in science? It seems an
extraordinarily inflated (and still to be proven consistent) solution to a comparatively
mundane physics problem: achieving consistency between the superposition princi-
ple and the individuality of events. That is not to say that the puzzle is not daunting,
but there should be proportion between problem and solution.Afashionable argument
often adduced in support of the many-worlds option is Occam’s razor, roughly the
claim that the “less” the premises the “better” the theory. Even if such a complex
business as the commensurability of theories could be formulated rigorously through
counting concepts, it is not clear why this should be the exclusive criterion, or even a
relevant one, in deciding the relative merit of theories, or why “less” means “better”
(one can conceive of theories of gravity much simpler than general relativity that are
wrong). For example, we might judge a theory’s value by examining the range of
questions it can answer. The real issue is the quality of explanation. A lesser count
could, after all, mean that we are operating with an inadequate system of concepts
relative to the full set of physical questions it is reasonable to pose.
This is, in fact, the starting point of the rational quantum mechanics initiated by de
Broglie and Bohm, in which the origin of the conceptual dilemmas posed by quantum
theory is located in the arbitrary rejection of the notion that a material system has
a well-defined space–time trajectory. The insistence that the wavefunction provides
the most complete physical description that is in principle possible has resulted in
nearly a century of confusion. It is as if one removed a key organ from the body and
struggled on in ill health insisting all the while that nothing was wrong.The simple step
of retaining the trajectory gives the quantum trajectorician a powerful constructive
tool to counter claims about the alleged incomprehensibility of the quantum world
(only one of which is now needed). We shall mention three key benefits of the idea
that offer clear advantages over other interpretations, such as many-worlds.
First, the trajectory opens a door—partially closed by Copenhagen—to a discourse
of conceptual precision about physics. In giving physical meaning to the wavefunction
the theory has recourse to a full range of physical concepts including energy, force,
momentum, mass, and associated notions of stability, structure, size, and form. This
ix
x Foreword
allows a significant improvement in the clarity and consistency with which language
is used in quantum theory, where concepts are now no longer just words attached to
symbols but are set in correspondence with the actual state of matter. In one way or
another, conventional quantum mechanics uses these terms and it is incumbent on a
serious interpretation to provide a benchmark against which to measure the efficacy
with which it does this. It is surely beneficial that we all agree on what we are talk-
ing about, as was unexceptional in pre-quantum physics. Central to this discourse is
Bohm’s perception that the novelty of quantum theory may be expressed through a
new notion of energy. The quantum potential (whose form for two-slit interference
is now an iconic figure in the subject) provides a post-mechanical mode of analysis
where the interactions between the parts of a system are a function of the whole that
contains them, a clear break with classical mechanism where the whole is no more
than the sum of parts whose interactions are prescribed uniquely by their intrinsic
properties. The relative magnitudes of the quantum potential and force can determine
the limits of applicability of the classical description.
In his well-known essays on the foundations of quantum theory, Bell advanced a
version of the de Broglie–Bohm theory that truncates its broad conceptual spectrum,
the theory comprising just a trajectory determined by a first-order guidance law with
an ensemble distributed according to the quantum formula. Certainly, if confronted by
a sceptical audience, tactical necessity may well dictate such a pared-down account.
Whether Bell regarded this emasculated version as the entire theory is not known but at
this primitive stage in the development of trajectory theories such an Occamist stance
would surely be misplaced if elevated to a principle. It is, of course, conceivable that,
in the quantum context, the deeper conceptual content we have alluded to above may
become inapplicable, or at least require modification. Such an analysis has, however,
never been given. Conceptual abstinence may even encourage a reactionary mecha-
nistic reading of de Broglie and Bohm that misses how far the old modes of thought
have been transcended. For, as indicated above, this is not a theory just about trajecto-
ries. Being able to say more—about, say, the meaning of operator eigenvalues, or the
physical content of a spinor field, or the energy changes that account for tunneling, or
the forces responsible for molecular stability—enhances rather than diminishes our
understanding and is hardly redundant knowledge. The particle trajectory realizes its
full constructive role only in this wider historical context of explanation.
The original de Broglie–Bohm guidance equation occupies a position somewhat
analogous to Einstein’s equations in gravitational theory. It was the first, it is battle-
tested, but it is only one of many possible laws that are empirically sufficient within
the domain of phenomena to which the theory applies. In the gravitational case the
contenders have been whittled down by consistency conditions and by enhanced
experimental constraints. In contrast, a comparable quantum-theoretical analysis has
begun only recently. Indeed, what the trajectory theory most needs is a universal and
necessary founding principle, analogous to the principle of equivalence of gravitation
and inertia. It is for this reason, perhaps, that this subject still lacks a generally agreed
name that captures its essence, a necessary milestone in the full emergence of a
scientific theory. And, of course, in the quantum case one cannot appeal to experiment
since the predictions of the formalism being interpreted do not depend on the existence