Quantum Dynamics with Trajectories

Introduction to Quantum Hydrodynamics

By: Robert E. Wyatt, Corey J. Trahan (Contribution by)

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Hardcover | 20 May 2005

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432 Pages

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Remarkable progress has recently been made in the development and application of quantum trajectories as the computational tool for solving the time dependent Schrodinger equation. Analogous methods for stationary bound states are also being developed. The purpose of this book is to present recent developments and applications of quantum trajectory methods in the broader context of the hydrodynamical formulation of quantum dynamics. While many chapters of the book deal with Lagrangian quantum trajectories in which the velocity matches that of the probability fluid, other chapters deal with what will be termed post-Lagrangian trajectories. There are also many state-of-the-art topics covered that are unique to this book. On the pedagogical side, a number of sections will be accessible to students who have had at least one course in quantum dynamics. There is also considerable material for advanced researchers, and chapters in the book cover both methodology and applications.

Industry Reviews

From the reviews:

"This excellent book covers a wide range of topics associated with Quantum Hydrodynamics. It's an excellent survey of the history, current state-of-the-field, and future research directions." (Brian Kendrick, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA)

"The book is unique in that it addresses with equal expertise, computational methodology and theoretical connections at the interface between de Broglie-Bohm theory and phase space moment methods. A highly didactic text, to be recommended to graduate students and researchers in physics and chemistry." (Irene Burghardt, Dept. de chimie, Ecole Normale Superieure, Paris, France)

"Wyatt shows how one can use the ideas drawn from Bohm's interpretation to develop new and efficient computational methods for both time dependent and time independent quantum mechanics.This is THE definitive text on practical Bohmian mechanics." (Eric Bittner, Dept. of Chemistry, University of Houston, TX, USA)

"The book under review is a good introduction to quantum hydrodynamics and an invaluable survey of the subject. It is written in clear and rigorous language and may be helpful both for novices having basic knowledge in quantum mechanics and experts. ... Numerous illustrations, results of numerical computations and historical comments make reading more fascinating." (Sergei A. Nemnyugin, Mathematical Reviews, Issue 2006 k)

"This book presents the state of the art in the quantum trajectory (QT) method and its applications in quantum hydrodynamics. ... In addition to advanced issues at the level of professional researchers, the book also contains many introductory elements that can nicely serve also to graduate students as a suitable extension of their textbook literature on quantum mechanics. Numerous boxes with outlines and summaries of basic results of chapter are particularly valuable to those who need a brief and conciseinformation on a particular item." (Vladimir Cadez, Zentralblatt MATH, Vol. 1107 (9), 2007)

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You Can Find This Book In

Non-Fiction Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Engineering & Technology Mechanical Engineering & Materials Materials Science Mechanics of Fluids Hydraulics & Pneumatics Mathematics Optimisation Calculus & Mathematical Analysis

Differential Calculus & Equations Chemistry Physical Chemistry Numerical Analysis Mechanical Engineering Nuclear Physics

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Paperback

Published: 1st December 2010

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Preface	p. v
Outline of Boxes	p. xv
Historical Comments with Portraits	p. xvii
Sources for Portraits of Physicists	p. xix
Permissions for Use of Figures	p. xxi
Introduction to Quantum Trajectories	p. 1
Dynamics with Quantum Trajectories	p. 1
Routes to Quantum Trajectories	p. 7
The Quantum Trajectory Method	p. 11
Derivative Evaluation on Unstructured Grids	p. 14
Applications of the Quantum Trajectory Method	p. 17
Beyond Bohm Trajectories: Adaptive Methods	p. 18
Approximations to the Quantum Force	p. 21
Propagation of Derivatives Along Quantum Trajectories	p. 22
Trajectories in Phase Space	p. 25
Mixed Quantum-Classical Dynamics	p. 27
Additional Topics in Quantum Hydrodynamics	p. 30
Quantum Trajectories for Stationary States	p. 32
Coping with Problems	p. 33
Topics Not Covered	p. 36
Reading Guide	p. 37
The Bohmian Route to the Hydrodynamic Equations	p. 40
Introduction	p. 40
The Madelung-Bohm Derivation of the Hydrodynamic Equations	p. 42
The Classical Hamilton-Jacobi Equation	p. 48
The Field Equations of Classical Dynamics	p. 52
The Quantum Potential	p. 53
The Quantum Hamilton-Jacobi Equation	p. 56
Pilot Waves, Hidden Variables, and Bohr	p. 59
The Phase Space Route to the Hydrodynamic Equations	p. 62
Introduction	p. 62
Classical Trajectories and Distribution Functions in Phase Space	p. 65
The Wigner Function	p. 68
Moments of the Wigner Function	p. 74
Equations of Motion for the Moments	p. 77
Moment Analysis for Classical Phase Space Distributions	p. 80
Time Evolution of Classical and Quantum Moments	p. 83
Comparison Between Liouville and Hydrodynamic Phase Spaces	p. 85
Discussion	p. 86
The Dynamics and Properties of Quantum Trajectories	p. 89
Introduction	p. 89
Equations of Motion for the Quantum Trajectories	p. 90
Wave Function Synthesis Along a Quantum Trajectory	p. 94
Bohm Trajectory Integral Versus Feynman Path Integral	p. 97
Wave Function Propagation and the Jacobian	p. 99
The Initial Value Representation for Quantum Trajectories	p. 101
The Trajectory Noncrossing Rules	p. 104
Dynamics of Quantum Trajectories Near Wave Function Nodes	p. 104
Chaotic Quantum Trajectories	p. 109
Examples of Chaotic Quantum Trajectories	p. 112
Chaos and the Role of Nodes in the Wave Function	p. 117
Why Weren't Quantum Trajectories Computed 50 Years Ago?	p. 119
Function and Derivative Approximation on Unstructured Grids	p. 123
Introduction	p. 123
Least Squares Fitting Algorithms	p. 127
Dynamic Least Squares	p. 132
Fitting with Distributed Approximating Functionals	p. 135
Derivative Computation via Tessellation and Fitting	p. 138
Finite Element Method for Derivative Computation	p. 141
Summary	p. 144
Applications of the Quantum Trajectory Method	p. 148
Introduction	p. 148
The Free Wave Packet	p. 150
The Anisotropic Harmonic Oscillator	p. 153
The Downhill Ramp Potential	p. 156
Scattering from the Eckart Barrier	p. 161
Discussion	p. 163
Adaptive Methods for Trajectory Dynamics	p. 166
Introduction	p. 166
Hydrodynamic Equations and Adaptive Grids	p. 167
Grid Adaptation with the ALE Method	p. 169
Grid Adaptation Using the Equidistribution Principle	p. 172
Adaptive Smoothing of the Quantum Force	p. 177
Adaptive Dynamics with Hybrid Algorithms	p. 182
Conclusions	p. 187
Quantum Trajectories for Multidimensional Dynamics	p. 190
Introduction	p. 190
Description of the Model for Decoherence	p. 191
Quantum Trajectory Results for the Decoherence Model	p. 194
Quantum Trajectory Results for the Decay of a Metastable State	p. 199
Quantum Trajectory equations for Electronic Nonadiabatic Dynamics	p. 203
Description of the Model for Electronic Nonadiabatic Dynamics	p. 211
Nonadiabatic Dynamics From Quantum Trajectory Propagation	p. 214
Conclusions	p. 215
Approximations to the Quantum Force	p. 218
Introduction	p. 218
Statistical Approach for Fitting the Density to Gaussians	p. 219
Determination of Parameters: Expectation-Maximization	p. 220
Computational Results: Ground Vibrational State of Methyl Iodide	p. 222
Fitting the Density Using Least Squares	p. 225
Global Fit to the Log Derivative of the Density	p. 227
Local Fit to the Log Derivative of the Density	p. 230
Conclusions	p. 233
Derivative Propagation Along Quantum Trajectories	p. 235
Introduction	p. 235
Review of the Hydrodynamic Equations	p. 236
The DPM Derivative Hierarchy	p. 237
Implementation of the DPM	p. 240
Two DPM Examples	p. 241
Multidimensional Extension of the DPM	p. 244
Propagation of the Trajectory Stability Matrix	p. 246
Application of the Trajectory Stability Method	p. 249
Comments and Comparisons	p. 250
Quantum Trajectories in Phase Space	p. 254
Introduction	p. 254
The Liouville, Langevin, and Kramers Equations	p. 255
The Wigner and Husimi Equations	p. 260
The Caldeira-Leggett Equation	p. 266
Phase Space Evolution with Entangled Trajectories	p. 270
Phase Space Evolution Using the Derivative Propagation Method	p. 271
Equations of Motion for Lagrangian Trajectories	p. 273
Examples of Quantum Phase Space Evolution	p. 275
Momentum Moments for Dissipative Dynamics	p. 285
Hydrodynamic Equations for Density Matrix Evolution	p. 288
Examples of Density Matrix Evolution with Trajectories	p. 292
Summary	p. 295
Mixed Quantum-Classical Dynamics	p. 300
Introduction	p. 300
The Ehrenfest Mean Field Approximation	p. 301
Hybrid Hydrodynamical-Liouville Phase Space Method	p. 302
Example of Mixed Quantum-Classical Dynamics	p. 307
The Mixed Quantum-Classical Bohmian Method (MQCB)	p. 308
Examples of the MQCB Method	p. 312
Backreaction Through the Bohmian Particle	p. 316
Discussion	p. 318
Topics in Quantum Hydrodynamics: The Stress Tensor and Vorticity	p. 322
Introduction	p. 322
Stress in the One-Dimensional Quantum Fluid	p. 323
Quantum Navier-Stokes Equation and the Stress Tensor	p. 328
A Stress Tensor Example	p. 329
Vortices in Quantum Dynamics	p. 334
Examples of Vortices in Quantum Dynamics	p. 336
Features of Dynamical Tunneling	p. 343
Vortices and Dynamical Tunneling in the Water Molecule	p. 344
Summary	p. 350
Quantum Trajectories for Stationary States	p. 354
Introduction	p. 354
Stationary Bound States and Bohmian Mechanics	p. 355
The Quantum Stationary Hamilton-Jacobi Equation: QSHJE	p. 356
Floydian Trajectories and Microstates	p. 357
The Equivalence Principle and Quantum Geometry	p. 363
Summary	p. 366
Challenges and Opportunities	p. 369
Introduction	p. 369
Coping with the Spatial Derivative Problem	p. 371
Coping with the Node Problem	p. 372
Decomposition of Wave Function into Counterpropagating Waves	p. 378
Applications of the Covering Function Method	p. 382
Quantum Trajectories and the Future	p. 387
Atomic Units	p. 389
Example QTM Program	p. 390
Index	p. 395
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Quantum Dynamics with Trajectories

Introduction to Quantum Hydrodynamics

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