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E-raamat: Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology

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Edited by Jordi Mompart (Universitaat Autņnoma de Barcelona, Bellaterra, Spain), Edited by Xavier Oriols Pladevall (Universitat Autņnoma de Barcelona, Bellaterra, Spain)

Formaat: 700 pages
Ilmumisaeg: 24-May-2019
Kirjastus: Pan Stanford Publishing Pte Ltd
Keel: eng
ISBN-13: 9781000650105

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Formaat: 700 pages
Ilmumisaeg: 24-May-2019
Kirjastus: Pan Stanford Publishing Pte Ltd
Keel: eng
ISBN-13: 9781000650105

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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 Schrödinger 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 Schrödinger equation, can be developed from it.

Foreword to the First Edition

xvii

Preface to the Second Edition

xix

Preface to the First Edition

xxiii

Introduction

(18)

1 Overview of Bohmian Mechanics

(148)

Xavier Oriols

Jordi Mompart

1.1 Historical Development of Bohmian Mechanics

(19)

1.1.1 Particles and Waves

(2)

1.1.2 Origins of the Quantum Theory

(2)

1.1.3 "Wave or Particle?" vs. "Wave and Particle"

(5)

1.1.4 Louis de Broglie and the Fifth Solvay Conference

(1)

1.1.5 Albert Einstein and Locality

(2)

1.1.6 David Bohm and Why the "Impossibility Proofs" were Wrong?

(4)

1.1.7 John Bell and Nonlocality

(2)

1.1.8 Quantum Hydrodynamics

(1)

1.1.9 Is Bohmian Mechanics a Useful Theory?

(1)

1.2 Bohmian Mechanics for a Single Particle

(24)

1.2.1 Preliminary Discussions

(1)

1.2.2 Creating a Wave Equation for Classical Mechanics

(8)

1.2.2.1 Newton's second law

(1)

1.2.2.2 Hamilton's principle

(2)

1.2.2.3 Lagrange's equation

(1)

1.2.2.4 Equation for an (infinite) ensemble of trajectories

(3)

1.2.2.5 Classical Hamilton-Jacobi equation

(1)

1.2.2.6 Local continuity equation for an (infinite) ensemble of classical particles

(1)

1.2.2.7 Classical wave equation

(1)

1.2.3 Trajectories for Quantum Systems

(5)

1.2.3.1 Schrodinger equation

(1)

1.2.3.2 Local conservation law for an (infinite) ensemble of quantum trajectories

(1)

1.2.3.3 Velocity of Bohmian particles

(1)

1.2.3.4 Quantum Hamilton-Jacobi equation

(2)

1.2.3.5 A quantum Newton-like equation

(1)

1.2.4 Similarities and Differences between Classical and Quantum Mechanics

(4)

1.2.5 Feynman Paths

(2)

1.2.6 Basic Postulates for a Single-Particle

(3)

1.3 Bohmian Mechanics for Many-Particle Systems

(30)

1.3.1 Preliminary Discussions: The Many Body Problem

(3)

1.3.2 Many-Particle Quantum Trajectories

(2)

1.3.2.1 Many-particle continuity equation

(1)

1.3.2.2 Many-particle quantum Hamilton-Jacobi equation

(1)

1.3.3 Factorizability, Entanglement, and Correlations

(3)

1.3.4 Spin and Identical Particles

(5)

1.3.4.1 Single-particle with s = 1/2

(3)

1.3.4.2 Many-particle system with s = 1/2 particles

(2)

1.3.5 Basic Postulates for Many-Particle Systems

(3)

1.3.6 The Conditional Wave Function: Many-Particle Bohmian Trajectories without the Many-Particle Wave Function

(14)

1.3.6.1 Single-particle pseudo-Schrodinger equation for many-particle systems

(3)

1.3.6.2 Example: Application in factorizable many-particle systems

(1)

1.3.6.3 Example: Application in interacting many-particle systems without exchange interaction

(2)

1.3.6.4 Example: Application in interacting many-particle systems with exchange interaction

(6)

1.4 Bohmian Explanation of the Measurement Process

(27)

1.4.1 The Measurement Problem

(6)

1.4.1.1 The orthodox measurement process

(3)

1.4.1.2 The Bohmian measurement process

(1)

1.4.2 Theory of the Bohmian Measurement Process

(15)

1.4.2.1 Example: Bohmian measurement of the momentum

106

(2)

1.4.2.2 Example: Sequential Bohmian measurement of the transmitted and reflected particles

108

(6)

1.4.3 The Evaluation of a Mean Value in Terms of Hermitian Operators

114

(21)

1.4.3.1 Why Hermitian operators in Bohmian mechanics?

114

(1)

1.4.3.2 Mean value from the list of outcomes and their probabilities

115

(1)

1.4.3.3 Mean value from the wave function and the operators

116

(1)

1.4.3.4 Mean value from Bohmian mechanics in the position representation

116

(1)

1.4.3.5 Mean value from Bohmian trajectories

117

(2)

1.4.3.6 On the meaning of local Bohmian operators AB(x)

119

(1)

1.5 Concluding Remarks

120

(2)

1.6 Problems and Solutions

122

(13)

A.1 Appendix: Numerical Algorithms for the Computation of Bohmian Mechanics

135

(33)

A.1.1 Analytical Computation of Bohmian Trajectories

138

(11)

A.1.1.1 Time-dependent Schrodinger equation for a 1D space (TDSE1D-BT) with an explicit method

138

(4)

A.1.1.2 Time-independent Schrodinger equation for a 1D space (TISE1D) with an implicit (matrix inversion) method

142

(3)

A.1.1.3 Time-independent Schrodinger equation for a 1D space (TISE1D) with an explicit method

145

(4)

A.1.2 Synthetic Computation of Bohmian Trajectories

149

(6)

A.1.2.1 Time-dependent quantum Hamilton-Jacobi equations (TDQHJE1D) with an implicit (Newton-like fixed Eulerian mesh) method

150

(3)

A.1.2.2 Time-dependent quantum Hamilton-Jacobi equations (TDQHJE1D) with an explicit (Lagrangian mesh) method

153

(2)

A.1.3 More Elaborated Algorithms

155

(12)

2 Hydrogen Photoionization with Strong Lasers

167

(44)

Albert Benseny

Antonio Picon

Jordi Mompart

Luis Plaja

Luis Roso

2.1 Introduction

168

(6)

2.1.1 A Brief Overview of Photoionization

168

(2)

2.1.2 The Computational Problem of Photoionization

170

(1)

2.1.3 Photoionization with Bohmian Trajectories

171

(3)

2.2 One-Dimensional Photoionization of Hydrogen

174

(13)

2.2.1 The Physical Model

174

(3)

2.2.2 Harmonic Generation

177

(5)

2.2.3 Above Threshold Ionization

182

(5)

2.3 Hydrogen Photoionization with Beams Carrying Orbital Angular Momentum

187

(15)

2.3.1 Physical System

187

(4)

2.3.2 Bohmian Equations in an Electromagnetic Field

191

(1)

2.3.3 Selection Rules

192

(1)

2.3.4 Numerical Simulations

193

(19)

2.3.4.1 Gaussian pulses

194

(2)

2.3.4.2 Laguerre-Gaussian pulses

196

(6)

2.4 Conclusions

202

(9)

3 Atomtronics: Coherent Control of Atomic Flow via Adiabatic Passage

211

(46)

Albert Benseny

Joan Baguda

Xavier Oriols

Gerhard Birkl

Jordi Mompart

3.1 Introduction

212

(8)

3.1.1 Atomtronics

212

(1)

3.1.2 Three-Level Atom Optics

213

(3)

3.1.3 Adiabatic Transport with Trajectories

216

(4)

3.2 Physical System: Neutral Atoms in Optical Microtraps

220

(2)

3.2.1 One-Dimensional Hamiltonian

221

(1)

3.3 Adiabatic Transport of a Single Atom

222

(6)

3.3.1 The Matter Wave STIRAP Paradox with Bohmian Trajectories

222

(2)

3.3.2 Velocities and Accelerations of Bohmian Trajectories

224

(4)

3.4 Adiabatic Transport of a Single Hole

228

(14)

3.4.1 Hole Transfer as an Array-Cleaning Technique

228

(1)

3.4.2 Adiabatic Transport of a Hole in an Array of Three Traps

229

(3)

3.4.2.1 Three-level approximation description

229

(3)

3.4.2.2 Numerical simulations

232

(1)

3.4.3 Hole Transport Fidelity

232

(3)

3.4.4 Bohmian Trajectories for the Hole Transport

235

(1)

3.4.5 Atomtronics with Holes

235

(7)

3.4.5.1 Single-hole diode

236

(3)

3.4.5.2 Single-hole transistor

239

(3)

3.5 Adiabatic Transport of a Bose-Einstein Condensate

242

(6)

3.5.1 Madelung Hydrodynamic Formulation

244

(1)

3.5.2 Numerical Simulations

244

(4)

3.6 Conclusions

248

(9)

4 Bohmian Pathways into Chemistry: A Brief Overview

257

(74)

Angel S. Sanz

4.1 Introduction

258

(5)

4.2 Approaching Molecular Systems at Different Levels

263

(14)

4.2.1 The Born-Oppenheimer Approximation

264

(4)

4.2.2 Electronic Configuration

268

(3)

4.2.3 Dynamics of "Small" Molecular Systems

271

(3)

4.2.4 Statistical Approach to Large (Complex) Molecular Systems

274

(3)

4.3 Bohmian Mechanics

277

(11)

4.3.1 Fundamentals

277

(5)

4.3.2 Nonlocality and Entanglement

282

(3)

4.3.3 Weak Values and Equations of Change

285

(3)

4.4 Applications

288

(21)

4.4.1 Time-Dependent DFT: The Quantum Hydrodynamic Route

288

(5)

4.4.2 Bound System Dynamics: Chemical Reactivity

293

(8)

4.4.3 Scattering Dynamics: Young's Two-Slit Experiment

301

(4)

4.4.4 Effective Dynamical Treatments: Decoherence and Reduced Bohmian Trajectories

305

(1)

4.4.5 Pathways to Complex Molecular Systems: Mixed Bohmian-Classical Mechanics

306

(3)

4.5 Concluding Remarks

309

(22)

5 Adaptive Quantum Monte Carlo Approach States for High-Dimensional Systems

331

(68)

Eric R. Bittner

Donald J. Kouri

Sean Derrickson

Jeremy B. Maddox

5.1 Introduction

332

(1)

5.2 Mixture Modeling Approach

333

(23)

5.2.1 Motivation for a Trajectory-Based Approach

334

(7)

5.2.1.1 Bohmian interpretation

336

(2)

5.2.1.2 Quantum hydrodynamic trajectories

338

(1)

5.2.1.3 Computational considerations

339

(2)

5.2.2 Density Estimation

341

(5)

5.2.2.1 The mixture model

341

(2)

5.2.2.2 Expectation maximization

343

(3)

5.2.3 Computational Results

346

(6)

5.2.3.1 Bivariate distribution with multiple nonseparable Gaussian components

346

(6)

5.2.4 The Ground State of Methyl Iodide

352

(4)

5.3 Quantum Effects in Atomic Clusters at Finite Temperature

356

(1)

5.4 Quantum Structures at Zero and Finite Temperature

357

(21)

5.4.1 Zero Temperature Theory

357

(2)

5.4.2 Finite Temperature Theory

359

(6)

5.4.2.1 Computational approach: The mixture model

362

(2)

5.4.2.2 Computational approach: Equations of motion for the sample points

364

(1)

5.4.3 Computational Studies

365

(13)

5.4.3.1 Zero temperature results

365

(5)

5.4.3.2 Finite temperature results

370

(8)

5.5 Overcoming the Node Problem

378

(10)

5.5.1 Supersymmetric Quantum Mechanics

380

(2)

5.5.2 Implementation of SUSY QM in an Adaptive Monte Carlo Scheme

382

(1)

5.5.3 Test Case: Tunneling in a Double-Well Potential

383

(4)

5.5.4 Extension to Higher Dimensions

387

(15)

5.5.4.1 Discussion

388

(1)

5.6 Summary

388

(11)

6 Nanoelectronics: Quantum Electron Transport

399

(64)

Enrique Colomes

Guillermo Albareda

Zhen Zhan

Devashish Pandey

Alfonso Alarcon

Fabio Traversa

Xavier Oriols

6.1 Introduction: From Electronics to Nanoelectronics

400

(2)

6.2 Evaluation of the Electrical Current and Its Fluctuations

402

(15)

6.2.1 Bohmian Measurement of the Current as a Function of the Particle Positions

403

(8)

6.2.1.1 Relationship between current in the ammeter Iammeter,g(t) and the current in the device-active region Ig(t)

405

(1)

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]}

406

(2)

6.2.1.3 Reducing the number of degrees of freedom of the whole circuit

408

(3)

6.2.2 Practical Computation of DC, AC, and Transient Currents

411

(2)

6.2.3 Practical Computation of Current Fluctuations and Higher Moments

413

(4)

6.2.3.1 Thermal and shot noise

414

(1)

6.2.3.2 Practical computation of current fluctuations

415

(2)

6.3 Solving Many-Particle Systems with Bohmian Trajectories

417

(5)

6.3.1 Coulomb Interaction Among Electrons

417

(2)

6.3.2 Exchange and Coulomb Interaction Among Electrons

419

(3)

6.3.2.1 Algorithm for spinless electrons

420

(1)

6.3.2.2 Algorithm for electrons with spins in arbitrary directions

421

(1)

6.4 Dissipation with Bohmian Mechanics

422

(3)

6.4.1 Parabolic Band Structures: Pseudo Schrodinger Equation

423

(1)

6.4.2 Linear Band Structures: Pseudo Dirac Equation

424

(1)

6.5 The BITLLES Simulator

425

(7)

6.5.1 Overall Charge Neutrality and Current Conservation

425

(3)

6.5.1.1 The Poisson equation in the simulation box

426

(1)

6.5.1.2 Time-dependent boundary conditions for the Poisson equation

427

(1)

6.5.2 Practical Computation of Time-Dependent Electrical Currents

428

(4)

6.5.2.1 The direct method for the computation of the total current

430

(1)

6.5.2.2 The Ramo-Shockley-Pellegrini method for the computation of the total current

430

(2)

6.6 Application of the BITLLES Simulator to Resonant Tunneling Diodes

432

(13)

6.6.1 Device Characteristics and Available Simulation Models

432

(3)

6.6.2 Numerical Results

435

(10)

6.6.2.1 Coulomb interaction in DC scenarios

435

(1)

6.6.2.2 Coulomb interaction in high-frequency scenarios

436

(5)

6.6.2.3 Current-current correlations

441

(2)

6.6.2.4 RTD with dissipation

443

(2)

6.7 Application of the BITLLES Simulator to Graphene and 2D Linear Band Structures

445

(7)

6.7.1 Bohmian Trajectories for Linear Band Structures

448

(1)

6.7.2 Numerical Results

448

(4)

6.8 Conclusions

452

(11)

7 Beyond the Eikonal Approximation in Classical Optics and Quantum Physics

463

(30)

Adriano Orefice

Raffaele Giovanelli

Domenico Ditto

7.1 Introduction

464

(2)

7.2 Helmholtz Equation and Geometrical Optics

466

(2)

7.3 Beyond the Geometrical Optics Approximation

468

(2)

7.4 The Time-Independent Schrodinger Equation

470

(2)

7.5 Hamiltonian Description of Quantum Particle Motion

472

(1)

7.6 The Unique Dimensionless Hamiltonian System

473

(3)

7.7 Wave-Like Features in Hamiltonian Form

476

(11)

7.8 Discussion and Conclusions

487

(2)

A.1 Appendix: The Paraxial Approach

489

(4)

8 Relativistic Quantum Mechanics and Quantum Field Theory

493

(52)

Hrvoje Nikolic

8.1 Introduction

494

(2)

8.2 Classical Relativistic Mechanics

496

(9)

8.2.1 Kinematics

496

(2)

8.2.2 Dynamics

498

(7)

8.2.2.1 Action and equations of motion

498

(3)

8.2.2.2 Canonical momentum and the Hamilton-Jacobi formulation

501

(1)

8.2.2.3 Generalization to many particles

502

(2)

8.2.2.4 Absolute time

504

(1)

8.3 Relativistic Quantum Mechanics

505

(17)

8.3.1 Wave Functions and Their Relativistic Probabilistic Interpretation

505

(3)

8.3.2 Theory of Quantum Measurements

508

(2)

8.3.3 Relativistic Wave Equations

510

(9)

8.3.3.1 Single particle without spin

511

(1)

8.3.3.2 Many particles without spin

512

(1)

8.3.3.3 Single particle with spin 1/2

513

(3)

8.3.3.4 Many particles with spin 1/2

516

(1)

8.3.3.5 Particles with spin 1

517

(2)

8.3.4 Bohmian Interpretation

519

(3)

8.4 Quantum Field Theory

522

(19)

8.4.1 Main Ideas of QFT and Its Bohmian Interpretation

522

(4)

8.4.2 Measurement in QFT as Entanglement with the Environment

526

(2)

8.4.3 Free Scalar QFT in the Particle-Position Picture

528

(5)

8.4.4 Generalization to Interacting QFT

533

(2)

8.4.5 Generalization to Other Types of Particles

535

(1)

8.4.6 Probabilistic Interpretation

536

(2)

8.4.7 Bohmian Interpretation

538

(3)

8.5 Conclusion

541

(4)

9 Quantum Accelerating Universe

545

(62)

Pedro F Gonzalez-Diaz

Alberto Rozas-Fernandez

9.1 Introduction

546

(3)

9.2 The Original Quantum Dark-Energy Model

549

(4)

9.3 Relativistic Bohmian Backgrounds

553

(7)

9.3.1 The Klein-Gordon Quantum Model

553

(1)

9.3.2 Quantum Theory of Special Relativity

554

(6)

9.4 Dark Energy Without Dark Energy

560

(9)

9.5 Benigner Phantom Cosmology

569

(9)

9.5.1 Thermodynamics

569

(4)

9.5.2 Violation of Classical NEC

573

(1)

9.5.3 Holographic Models

574

(3)

9.5.4 Quantum Cosmic Models and Entanglement Entropy

577

(1)

9.6 Generalized Cosmic Solutions

578

(5)

9.7 Gravitational Waves and Semiclassical Instability

583

(4)

9.8 On the Onset of the Cosmic Accelerating Phase

587

(7)

9.9 Conclusions and Comments

594

(13)

10 Bohmian Quantum Gravity and Cosmology

607

(58)

Nelson Pinto-Neto

Ward Struyve

10.1 Introduction

608

(2)

10.2 Nonrelativistic Bohmian Mechanics

610

(3)

10.3 Canonical Quantum Gravity

613

(3)

10.4 Bohmian Canonical Quantum Gravity

616

(3)

10.5 Minisuperspace

619

(2)

10.6 Space-Time Singularities

621

(17)

10.6.1 Minisuperspace: Canonical Scalar Field

622

(8)

10.6.1.1 Free massless scalar field

623

(2)

10.6.1.2 The exponential potential

625

(5)

10.6.2 Minisuperspace: Perfect Fluid

630

(4)

10.6.3 Loop Quantum Cosmology

634

(4)

10.7 Cosmological Perturbations

638

(14)

10.7.1 Cosmological Perturbations in a Quantum Cosmological Background

640

(2)

10.7.2 Bunch-Davies Vacuum and Power Spectrum

642

(2)

10.7.3 Power Spectrum and Cosmic Microwave Background

644

(2)

10.7.4 Quantum-to-Classical Transition in Inflation Theory

646

(2)

10.7.5 Observational Aspects for Matter Bounces

648

(4)

10.8 Semiclassical Gravity

652

(4)

10.9 Conclusion

656

(9)

Index

665

Xavier Oriols is an associate professor at the UAB. His research interests range from quantum foundations to practical engineering of electron devices. He is the author or coauthor of more than 140 papers and has developed the quantum electron transport simulator, named BITLLES, based on Bohmian mechanics.

Jordi Mompart, after a postdoctoral stay at the Leibniz Universität Hannover, he became an associate professor at the UAB.

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