Dürr / Teufel Bohmian Mechanics

The Physics and Mathematics of Quantum Theory
1. Auflage 2009
ISBN: 978-3-540-89344-8
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 393 Seiten

Bohmian Mechanics was formulated in 1952 by David Bohm as a complete theory of quantum phenomena based on a particle picture. It was promoted some decades later by John S. Bell, who, intrigued by the manifestly nonlocal structure of the theory, was led to his famous Bell's inequalities. Experimental tests of the inequalities verified that nature is indeed nonlocal. Bohmian mechanics has since then prospered as the straightforward completion of quantum mechanics. This book provides a systematic introduction to Bohmian mechanics and to the mathematical abstractions of quantum mechanics, which range from the self-adjointness of the Schrödinger operator to scattering theory. It explains how the quantum formalism emerges when Boltzmann's ideas about statistical mechanics are applied to Bohmian mechanics. The book is self-contained, mathematically rigorous and an ideal starting point for a fundamental approach to quantum mechanics. It will appeal to students and newcomers to the field, as well as to established scientists seeking a clear exposition of the theory.

Detlef Dürr studied physics in Münster, Germany, where he obtained his PhD in physics in 1978. After four post-doc years at Rutgers in the group of Joel Lebowitz working with Sheldon Goldstein, he was awarded a Heisenberg fellowship (1985-1989), during which he joined forces with Sheldon Goldstein and Nino Zanghì to develop the statistical analysis of Bohmian mechanics - a cooperation which continues to this day. In 1989 he became professor of mathematics at the University of Munich. His research interests are non-equilibrium statistical mechanics, foundations of statistical mechanics, Bohmian mechanics and the foundations of quantum theory.Stefan Teufel studied physics in Munich, Germany, where he was awarded his PhD in mathematics in 1998. His PhD advisor was Detlef Dürr. After one year as a post doc at Rutgers with Sheldon Goldstein he joined the group of Herbert Spohn at the Technical University of Munich. In 2004 he became lecturer in mathematics at Warwick University, UK. Since 2005 he has been full professor of mathematics at the University of Tübingen. His research interests include adiabatic and semiclassical problems in quantum dynamics, exponential asymptotics and Bohmian mechanics.

Autoren/Hrsg.

Dürr, Detlef

Teufel, Stefan

Inhaltsverzeichnis

1;Preface;5
2;Contents;9
3;Introduction;13
3.1;Ontology: What There Is;13
3.1.1;Extracts;13
3.1.2;In Brief: The Problem of Quantum Mechanics;16
3.1.3;In Brief: Bohmian Mechanics;18
3.2;Determinism and Realism;21
3.3;References;22
4;Classical Physics;23
4.1;Newtonian Mechanics;24
4.2;Hamiltonian Mechanics;25
4.3;Hamilton--Jacobi Formulation;36
4.4;Fields and Particles: Electromagnetism;38
4.5;No fields, Only Particles: Electromagnetism;46
4.6;On the Symplectic Structure of the Phase Space;50
4.7;References;54
5;Symmetry;55
6;Chance;60
6.1;Typicality;62
6.1.1;Typical Behavior. The Law of Large Numbers;65
6.1.2;Statistical Hypothesis and Its Justification;74
6.1.3;Typicality in Subsystems:Microcanonical and Canonical Ensembles;77
6.2;Irreversibility;91
6.2.1;Typicality Within Atypicality;92
6.2.2;Our Atypical Universe;100
6.2.3;Ergodicity and Mixing;101
6.3;Probability Theory;107
6.3.1;Lebesgue Measure and Coarse-Graining;107
6.3.2;The Law of Large Numbers;113
6.4;References;118
7;Brownian motion;119
7.1;Einstein's Argument;120
7.2;On Smoluchowski's Microscopic Derivation;124
7.3;Path Integration;128
7.4;References;129
8;The Beginning of Quantum Theory;130
8.1;References;136
9;Schrödinger's Equation;137
9.1;The Equation;137
9.2;What Physics Must Not Be;143
9.3;Interpretation, Incompleteness, and =||2;147
9.4;References;151
10;Bohmian Mechanics;152
10.1;Derivation of Bohmian Mechanics;154
10.2;Bohmian Mechanics and Typicality;158
10.3;Electron Trajectories;160
10.4;Spin;165
10.5;A Topological View of Indistinguishable Particles;173
10.6;References;178
11;The Macroscopic World;179
11.1;Pointer Positions;179
11.2;Effective Collapse;185
11.3;Centered Wave packets;189
11.4;The Classical Limit of Bohmian Mechanics;192
11.5;Some Further Observations;197
11.5.1;Dirac Formalism, Density Matrix,Reduced Density Matrix, and Decoherence;197
11.5.2;Poincaré Recurrence;204
11.6;References;206
12;Nonlocality;207
12.1;Singlet State and Probabilities for Anti-Correlations;211
12.2;Faster Than Light Signals?;214
12.3;References;215
13;The Wave Function and Quantum Equilibrium;216
13.1;Measure of Typicality;216
13.2;Conditional Wave Function;218
13.3;Effective Wave function;221
13.4;Typical Empirical Distributions;223
13.5;Misunderstandings;228
13.6;Quantum Nonequilibrium;229
13.7;References;230
14;From Physics to Mathematics;231
14.1;Observables. An Unhelpful Notion;231
14.2;Who Is Afraid of PVMs and POVMs?;237
14.2.1;The Theory Decides What Is Measurable;245
14.2.2;Joint Probabilities;246
14.2.3;Naive Realism about Operators;248
14.3;Schrödinger's Equation Revisited;249
14.4;What Comes Next?;252
14.5;References;253
15;Hilbert Space;254
15.1;The Hilbert Space L2;256
15.1.1;The Coordinate Space 2;258
15.1.2;Fourier Transformation on L2;261
15.2;Bilinear Forms and Bounded Linear Operators;271
15.3;Tensor Product Spaces;274
15.4;References;281
16;The Schrödinger Operator;282
16.1;Unitary Groups and Their Generators;282
16.2;Self-Adjoint Operators;287
16.3;The Atomistic Schrödinger Operator;297
16.4;References;301
17;Measures and Operators;302
17.1;Examples of PVMs and Their Operators;306
17.1.1;Heisenberg Operators;308
17.1.2;Asymptotic Velocity and the Momentum Operator;309
17.2;The Spectral Theorem;314
17.2.1;The Dirac Formalism;314
17.2.2;Mathematics of the Spectral Theorem;316
17.2.3;Spectral Representations;325
17.2.4;Unbounded Operators;327
17.2.5;Unitary Groups;335
17.2.6;H0=-/2;336
17.2.7;The Spectrum;344
17.3;References;347
18;Bohmian Mechanics on Scattering Theory;348
18.1;Exit Statistics;349
18.2;Asymptotic Exits;356
18.3;Scattering Theory and Exit Distribution;359
18.4;More on Abstract Scattering Theory;361
18.5;Generalized Eigenfunctions;364
18.6;Towards the Scattering Cross-Section;371
18.7;The Scattering Cross-Section;372
18.7.1;Born's Formula;373
18.7.2;Time-Dependent Scattering;375
18.8;References;381
19;Epilogue;382
19.1;References;383
20;Bibliography;384
21;Index;389