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E-grāmata: Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics

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Remarkable progress has recently been made in the application of quantumtrajectories as the computational tool for solving quantum mechanical problems. This is the first book to present these developments in the broader context of the hydrodynamical formulation of quantum dynamics. In addition to a thorough discussion of the quantum trajectory equations of motion, there is considerable material that deals with phase space dynamics, adaptive moving grids, electronic energy transfer, and trajectories for stationary states.On the pedagogical side, a number of sections of this book will be accessible to students who have had an introductory quantum mechanics course. There is also considerable material for advanced researchers, and chapters in the book cover both methodology and applications. The book will be useful to students and researchers in physics, chemistry, applied math, and computational dynamics.

This book presents recent developments and applications of quantum trajectory methods in the broader context of the hydrodynamical formulation of quantum dynamics. While many chapters 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.

Recenzijas

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, Dépt. de chimie, Ecole Normale Supérieure, 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 Cade, Zentralblatt MATH, Vol. 1107 (9), 2007)

Preface v
Outline of Boxes xv
Historical Comments with Portraits xvii
Sources for Portraits of Physicists xix
Permissions for Use of Figures xxi
Introduction to Quantum Trajectories
1(39)
Dynamics with Quantum Trajectories
1(6)
Routes to Quantum Trajectories
7(4)
The Quantum Trajectory Method
11(3)
Derivative Evaluation on Unstructured Grids
14(3)
Applications of the Quantum Trajectory Method
17(1)
Beyond Bohm Trajectories: Adaptive Methods
18(3)
Approximations to the Quantum Force
21(1)
Propagation of Derivatives Along Quantum Trajectories
22(3)
Trajectories in Phase Space
25(2)
Mixed Quantum--Classical Dynamics
27(3)
Additional Topics in Quantum Hydrodynamics
30(2)
Quantum Trajectories for Stationary States
32(1)
Coping with Problems
33(3)
Topics Not Covered
36(1)
Reading Guide
37(3)
The Bohmian Route to the Hydrodynamic Equations
40(22)
Introduction
40(2)
The Madelung--Bohm Derivation of the Hydrodynamic Equations
42(6)
The Classical Hamilton--Jacobi Equation
48(4)
The Field Equations of Classical Dynamics
52(1)
The Quantum Potential
53(3)
The Quantum Hamilton--Jacobi Equation
56(3)
Pilot Waves, Hidden Variables, and Bohr
59(3)
The Phase Space Route to the Hydrodynamic Equations
62(27)
Introduction
62(3)
Classical Trajectories and Distribution Functions in Phase Space
65(3)
The Wigner Function
68(6)
Moments of the Wigner Function
74(3)
Equations of Motion for the Moments
77(3)
Moment Analysis for Classical Phase Space Distributions
80(3)
Time Evolution of Classical and Quantum Moments
83(2)
Comparison Between Liouville and Hydrodynamic Phase Spaces
85(1)
Discussion
86(3)
The Dynamics and Properties of Quantum Trajectories
89(34)
Introduction
89(1)
Equations of Motion for the Quantum Trajectories
90(4)
Wave Function Synthesis Along a Quantum Trajectory
94(3)
Bohm Trajectory Integral Versus Feynman Path Integral
97(2)
Wave Function Propagation and the Jacobian
99(2)
The Initial Value Representation for Quantum Trajectories
101(3)
The Trajectory Noncrossing Rules
104(1)
Dynamics of Quantum Trajectories Near Wave Function Nodes
104(5)
Chaotic Quantum Trajectories
109(3)
Examples of Chaotic Quantum Trajectories
112(5)
Chaos and the Role of Nodes in the Wave Function
117(2)
Why Weren't Quantum Trajectories Computed 50 Years Ago?
119(4)
Function and Derivative Approximation on Unstructured Grids
123(25)
Introduction
123(4)
Least Squares Fitting Algorithms
127(5)
Dynamic Least Squares
132(3)
Fitting with Distributed Approximating Functionals
135(3)
Derivative Computation via Tessellation and Fitting
138(3)
Finite Element Method for Derivative Computation
141(3)
Summary
144(4)
Applications of the Quantum Trajectory Method
148(18)
Corey J. Trahan
Introduction
148(2)
The Free Wave Packet
150(3)
The Anisotropic Harmonic Oscillator
153(3)
The Downhill Ramp Potential
156(5)
Scattering from the Eckart Barrier
161(2)
Discussion
163(3)
Adaptive Methods for Trajectory Dynamics
166(24)
Corey J. Trahan
Introduction
166(1)
Hydrodynamic Equations and Adaptive Grids
167(2)
Grid Adaptation with the ALE Method
169(3)
Grid Adaptation Using the Equidistribution Principle
172(5)
Adaptive Smoothing of the Quantum Force
177(5)
Adaptive Dynamics with Hybrid Algorithms
182(5)
Conclusions
187(3)
Quantum Trajectories for Multidimensional Dynamics
190(28)
Introduction
190(1)
Description of the Model for Decoherence
191(3)
Quantum Trajectory Results for the Decoherence Model
194(5)
Quantum Trajectory Results for the Decay of a Metastable State
199(4)
Quantum Trajectory equations for Electronic Nonadiabatic Dynamics
203(8)
Description of the Model for Electronic Nonadiabatic Dynamics
211(3)
Nonadiabatic Dynamics From Quantum Trajectory Propagation
214(1)
Conclusions
215(3)
Approximations to the Quantum Force
218(17)
Introduction
218(1)
Statistical Approach for Fitting the Density to Gaussians
219(1)
Determination of Parameters: Expectation-Maximization
220(2)
Computational Results: Ground Vibrational State of Methyl Iodide
222(3)
Fitting the Density Using Least Squares
225(2)
Global Fit to the Log Derivative of the Density
227(3)
Local Fit to the Log Derivative of the Density
230(3)
Conclusions
233(2)
Derivative Propagation Along Quantum Trajectories
235(19)
Introduction
235(1)
Review of the Hydrodynamic Equations
236(1)
The DPM Derivative Hierarchy
237(3)
Implementation of the DPM
240(1)
Two DPM Examples
241(3)
Multidimensional Extension of the DPM
244(2)
Propagation of the Trajectory Stability Matrix
246(3)
Application of the Trajectory Stability Method
249(1)
Comments and Comparisons
250(4)
Quantum Trajectories in Phase Space
254(46)
Introduction
254(1)
The Liouville, Langevin, and Kramers Equations
255(5)
The Wigner and Husimi Equations
260(6)
The Caldeira--Leggett Equation
266(4)
Phase Space Evolution with Entangled Trajectories
270(1)
Phase Space Evolution Using the Derivative Propagation Method
271(2)
Equations of Motion for Lagrangian Trajectories
273(2)
Examples of Quantum Phase Space Evolution
275(10)
Momentum Moments for Dissipative Dynamics
285(3)
Hydrodynamic Equations for Density Matrix Evolution
288(4)
Examples of Density Matrix Evolution with Trajectories
292(3)
Summary
295(5)
Mixed Quantum--Classical Dynamics
300(22)
Introduction
300(1)
The Ehrenfest Mean Field Approximation
301(1)
Hybrid Hydrodynamical--Liouville Phase Space Method
302(5)
Example of Mixed Quantum--Classical Dynamics
307(1)
The Mixed Quantum--Classical Bohmian Method (MQCB)
308(4)
Examples of the MQCB Method
312(4)
Backreaction Through the Bohmian Particle
316(2)
Discussion
318(4)
Topics in Quantum Hydrodynamics: The Stress Tensor and Vorticity
322(32)
Introduction
322(1)
Stress in the One-Dimensional Quantum Fluid
323(5)
Quantum Navier-Stokes Equation and the Stress Tensor
328(1)
A Stress Tensor Example
329(5)
Vortices in Quantum Dynamics
334(2)
Examples of Vortices in Quantum Dynamics
336(7)
Features of Dynamical Tunneling
343(1)
Vortices and Dynamical Tunneling in the Water Molecule
344(6)
Summary
350(4)
Quantum Trajectories for Stationary States
354(15)
Introduction
354(1)
Stationary Bound States and Bohmian Mechanics
355(1)
The Quantum Stationary Hamilton--Jacobi Equation: QSHJE
356(1)
Floydian Trajectories and Microstates
357(6)
The Equivalence Principle and Quantum Geometry
363(3)
Summary
366(3)
Challenges and Opportunities
369(20)
Introduction
369(2)
Coping with the Spatial Derivative Problem
371(1)
Coping with the Node Problem
372(6)
Decomposition of Wave Function into Counterpropagating Waves
378(4)
Applications of the Covering Function Method
382(5)
Quantum Trajectories and the Future
387(2)
Appendix 1: Atomic Units 389(1)
Appendix 2: Example QTM Program 390(5)
Index 395


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