Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics

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Springer Science & Business Media, May 20, 2005 - Mathematics - 408 pages
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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.

 

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Contents

Introduction to Quantum Trajectories
1
12 Routes to Quantum Trajectories
7
13 The Quantum Trajectory Method
11
14 Derivative Evaluation on Unstructured Grids
14
15 Applications of the Quantum Trajectory Method
17
Adaptive Methods
18
17 Approximations to the Quantum Force
21
18 Propagation of Derivatives Along Quantum Trajectories
22
85 Quantum Trajectory equations for Electronic Nonadiabatic Dynamics
203
86 Description of the Model for Electronic Nonadiabatic Dynamics
211
87 Nonadiabatic Dynamics From Quantum Trajectory Propagation
214
88 Conclusions
215
Approximations to the Quantum Force
218
92 Statistical Approach for Fitting the Density to Gaussians
219
ExpectationMaximization
220
Ground Vibrational State of Methyl Iodide
222

19 Trajectories in Phase Space
25
110 Mixed QuantumClassical Dynamics
27
111 Additional Topics in Quantum Hydrodynamics
30
112 Quantum Trajectories for Stationary States
32
113 Coping with Problems
33
114 Topics Not Covered
36
115 Reading Guide
37
The Bohmian Route to the Hydrodynamic Equations
40
22 The MadelungBohm Derivation of the Hydrodynamic Equations
42
23 The Classical HamiltonJacobi Equation
48
24 The Field Equations of Classical Dynamics
52
25 The Quantum Potential
53
26 The Quantum HamiltonJacobi Equation
56
27 Pilot Waves Hidden Variables and Bohr
59
The Phase Space Route to the Hydrodynamic Equations
62
32 Classical Trajectories and Distribution Functions in Phase Space
65
33 The Wigner Function
68
34 Moments of the Wigner Function
74
35 Equations of Motion for the Moments
77
36 Moment Analysis for Classical Phase Space Distributions
80
37 Time Evolution of Classical and Quantum Moments
83
38 Comparison Between Liouville and Hydrodynamic Phase Spaces
85
39 Discussion
86
The Dynamics and Properties of Quantum Trajectories
89
42 Equations of Motion for the Quantum Trajectories
90
43 Wave Function Synthesis Along a Quantum Trajectory
94
44 Bohm Trajectory Integral Versus Feynman Path Integral
97
45 Wave Function Propagation and the Jacobian
99
46 The Initial Value Representation for Quantum Trajectories
101
47 The Trajectory Noncrossing Rules
104
49 Chaotic Quantum Trajectories
109
410 Examples of Chaotic Quantum Trajectories
112
411 Chaos and the Role of Nodes in the Wave Function
117
412 Why Werent Quantum Trajectories Computed 50 Years Ago?
119
Function and Derivative Approximation on Unstructured Grids
123
52 Least Squares Fitting Algorithms
127
53 Dynamic Least Squares
132
54 Fitting with Distributed Approximating Functionals
135
55 Derivative Computation via Tessellation and Fitting
138
56 Finite Element Method for Derivative Computation
141
57 Summary
144
Applications of the Quantum Trajectory Method
148
62 The Free Wave Packet
150
63 The Anisotropic Harmonic Oscillator
153
64 The Downhill Ramp Potential
156
65 Scattering from the Eckart Barrier
161
66 Discussion
163
Adaptive Methods for Trajectory Dynamics
166
72 Hydrodynamic Equations and Adaptive Grids
167
73 Grid Adaptation with the ALE Method
169
74 Grid Adaptation Using the Equidistribution Principle
172
75 Adaptive Smoothing of the Quantum Force
177
76 Adaptive Dynamics with Hybrid Algorithms
182
77 Conclusions
187
Quantum Trajectories for Multidimensional Dynamics
190
82 Description of the Model for Decoherence
191
83 Quantum Trajectory Results for the Decoherence Model
194
84 Quantum Trajectory Results for the Decay of a Metastable State
199
95 Fitting the Density Using Least Squares
225
96 Global Fit to the Log Derivative of the Density
227
97 Local Fit to the Log Derivative of the Density
230
98 Conclusions
233
Derivative Propagation Along Quantum Trajectories
235
102 Review of the Hydrodynamic Equations
236
103 The DPM Derivative Hierarchy
237
104 Implementation of the DPM
240
105 Two DPM Examples
241
106 Multidimensional Extension of the DPM
244
107 Propagation of the Trajectory Stability Matrix
246
108 Application of the Trajectory Stability Method
249
109 Comments and Comparisons
250
Quantum Trajectories in Phase Space
254
112 The Liouville Langevin and Kramers Equations
255
113 The Wigner and Husimi Equations
260
114 The CaldeiraLeggett Equation
266
115 Phase Space Evolution with Entangled Trajectories
270
116 Phase Space Evolution Using the Derivative Propagation Method
271
117 Equations of Motion for Lagrangian Trajectories
273
118 Examples of Quantum Phase Space Evolution
275
119 Momentum Moments for Dissipative Dynamics
285
1110 Hydrodynamic Equations for Density Matrix Evolution
288
1111 Examples of Density Matrix Evolution with Trajectories
292
1112 Summary
295
Mixed QuantumClassical Dynamics
300
122 The Ehrenfest Mean Field Approximation
301
123 Hybrid HydrodynamicalLiouville Phase Space Method
302
124 Example of Mixed QuantumClassical Dynamics
307
125 The Mixed QuantumClassical Bohmian Method MQCB
308
126 Examples of the MQCB Method
312
127 Backreaction Through the Bohmian Particle
316
128 Discussion
318
Topics in Quantum Hydrodynamics The Stress Tensor and Vorticity
322
132 Stress in the OneDimensional Quantum Fluid
323
133 Quantum NavierStokes Equation and the Stress Tensor
328
134 A Stress Tensor Example
329
135 Vortices in Quantum Dynamics
334
136 Examples of Vortices in Quantum Dynamics
336
137 Features of Dynamical Tunneling
343
138 Vortices and Dynamical Tunneling in the Water Molecule
344
139 Summary
350
Quantum Trajectories for Stationary States
354
142 Stationary Bound States and Bohmian Mechanics
355
QSHJE
356
144 Floydian Trajectories and Microstates
357
145 The Equivalence Principle and Quantum Geometry
363
146 Summary
366
Challenges and Opportunities
369
152 Coping with the Spatial Derivative Problem
371
153 Coping with the Node Problem
372
154 Decomposition of Wave Function into Counterpropagating Waves
378
155 Applications of the Covering Function Method
382
156 Quantum Trajectories and the Future
387
Atomic Units
389
Example QTM Program
390
Index
395
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