## Quantum Dynamics with Trajectories: Introduction to Quantum HydrodynamicsRemarkable 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 |

395 | |

### Other editions - View all

Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics Robert E. Wyatt Limited preview - 2006 |

Quantum Dynamics with Trajectories: Introduction to Quantum Hydrodynamics Robert E. Wyatt No preview available - 2010 |