Most textbooks explain quantum mechanics as a story where each step follows naturally from the one preceding it. However, the development of quantum mechanics was exactly the opposite. It was a zigzag route, full of personal disputes where scientists were forced to abandon well-established classical concepts and to explore new and imaginative pathways. Some of the explored routes were successful in providing new mathematical formalisms capable of predicting experiments at the atomic scale. However, even such successful routes were painful enough, so that relevant scientists like Albert Einstein and Erwin Schrodinger decided not to support them.
In this book, the authors demonstrate the huge practical utility of another of these routes in explaining quantum phenomena in many different research fields. Bohmian mechanics, the formulation of the quantum theory pioneered by Louis de Broglie and David Bohm, offers an alternative mathematical formulation of quantum phenomena in terms of quantum trajectories. Novel computational tools to explore physical scenarios that are currently computationally inaccessible, such as many-particle solutions of the Schrodinger equation, can be developed from it.
Author(s): Xavier Oriols Pladevall; Jordi Mompart
Publisher: Jenny Stanford Publishing
Year: 2019
Language: English
Pages: 674
Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Table of Contents......Page 6
Foreword to the First Edition......Page 18
Preface to the Second Edition......Page 20
Preface to the First Edition......Page 24
Introduction......Page 28
1: Overview of Bohmian Mechanics......Page 46
1.1.1 Particles and Waves......Page 47
1.1.2 Origins of the Quantum Theory......Page 49
1.1.3 “Wave or Particle?” vs. “Wave and Particle”......Page 51
1.1.5 Albert Einstein and Locality......Page 56
1.1.6 David Bohm and Why the “Impossibility Proofs” were Wrong?......Page 58
1.1.7 John Bell and Nonlocality......Page 62
1.1.8 Quantum Hydrodynamics......Page 64
1.1.9 Is Bohmian Mechanics a Useful Theory?......Page 65
1.2 Bohmian Mechanics for a Single Particle......Page 66
1.2.1 Preliminary Discussions......Page 67
1.2.2.2 Hamilton’s principle......Page 68
1.2.2.3 Lagrange’s equation......Page 70
1.2.2.4 Equation for an (infinite) ensemble of trajectories......Page 71
1.2.2.6 Local continuity equation for an (infinite) ensemble of classical particles......Page 74
1.2.2.7 Classical wave equation......Page 75
1.2.3.1 Schrödinger equation......Page 76
1.2.3.2 Local conservation law for an (infinite) ensemble of quantum trajectories......Page 77
1.2.3.4 Quantum Hamilton–Jacobi equation......Page 78
1.2.3.5 A quantum Newton-like equation......Page 80
1.2.4 Similarities and Differences between Classical and Quantum Mechanics......Page 81
1.2.5 Feynman Paths......Page 85
1.2.6 Basic Postulates for a Single-Particle......Page 87
1.3.1 Preliminary Discussions: The Many Body Problem......Page 90
1.3.2.1 Many-particle continuity equation......Page 93
1.3.2.2 Many-particle quantum Hamilton–Jacobi equation......Page 94
1.3.3 Factorizability, Entanglement, and Correlations......Page 95
1.3.4.1 Single-particle with s = 1/2......Page 98
1.3.4.2 Many-particle system with s = 1/2 particles......Page 101
1.3.5 Basic Postulates for Many-Particle Systems......Page 103
1.3.6 The Conditional Wave Function: Many-Particle Bohmian Trajectories without the Many-Particle Wave Function......Page 106
1.3.6.1 Single-particle pseudo-Schrödinger equation for many-particle systems......Page 108
1.3.6.2 Example: Application in factorizable many-particle systems......Page 111
1.3.6.3 Example: Application in interacting many-particle systems without exchange interaction......Page 112
1.3.6.4 Example: Application in interacting many-particle systems with exchange interaction......Page 114
1.4.1 The Measurement Problem......Page 120
1.4.1.1 The orthodox measurement process......Page 122
1.4.1.2 The Bohmian measurement process......Page 125
1.4.2 Theory of the Bohmian Measurement Process......Page 126
1.4.2.1 Example: Bohmian measurement of the momentum......Page 133
1.4.2.2 Example: Sequential Bohmian measurement of the transmitted and reflected particles......Page 135
1.4.3.1 Why Hermitian operators in Bohmian mechanics?......Page 141
1.4.3.2 Mean value from the list of outcomes and their probabilities......Page 142
1.4.3.4 Mean value from Bohmian mechanics in the position representation......Page 143
1.4.3.5 Mean value from Bohmian trajectories......Page 144
1.4.3.6 On the meaning of local Bohmian operators AB(x)......Page 146
1.5 Concluding Remarks......Page 147
1.6 Problems and Solutions......Page 149
A.1 Appendix: Numerical Algorithms for the Computation of Bohmian Mechanics......Page 162
A.1.1.1 Time-dependent Schrödinger equation for a 1D space (TDSE1D-BT) with an explicit method......Page 165
A.1.1.2 Time-independent Schrödinger equation for a 1D space (TISE1D) with an implicit (matrix inversion) method......Page 169
A.1.1.3 Time-independent Schrödinger equation for a 1D space (TISE1D) with an explicit method......Page 172
A.1.2 Synthetic Computation of Bohmian Trajectories......Page 176
A.1.2.1 Time-dependent quantum Hamilton–Jacobi equations (TDQHJE1D) with an implicit (Newton-like fixed Eulerian mesh) method......Page 177
A.1.2.2 Time-dependent quantum Hamilton–Jacobi equations (TDQHJE1D) with an explicit (Lagrangian mesh) method......Page 180
A.1.3 More Elaborated Algorithms......Page 182
2: Hydrogen Photoionization with Strong Lasers......Page 194
2.1.1 A Brief Overview of Photoionization......Page 195
2.1.2 The Computational Problem of Photoionization......Page 197
2.1.3 Photoionization with Bohmian Trajectories......Page 198
2.2.1 The Physical Model......Page 201
2.2.2 Harmonic Generation......Page 204
2.2.3 Above Threshold Ionization......Page 209
2.3.1 Physical System......Page 214
2.3.2 Bohmian Equations in an Electromagnetic Field......Page 218
2.3.3 Selection Rules......Page 219
2.3.4 Numerical Simulations......Page 220
2.3.4.1 Gaussian pulses......Page 221
2.3.4.2 Laguerre–Gaussian pulses......Page 223
2.4 Conclusions......Page 229
3: Atomtronics: Coherent Control of Atomic Flow via Adiabatic Passage......Page 238
3.1.1 Atomtronics......Page 239
3.1.2 Three-Level Atom Optics......Page 240
3.1.3 Adiabatic Transport with Trajectories......Page 243
3.2 Physical System: Neutral Atoms in Optical Microtraps......Page 247
3.2.1 One-Dimensional Hamiltonian......Page 248
3.3.1 The Matter Wave STIRAP Paradox with Bohmian Trajectories......Page 249
3.3.2 Velocities and Accelerations of Bohmian Trajectories......Page 251
3.4.1 Hole Transfer as an Array-Cleaning Technique......Page 255
3.4.2.1 Three-level approximation description......Page 256
3.4.3 Hole Transport Fidelity......Page 259
3.4.5 Atomtronics with Holes......Page 262
3.4.5.1 Single-hole diode......Page 263
3.4.5.2 Single-hole transistor......Page 266
3.5 Adiabatic Transport of a Bose–Einstein Condensate......Page 269
3.5.2 Numerical Simulations......Page 271
3.6 Conclusions......Page 275
4: Bohmian Pathways into Chemistry: A Brief Overview......Page 284
4.1 Introduction......Page 285
4.2 Approaching Molecular Systems at Different Levels......Page 290
4.2.1 The Born–Oppenheimer Approximation......Page 291
4.2.2 Electronic Configuration......Page 295
4.2.3 Dynamics of “Small” Molecular Systems......Page 298
4.2.4 Statistical Approach to Large (Complex) Molecular Systems......Page 301
4.3.1 Fundamentals......Page 304
4.3.2 Nonlocality and Entanglement......Page 309
4.3.3 Weak Values and Equations of Change......Page 312
4.4.1 Time-Dependent DFT: The Quantum Hydrodynamic Route......Page 315
4.4.2 Bound System Dynamics: Chemical Reactivity......Page 320
4.4.3 Scattering Dynamics: Young’s Two-Slit Experiment......Page 328
4.4.4 Effective Dynamical Treatments: Decoherence and Reduced Bohmian Trajectories......Page 332
4.4.5 Pathways to Complex Molecular Systems: Mixed Bohmian-Classical Mechanics......Page 333
4.5 Concluding Remarks......Page 336
5: Adaptive Quantum Monte Carlo Approach States for High-Dimensional Systems......Page 358
5.1 Introduction......Page 359
5.2 Mixture Modeling Approach......Page 360
5.2.1 Motivation for a Trajectory-Based Approach......Page 361
5.2.1.1 Bohmian interpretation......Page 363
5.2.1.2 Quantum hydrodynamic trajectories......Page 365
5.2.1.3 Computational considerations......Page 366
5.2.2.1 The mixture model......Page 368
5.2.2.2 Expectation maximization......Page 370
5.2.3.1 Bivariate distribution with multiple nonseparable Gaussian components......Page 373
5.2.4 The Ground State of Methyl Iodide......Page 379
5.3 Quantum Effects in Atomic Clusters at Finite Temperature......Page 383
5.4.1 Zero Temperature Theory......Page 384
5.4.2 Finite Temperature Theory......Page 386
5.4.2.1 Computational approach: The mixture model......Page 389
5.4.2.2 Computational approach: Equations of motion for the sample points......Page 391
5.4.3.1 Zero temperature results......Page 392
5.4.3.2 Finite temperature results......Page 397
5.5 Overcoming the Node Problem......Page 405
5.5.1 Supersymmetric Quantum Mechanics......Page 407
5.5.2 Implementation of SUSY QM in an Adaptive Monte Carlo Scheme......Page 409
5.5.3 Test Case: Tunneling in a Double-Well Potential......Page 410
5.5.4 Extension to Higher Dimensions......Page 414
5.6 Summary......Page 415
6: Nanoelectronics: Quantum Electron Transport......Page 426
6.1 Introduction: From Electronics to Nanoelectronics......Page 427
6.2 Evaluation of the Electrical Current and Its Fluctuations......Page 429
6.2.1 Bohmian Measurement of the Current as a Function of the Particle Positions......Page 430
6.2.1.1 Relationship between current in the ammeter Iammeter, g(t) and the current in the device-active region Ig(t)......Page 432
6.2.1.2 Relationship between the current on the device-active region Ig(t) and the Bohmian trajectories {r1,g[t], . . . , rMP,g[t]}......Page 433
6.2.1.3 Reducing the number of degrees of freedom of the whole circuit......Page 435
6.2.2 Practical Computation of DC, AC, and Transient Currents......Page 438
6.2.3 Practical Computation of Current Fluctuations and Higher Moments......Page 440
6.2.3.1 Thermal and shot noise......Page 441
6.2.3.2 Practical computation of current fluctuations......Page 442
6.3.1 Coulomb Interaction Among Electrons......Page 444
6.3.2 Exchange and Coulomb Interaction Among Electrons......Page 446
6.3.2.1 Algorithm for spinless electrons......Page 447
6.3.2.2 Algorithm for electrons with spins in arbitrary directions......Page 448
6.4 Dissipation with Bohmian Mechanics......Page 449
6.4.1 Parabolic Band Structures: Pseudo Schrödinger Equation......Page 450
6.4.2 Linear Band Structures: Pseudo Dirac Equation......Page 451
6.5.1 Overall Charge Neutrality and Current Conservation......Page 452
6.5.1.1 The Poisson equation in the simulation box......Page 453
6.5.1.2 Time-dependent boundary conditions for the Poisson equation......Page 454
6.5.2 Practical Computation of Time-Dependent Electrical Currents......Page 455
6.5.2.2 The Ramo-Shockley-Pellegrini method for the computation of the total current......Page 457
6.6.1 Device Characteristics and Available Simulation Models......Page 459
6.6.2.1 Coulomb interaction in DC scenarios......Page 462
6.6.2.2 Coulomb interaction in high-frequency scenarios......Page 463
6.6.2.3 Current-current correlations......Page 468
6.6.2.4 RTD with dissipation......Page 470
6.7 Application of the BITLLES Simulator to Graphene and 2D Linear Band Structures......Page 472
6.7.2 Numerical Results......Page 475
6.8 Conclusions......Page 479
7: Beyond the Eikonal Approximation in Classical Optics and Quantum Physics......Page 490
7.1 Introduction......Page 491
7.2 Helmholtz Equation and Geometrical Optics......Page 493
7.3 Beyond the Geometrical Optics Approximation......Page 495
7.4 The Time-Independent Schrödinger Equation......Page 497
7.5 Hamiltonian Description of Quantum Particle Motion......Page 499
7.6 The Unique Dimensionless Hamiltonian System......Page 500
7.7 Wave-Like Features in Hamiltonian Form......Page 503
7.8 Discussion and Conclusions......Page 514
A.1 Appendix: The Paraxial Approach......Page 516
8: Relativistic Quantum Mechanics and Quantum Field Theory......Page 520
8.1 Introduction......Page 521
8.2.1 Kinematics......Page 523
8.2.2.1 Action and equations of motion......Page 525
8.2.2.2 Canonical momentum and the Hamilton–Jacobi formulation......Page 528
8.2.2.3 Generalization to many particles......Page 529
8.2.2.4 Absolute time......Page 531
8.3.1 Wave Functions and Their Relativistic Probabilistic Interpretation......Page 532
8.3.2 Theory of Quantum Measurements......Page 535
8.3.3 Relativistic Wave Equations......Page 537
8.3.3.1 Single particle without spin......Page 538
8.3.3.2 Many particles without spin......Page 539
8.3.3.3 Single particle with spin 1/2......Page 540
8.3.3.4 Many particles with spin 1/2......Page 543
8.3.3.5 Particles with spin 1......Page 544
8.3.4 Bohmian Interpretation......Page 546
8.4.1 Main Ideas of QFT and Its Bohmian Interpretation......Page 549
8.4.2 Measurement in QFT as Entanglement with the Environment......Page 553
8.4.3 Free Scalar QFT in the Particle-Position Picture......Page 555
8.4.4 Generalization to Interacting QFT......Page 560
8.4.5 Generalization to Other Types of Particles......Page 562
8.4.6 Probabilistic Interpretation......Page 563
8.4.7 Bohmian Interpretation......Page 565
8.5 Conclusion......Page 568
9: Quantum Accelerating Universe......Page 572
9.1 Introduction......Page 573
9.2 The Original Quantum Dark-Energy Model......Page 576
9.3.1 The Klein–Gordon Quantum Model......Page 580
9.3.2 Quantum Theory of Special Relativity......Page 581
9.4 Dark Energy Without Dark Energy......Page 587
9.5.1 Thermodynamics......Page 596
9.5.2 Violation of Classical NEC......Page 600
9.5.3 Holographic Models......Page 601
9.5.4 Quantum Cosmic Models and Entanglement Entropy......Page 604
9.6 Generalized Cosmic Solutions......Page 605
9.7 Gravitational Waves and Semiclassical Instability......Page 610
9.8 On the Onset of the Cosmic Accelerating Phase......Page 614
9.9 Conclusions and Comments......Page 621
10: Bohmian Quantum Gravity and Cosmology......Page 634
10.1 Introduction......Page 635
10.2 Nonrelativistic Bohmian Mechanics......Page 637
10.3 Canonical Quantum Gravity......Page 640
10.4 Bohmian Canonical Quantum Gravity......Page 643
10.5 Minisuperspace......Page 646
10.6 Space-Time Singularities......Page 648
10.6.1 Minisuperspace: Canonical Scalar Field......Page 649
10.6.1.1 Free massless scalar field......Page 650
10.6.1.2 The exponential potential......Page 652
10.6.2 Minisuperspace: Perfect Fluid......Page 657
10.6.3 Loop Quantum Cosmology......Page 661
10.7 Cosmological Perturbations......Page 665
10.7.1 Cosmological Perturbations in a Quantum Cosmological Background......Page 667
10.7.2 Bunch–Davies Vacuum and Power Spectrum......Page 669
10.7.3 Power Spectrum and Cosmic Microwave Background......Page 671
10.7.4 Quantum-to-Classical Transition in Inflation Theory......Page 673
10.7.5 Observational Aspects for Matter Bounces......Page 675
10.8 Semiclassical Gravity......Page 679
10.9 Conclusion......Page 683
Index......Page 692
