Jean-Louis Basdevant Jean DalibardThe Quantum Mechanics Solver How to Apply Quantum Theory to Modern Physics Second Edition With 59 Figures, Numerous Problems and Solutions ABC... Profes
Trang 1Jean-Louis Basdevant Jean Dalibard
The Quantum
Mechanics Solver
How to Apply Quantum Theory
to Modern Physics
Second Edition
With 59 Figures, Numerous Problems and Solutions
ABC
Trang 2Professor Jean-Louis Basdevant
Department of Physics
Laboratoire Leprince-Ringuet
Ecole Polytechnique
91128 Palaisseau Cedex
France
E-mail: jean-louis.basdevant@
polytechnique.edu
Professor Jean Dalibard Ecole Normale Superieure Laboratoire Kastler Brossel rue Lhomond 24, 75231 Paris, CX 05
France E-mail: jean.dalibard@lkb.ens.fr
Library of Congress Control Number: 2005930228
ISBN-10 3-540-27721-8 (2nd Edition) Springer Berlin Heidelberg New York
ISBN-13 978-3-540-27721-7 (2nd Edition) Springer Berlin Heidelberg New York ISBN-10 3-540-63409-6 (1st Edition) Springer Berlin Heidelberg New York
ISBN-13 978-3-540-63409-6 (1st Edition) Springer Berlin Heidelberg New York
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Trang 3Preface to the Second Edition
Quantum mechanics is an endless source of new questions and fascinating observations Examples can be found in fundamental physics and in applied physics, in mathematical questions as well as in the currently popular debates
on the interpretation of quantum mechanics and its philosophical implications Teaching quantum mechanics relies mostly on theoretical courses, which are illustrated by simple exercises often of a mathematical character Reduc-ing quantum physics to this type of problem is somewhat frustratReduc-ing since very few, if any, experimental quantities are available to compare the results with For a long time, however, from the 1950s to the 1970s, the only alterna-tive to these basic exercises seemed to be restricted to questions originating from atomic and nuclear physics, which were transformed into exactly soluble problems and related to known higher transcendental functions
In the past ten or twenty years, things have changed radically The devel-opment of high technologies is a good example The one-dimensional square-well potential used to be a rather academic exercise for beginners The emer-gence of quantum dots and quantum wells in semiconductor technologies has changed things radically Optronics and the associated developments in infra-red semiconductor and laser technologies have considerably elevated the social rank of the square-well model As a consequence, more and more emphasis is given to the physical aspects of the phenomena rather than to analytical or computational considerations
Many fundamental questions raised since the very beginnings of quantum theory have received experimental answers in recent years A good example
is the neutron interference experiments of the 1980s, which gave experimental answers to 50 year old questions related to the measurability of the phase of the wave function Perhaps the most fundamental example is the experimen-tal proof of the violation of Bell’s inequality, and the properties of entangled states, which have been established in decisive experiments since the late 1970s More recently, the experiments carried out to quantitatively verify de-coherence effects and “Schr¨odinger-cat” situations have raised considerable
Trang 4VI Preface to the Second Edition
interest with respect to the foundations and the interpretation of quantum mechanics
This book consists of a series of problems concerning present-day experi-mental or theoretical questions on quantum mechanics All of these problems are based on actual physical examples, even if sometimes the mathematical structure of the models under consideration is simplified intentionally in order
to get hold of the physics more rapidly
The problems have all been given to our students in the ´Ecole Polytech-nique and in the ´Ecole Normale Sup´erieure in the past 15 years or so A special feature of the ´Ecole Polytechnique comes from a tradition which has been kept for more than two centuries, and which explains why it is necessary to devise original problems each year The exams have a double purpose On one hand, they are a means to test the knowledge and ability of the students On the other hand, however, they are also taken into account as part of the entrance examinations to public office jobs in engineering, administrative and military careers Therefore, the traditional character of stiff competitive examinations and strict meritocracy forbids us to make use of problems which can be found
in the existing literature We must therefore seek them among the forefront of present research This work, which we have done in collaboration with many colleagues, turned out to be an amazing source of discussions between us We all actually learned very many things, by putting together our knowledge in our respective fields of interest
Compared to the first version of this book, which was published by Springer-Verlag in 2000, we have made several modifications First of all,
we have included new themes, such as the progress in measuring neutrino oscillations, quantum boxes, the quantum thermometer etc Secondly, it has appeared useful to include, at the beginning, a brief summary on the basics of quantum mechanics and the formalism we use Finally, we have grouped the problems under three main themes The first (Part A) deals with Elementary Particles, Nuclei and Atoms, the second (Part B) with Quantum Entangle-ment and MeasureEntangle-ment, and the third (Part C) with Complex Systems
We are indebted to many colleagues who either gave us driving ideas, or wrote first drafts of some of the problems presented here We want to pay a tribute to the memory of Gilbert Grynberg, who wrote the first versions of
“The hydrogen atom in crossed fields”, “Hidden variables and Bell’s inequal-ities” and “Spectroscopic measurement on a neutron beam” We are particu-larly grateful to Fran¸cois Jacquet, Andr´e Roug´e and Jim Rich for illuminating discussions on “Neutrino oscillations” Finally we thank Philippe Grangier, who actually conceived many problems among which the “Schr¨odinger’s cat”, the “Ideal quantum measurement” and the “Quantum thermometer”, G´erald Bastard for “Quantum boxes”, Jean-No¨el Chazalviel for “Hyperfine struc-ture in electron spin resonance”, Thierry Jolicoeur for “Magnetic excitons”, Bernard Equer for “Probing matter with positive muons”, Vincent Gillet for
“Energy loss of ions in matter”, and Yvan Castin, Jean-Michel Courty and
Trang 5Do-Preface to the Second Edition VII minique Delande for “Quantum reflection of atoms on a surface” and “Quan-tum motion in a periodic potential”
Jean Dalibard
Trang 6Summary of Quantum Mechanics 1
1 Principles 1
2 General Results 4
3 The Particular Case of a Point-Like Particle; Wave Mechanics 4 4 Angular Momentum and Spin 6
5 Exactly Soluble Problems 7
6 Approximation Methods 9
7 Identical Particles 10
8 Time-Evolution of Systems 11
9 Collision Processes 12
Part I Elementary Particles, Nuclei and Atoms 1 Neutrino Oscillations 17
1.1 Mechanism of the Oscillations; Reactor Neutrinos 18
1.2 Oscillations of Three Species; Atmospheric Neutrinos 20
1.3 Solutions 23
1.4 Comments 27
2 Atomic Clocks 29
2.1 The Hyperfine Splitting of the Ground State 29
2.2 The Atomic Fountain 31
2.3 The GPS System 32
2.4 The Drift of Fundamental Constants 32
2.5 Solutions 33
3 Neutron Interferometry 37
3.1 Neutron Interferences 38
3.2 The Gravitational Effect 39
3.3 Rotating a Spin 1/2 by 360 Degrees 40
Trang 7X Contents
3.4 Solutions 42
4 Spectroscopic Measurement on a Neutron Beam 47
4.1 Ramsey Fringes 47
4.2 Solutions 49
5 Analysis of a Stern–Gerlach Experiment 55
5.1 Preparation of the Neutron Beam 55
5.2 Spin State of the Neutrons 57
5.3 The Stern–Gerlach Experiment 57
5.4 Solutions 59
6 Measuring the Electron Magnetic Moment Anomaly 65
6.1 Spin and Momentum Precession of an Electron in a Magnetic Field 65
6.2 Solutions 66
7 Decay of a Tritium Atom 69
7.1 The Energy Balance in Tritium Decay 69
7.2 Solutions 70
7.3 Comments 71
8 The Spectrum of Positronium 73
8.1 Positronium Orbital States 73
8.2 Hyperfine Splitting 73
8.3 Zeeman Effect in the Ground State 74
8.4 Decay of Positronium 75
8.5 Solutions 77
9 The Hydrogen Atom in Crossed Fields 81
9.1 The Hydrogen Atom in Crossed Electric and Magnetic Fields 82
9.2 Pauli’s Result 82
9.3 Solutions 83
10 Energy Loss of Ions in Matter 87
10.1 Energy Absorbed by One Atom 87
10.2 Energy Loss in Matter 88
10.3 Solutions 90
10.4 Comments 94
Trang 8Contents XI
Part II Quantum Entanglement and Measurement
11 The EPR Problem and Bell’s Inequality 99
11.1 The Electron Spin 99
11.2 Correlations Between the Two Spins 100
11.3 Correlations in the Singlet State 100
11.4 A Simple Hidden Variable Model 101
11.5 Bell’s Theorem and Experimental Results 102
11.6 Solutions 103
12 Schr¨ odinger’s Cat 109
12.1 The Quasi-Classical States of a Harmonic Oscillator 109
12.2 Construction of a Schr¨odinger-Cat State 111
12.3 Quantum Superposition Versus Statistical Mixture 111
12.4 The Fragility of a Quantum Superposition 112
12.5 Solutions 114
12.6 Comments 119
13 Quantum Cryptography 121
13.1 Preliminaries 121
13.2 Correlated Pairs of Spins 122
13.3 The Quantum Cryptography Procedure 125
13.4 Solutions 126
14 Direct Observation of Field Quantization 131
14.1 Quantization of a Mode of the Electromagnetic Field 131
14.2 The Coupling of the Field with an Atom 133
14.3 Interaction of the Atom with an “Empty” Cavity 134
14.4 Interaction of an Atom with a Quasi-Classical State 135
14.5 Large Numbers of Photons: Damping and Revivals 136
14.6 Solutions 137
14.7 Comments 144
15 Ideal Quantum Measurement 147
15.1 Preliminaries: a von Neumann Detector 147
15.2 Phase States of the Harmonic Oscillator 148
15.3 The Interaction between the System and the Detector 149
15.4 An “Ideal” Measurement 149
15.5 Solutions 150
Trang 9XII Contents
15.6 Comments 153
16 The Quantum Eraser 155
16.1 Magnetic Resonance 155
16.2 Ramsey Fringes 156
16.3 Detection of the Neutron Spin State 158
16.4 A Quantum Eraser 159
16.5 Solutions 160
16.6 Comments 166
17 A Quantum Thermometer 169
17.1 The Penning Trap in Classical Mechanics 169
17.2 The Penning Trap in Quantum Mechanics 170
17.3 Coupling of the Cyclotron and Axial Motions 172
17.4 A Quantum Thermometer 173
17.5 Solutions 174
Part III Complex Systems 18 Exact Results for the Three-Body Problem 185
18.1 The Two-Body Problem 185
18.2 The Variational Method 186
18.3 Relating the Three-Body and Two-Body Sectors 186
18.4 The Three-Body Harmonic Oscillator 187
18.5 From Mesons to Baryons in the Quark Model 187
18.6 Solutions 188
19 Properties of a Bose–Einstein Condensate 193
19.1 Particle in a Harmonic Trap 193
19.2 Interactions Between Two Confined Particles 194
19.3 Energy of a Bose–Einstein Condensate 195
19.4 Condensates with Repulsive Interactions 195
19.5 Condensates with Attractive Interactions 196
19.6 Solutions 197
19.7 Comments 202
20 Magnetic Excitons 203
20.1 The Molecule CsFeBr3 203
20.2 Spin–Spin Interactions in a Chain of Molecules 204
20.3 Energy Levels of the Chain 204
20.4 Vibrations of the Chain: Excitons 206
20.5 Solutions 208
Trang 10Contents XIII
21 A Quantum Box 215
21.1 Results on the One-Dimensional Harmonic Oscillator 216
21.2 The Quantum Box 217
21.3 Quantum Box in a Magnetic Field 218
21.4 Experimental Verification 219
21.5 Anisotropy of a Quantum Box 220
21.6 Solutions 221
21.7 Comments 229
22 Colored Molecular Ions 231
22.1 Hydrocarbon Ions 231
22.2 Nitrogenous Ions 232
22.3 Solutions 233
22.4 Comments 235
23 Hyperfine Structure in Electron Spin Resonance 237
23.1 Hyperfine Interaction with One Nucleus 238
23.2 Hyperfine Structure with Several Nuclei 238
23.3 Experimental Results 239
23.4 Solutions 240
24 Probing Matter with Positive Muons 245
24.1 Muonium in Vacuum 246
24.2 Muonium in Silicon 247
24.3 Solutions 249
25 Quantum Reflection of Atoms from a Surface 255
25.1 The Hydrogen Atom–Liquid Helium Interaction 255
25.2 Excitations on the Surface of Liquid Helium 257
25.3 Quantum Interaction Between H and Liquid He 258
25.4 The Sticking Probability 258
25.5 Solutions 259
25.6 Comments 265
26 Laser Cooling and Trapping 267
26.1 Optical Bloch Equations for an Atom at Rest 267
26.2 The Radiation Pressure Force 268
26.3 Doppler Cooling 269
26.4 The Dipole Force 270
26.5 Solutions 270
26.6 Comments 276