Quantum Trajectories (Atoms, Molecules, and Clusters)

Contents......Page 6
Foreword......Page 9
Preface......Page 12
Editor......Page 14
List of Contributors......Page 15
1.1 Bohmian Trajectories......Page 18
1.2 Bohmian Mechanics......Page 20
1.4 The Quantum Potential......Page 21
1.5 Connection with Numerical Methods......Page 22
1.7 Wave–Particle Duality......Page 24
1.8 The Phase Function S(q, t )......Page 25
1.9 Predictions and the Quantum Formalism......Page 26
1.10 The Generalized Quantum Formalism......Page 27
1.11 What is Unsatisfactory About Standard Quantum Mechanics?......Page 28
1.12 Spin......Page 29
1.13 The Symmetrization Postulate......Page 30
1.14 Further Reading......Page 31
Bibliography......Page 32
2.1 The Hamilton–Jacobi Equation and Coordinate Transformations......Page 34
2.2 The Equivalence Postulate......Page 36
2.3 Proof of Theorem 1......Page 41
2.4 Proof of Theorem 2......Page 47
Bibliography......Page 54
3.1 Introduction......Page 58
3.2 Recipe......Page 59
3.3.1 Wave Packet Spreading......Page 63
3.3.2 Simplified Double-Slit Experiment......Page 64
3.3.3 Quantum Trajectory for EPR-Molecule......Page 66
Bibliography......Page 67
4.1 A First Date......Page 70
4.2.1 Hamiltonian Formulation of SUSY......Page 71
4.2.2 SUSY Algebra......Page 72
4.2.3 Scattering in One Dimension......Page 73
4.2.4 Non-Stationary States......Page 74
4.3.1 Using SUSY to Improve Quality of Variational Calculations......Page 75
4.3.2 Monte Carlo SUSY......Page 77
4.4 Test Case: Tunneling in a Double Well Potential......Page 78
4.5 Extension of SUSY to Multiple Dimensions......Page 81
4.5.1 Difficulties in Extending Beyond One Dimension......Page 82
4.5.2 Vector SUSY......Page 83
Bibliography......Page 87
5.1 Introduction......Page 90
5.2 Spin and Trajectories......Page 91
5.3.1 Lagrangian Picture......Page 93
5.3.2 Initial Conditions......Page 95
5.3.3 Eulerian Picture......Page 96
5.4 Bosons......Page 98
5.5 Fermions......Page 99
Bibliography......Page 102
6.1 Introduction......Page 104
6.2 From Schrödinger to Bohm......Page 105
6.3.1 General Remarks......Page 106
6.3.3 Nature of the Quantum Force......Page 107
6.4.1 Uniform Forces......Page 108
6.4.3 Linear Forces......Page 109
Bibliography......Page 111
7.1 Introduction......Page 112
7.2 Dynamics in Phase Space......Page 113
7.3 Quantum Trajectories......Page 116
7.4 Entangled Trajectory Molecular Dynamics......Page 118
7.5 Husimi Representation......Page 122
7.6 Integrodifferential Equation Form......Page 124
7.7 Gauge Freedom in Phase Space......Page 127
Bibliography......Page 128
8.1 Introduction and Motivation......Page 130
8.2 The Setup......Page 134
8.3 The Testable Probabilities......Page 135
8.4 The Evaluation of Π(t) and Π(φ) Using the Bohmian Model......Page 136
8.5 Subtleties in the Bohmian Calculation of Π(t) and Π(φ)......Page 141
8.6 Summary and Outlook......Page 143
Appendix A......Page 144
Bibliography......Page 148
9.1 Introduction......Page 152
9.2 Globally Defined Approximate Quantum Potential (AQP)......Page 155
9.3 AQP Dynamics with Balanced Approximation Errors......Page 158
9.4 Conclusions......Page 163
Bibliography......Page 165
10.1 Introduction......Page 166
10.2 Bohmian Trajectories and the MQCB Method......Page 168
10.3 Stochastic DVR and MQCS......Page 171
10.4.1 Vink's Formulation......Page 173
10.4.2 N-Point Formula......Page 175
Bibliography......Page 177
11.1 Introduction......Page 180
11.2.1 System-Bath Hamiltonian......Page 182
11.2.2 Effective-Mode Representation......Page 183
11.2.3 Effective-Mode Chain Including Markovian Closure......Page 184
11.2.4 Spectral Densities......Page 185
11.2.5 Reduced-Dimensional Effective-Mode Chain Including Markoviant Closure......Page 186
11.2.6 Partial Moments......Page 187
11.2.7 Lagrangian Trajectories......Page 189
11.3 Results and Discussion......Page 190
11.4 Conclusion......Page 195
Bibliography......Page 196
12.1 Introduction......Page 200
12.2 Quantum Fluid Dynamical Transcription of Quantum Mechanics (QM) of a Single Particle......Page 202
12.3 Quantum Fluid Dynamical Equations for Many-Particle Systems in Configuration Space......Page 203
12.4 Density Functional Theory of Many-Electron Systems: A Single-Particle Framework for QFD in 3-D Space......Page 205
12.5 Quantum Fluid Dynamical Equations within a Density Functional Framework......Page 208
12.6 Miscellaneous Aspects of the Quantum Fluid Dynamical Approach......Page 210
Bibliography......Page 211
13.1 Introduction......Page 214
13.2.1 Some Basic Theoretical Background......Page 216
13.2.2 The Two-Slit Experiment......Page 218
13.2.3 Many Slits: The Talbot Effect and Multimode Cavities......Page 221
13.2.4 Interference in Two-Dimensions: Vortical Dynamics......Page 222
13.3 Quantum Flux Analysis......Page 223
13.4 Effective Interference Potentials......Page 226
13.5 Quantum Interference in the Complex Plane......Page 231
13.6 Concluding Remarks......Page 234
Bibliography......Page 236
14.1 Quantum Fluid Density Functional Theory......Page 240
14.2 Chemical Reactivity Dynamics......Page 242
Bibliography......Page 249
15.1 Introduction......Page 252
15.2 The Bipolar Decomposition......Page 255
15.3 Stationary Bound States in 1D......Page 256
15.4 Stationary Scattering States in 1D......Page 259
15.5 Wavepacket Scattering Dynamics in 1D......Page 260
15.7 Numerical Implementations......Page 263
15.8 Conclusions......Page 265
Bibliography......Page 266
16.1 Introduction......Page 268
16.2 Bohmian Evolution for ψ ∈ C2......Page 269
16.3 Bohmian Evolution for Ψt ∈ C∞h......Page 271
16.4 Bohmian Evolution for Ψt ∈ L2 \ C∞h......Page 272
16.5 Hall's Counterexample......Page 274
Bibliography......Page 278
17.1 Introduction......Page 280
17.2.1 Quantum Trajectory Equations for Nonadiabatic Dynamics......Page 282
17.2.2 Algorithm and Numerical Details......Page 284
17.3.1 Application of the Exact Multistate Quantum Trajectory Method......Page 286
17.3.2 Mixed Representations......Page 288
17.3.3 Bipolar Wave Packets......Page 293
17.4 Summary and Conclusion......Page 296
Bibliography......Page 297
18.1 Introduction......Page 300
18.2.1 Time-Dependent Problems......Page 302
18.2.2 Time-Independent Problems......Page 303
18.3.1 Quantum Trajectories for One-Dimensional Stationary Scattering Problems......Page 304
18.3.2 Quantum Vortices and Streamlines within the Complex Quantum Hamilton–Jacobi Formalism......Page 307
18.3.3 Quantum Caves and Wave-Packet Interference......Page 310
18.3.4 Complex-Extended Born Probability Density......Page 313
Acknowledgments......Page 314
Bibliography......Page 315
19.1 Historical Introduction......Page 318
19.2 Complex Quantum Trajectories......Page 320
19.3 Probability from the Velocity Field......Page 322
19.4 Probability Distribution for the Subnests......Page 326
19.5 Discussion......Page 327
Bibliography......Page 328
20.1 Introduction......Page 330
20.2.1 Variational Principle......Page 331
20.2.2 Variational Multiconfigurational Gaussians (vMCG)......Page 332
20.2.3 Variational Single Configurational Gaussian (vSCG)......Page 333
20.2.4 Coupled Coherent States (CCS)......Page 334
20.2.6 The Ehrenfest Method......Page 335
20.2.7 The Multiconfigurational Ehrenfest Method......Page 336
Appendix: Variational Principle and the Equations of G-MCTDH......Page 337
Bibliography......Page 339
21.1 Introduction......Page 342
21.2 The Quantum Hydrodynamic Equations of Motion......Page 345
21.3 The MLS/ALE Method......Page 346
21.4.1 One to Four-Dimensional Scattering Off an Eckart Barrier......Page 348
21.4.2 One-Dimensional Scattering Off a Rounded Square Barrier......Page 351
21.4.3 Dynamical Approximations for Treating Quantum Many-Body Systems......Page 354
21.5 The Iterative Finite Difference Method......Page 356
21.6 Concluding Remarks and Future Research......Page 359
Bibliography......Page 360
22.1 Introduction......Page 362
22.2 Problems of the Numerical Implementation......Page 366
22.3 Fitting Algorithms: What They Can Do and What They Can't......Page 369
22.4 A Numerical Example in MATLAB®......Page 372
Bibliography......Page 376
23.1 Introduction......Page 378
23.2 Calculation of N(E) Within the Flux Operator Formalism With Approximate Quantum Trajectories......Page 380
23.2.1 Real-Time Variational Approximate Quantum Potential......Page 381
23.2.3 Numerical Illustration......Page 383
23.3.1 Equations of Motion......Page 385
23.3.2 Approximate Implementation and Excited States......Page 386
23.3.3 Calculation of the Energy Levels......Page 388
23.4 Computation of Reaction Rates From the Imaginary and Real-Time Quantum Trajectory Dynamics......Page 390
23.5 Summary......Page 392
Bibliography......Page 394
24.1 Introduction......Page 398
24.2.1 One-Dimensional Hamiltonian Systems......Page 399
24.2.2 Two-Dimensional Hamiltonian Systems......Page 400
24.2.3 Time-Reversible Systems......Page 402
24.3.2 Results......Page 403
Acknowledgments......Page 405
Bibliography......Page 406