Lectures on quantum mechanics

322 Pages·2007·3.271 MB·English

by  Jean-Louis Basdevant

Lectures on Quantum Mechanics
Jean-Louis Basdevant
Lectures on
Quantum Mechanics
Professor Jean-Louis Basdevant
Physics Department
École Polytechnique
91128 Palaiseau
France
[e-mail]
Cover illustration:Siné,Schrödinger’s cat.
Library ofCongress Control Number:2006936625
ISBN-10:0-387-37742-5 e-ISBN-10:0-387-37744-1
ISBN-13:978-0-387-37742-1 e-ISBN-13:978-0-387-37744-5
Printed on acid-free paper.
© 2007 Springer Science+Business Media,LLC
All rights reserved. This work may not be translated or copied in whole or in part without the
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9 8 7 6 5 4 3 2 1
springer.com
To my son Nicolas
who shared this for years since he was a child,
and who taught me that people who can’t laugh
are not serious people
Contents
Preface ........................................................ xv
1 Praise of physics........................................... 1
1.1 The interplay of the eye and the mind...................... 1
1.2 Advanced technologies ................................... 5
1.3 The pillars of contemporary physics........................ 6
1.3.1 Mysteries of light.................................. 6
1.3.2 Fundamental structure of matter .................... 8
1.4 The infinitely complex ................................... 9
1.5 The Universe ........................................... 12
2 A quantum phenomenon................................... 13
2.1 Wave behavior of particles................................ 16
2.1.1 Interferences...................................... 16
2.1.2 Wave behavior of matter ........................... 17
2.1.3 Analysis of the phenomenon ........................ 18
2.2 Probabilistic nature of quantum phenomena ................ 20
2.2.1 Random behavior of particles ....................... 20
2.2.2 A nonclassical probabilistic phenomenon ............. 20
2.3 Conclusions............................................. 21
2.4 Phenomenological description ............................. 23
3 Wave function, Schr¨odinger equation ...................... 25
3.1 Terminology and methodology ............................ 25
3.1.1 Terminology ...................................... 25
3.1.2 Methodology...................................... 26
3.2 Principles of wave mechanics.............................. 27
3.2.1 The interference experiment ........................ 27
3.2.2 Wave function .................................... 27
3.2.3 Schro¨dinger equation .............................. 29
3.3 Superposition principle................................... 30
viii Contents
3.4 Wave packets ........................................... 31
3.4.1 Free wave packets ................................. 31
3.4.2 Fourier transformation ............................. 32
3.4.3 Shape of wave packets ............................. 33
3.5 Historical landmarks..................................... 33
3.6 Momentum probability law ............................... 35
3.6.1 Free particle ...................................... 35
3.6.2 General case...................................... 36
3.7 Heisenberg uncertainty relations........................... 36
3.7.1 Size and energy of a quantum system ................ 37
3.7.2 Stability of matter................................. 38
3.8 Controversies and paradoxes .............................. 40
3.8.1 The 1927 Solvay Congress .......................... 40
3.8.2 The EPR paradox ................................. 41
3.8.3 Hidden variables, Bell’s inequalities.................. 41
3.8.4 The experimental test.............................. 42
4 Physical quantities......................................... 45
4.1 Statement of the problem................................. 46
4.1.1 Physical quantities ................................ 46
4.1.2 Position and momentum ........................... 47
4.2 Observables............................................. 48
4.2.1 Position observable ................................ 49
4.2.2 Momentum observable ............................. 49
4.2.3 Correspondence principle........................... 50
4.2.4 Historical landmarks............................... 50
4.3 A counterexample of Einstein and its consequences .......... 51
4.3.1 What do we know after a measurement? ............. 53
4.3.2 Eigenstates and eigenvalues of an observable .......... 54
4.3.3 Wave packet reduction ............................. 55
4.4 The specific role of energy ................................ 56
4.4.1 The Hamiltonian .................................. 56
4.4.2 The Schro¨dinger equation, time and energy ........... 57
4.4.3 Stationary states .................................. 58
4.4.4 Motion: Interference of stationary states.............. 59
4.5 Schro¨dinger’s cat ........................................ 60
4.5.1 The dreadful idea ................................. 60
4.5.2 The classical world ................................ 63
5 Energy quantization ....................................... 65
5.1 Methodology............................................ 65
5.1.1 Bound states and scattering states................... 66
5.1.2 One-dimensional problems.......................... 67
5.2 The harmonic oscillator .................................. 67
5.2.1 Harmonic potential ................................ 67
Contents ix
5.2.2 Energy levels, eigenfunctions........................ 68
5.3 Square well potentials.................................... 69
5.3.1 Square potentials.................................. 69
5.3.2 Symmetric square well ............................. 70
5.3.3 Infinite well, particle in a box ....................... 73
5.4 Double well, the ammonia molecule........................ 74
5.4.1 The model........................................ 74
5.4.2 Stationary states, the tunnel effect................... 75
5.4.3 Energy levels ..................................... 76
5.4.4 Wave functions.................................... 78
5.4.5 Inversion of the molecule ........................... 79
5.5 Illustrations and applications of the tunnel effect ............ 81
5.5.1 Sensitivity to the parameters ....................... 81
5.5.2 Molecular structure................................ 82
5.6 Tunneling microscopy, nanotechnologies .................... 84
5.6.1 Nanotechnologies.................................. 84
5.6.2 Classical limit..................................... 85
6 Principles of quantum mechanics .......................... 87
6.1 Hilbert space ........................................... 88
6.1.1 Two-dimensional space............................. 89
6.1.2 Square integrable functions ......................... 89
6.2 Dirac formalism ......................................... 92
6.2.1 Notations ........................................ 92
6.2.2 Operators ........................................ 93
6.2.3 Syntax rules ...................................... 95
6.2.4 Projectors; decomposition of the identity ............. 95
6.3 Measurement results ..................................... 96
6.3.1 Eigenvectors and eigenvalues of an observable ......... 96
6.3.2 Results of the measurement of a physical quantity ..... 97
6.3.3 Probabilities...................................... 98
6.3.4 The Riesz spectral theorem ......................... 98
6.3.5 Physical meaning of various representations...........100
6.4 Principles of quantum mechanics ..........................101
6.4.1 The principles ....................................101
6.4.2 The case of a continuous spectrum...................102
6.4.3 Interest of this synthetic formulation.................102
6.5 Heisenberg’s matrices ....................................103
6.5.1 Matrix representation of operators...................103
6.5.2 Matrices X and P .................................104
6.5.3 Heisenberg’s thoughts..............................104
6.6 The polarization of light, quantum “logic” ..................107
x Contents
7 Two-state systems .........................................113
7.1 The NH3 molecule.......................................113
7.2 “Two-state” system......................................114
7.3 Matrix quantum mechanics ...............................116
7.3.1 Vectors ..........................................116
7.3.2 Hamiltonian ......................................117
7.3.3 Observables ......................................117
7.3.4 Examples ........................................119
7.3.5 Basis of classical configurations .....................119
7.3.6 Interference and measurement.......................120
7.4 NH3 in an electric field...................................120
7.4.1 Uniform constant field .............................121
7.4.2 Weak and strong field regimes ......................122
7.4.3 Other two-state systems............................123
7.5 The ammonia molecule in an inhomogeneous field ...........123
7.5.1 Force on the molecule in an inhomogeneous field ......124
7.5.2 Population inversion ...............................126
7.6 Reaction to an oscillating field, the maser ..................126
7.7 Principle and applications of the maser.....................128
7.7.1 Amplifiers ........................................129
7.7.2 Oscillators........................................130
7.7.3 Atomic clocks.....................................130
7.7.4 Tests of relativity .................................132
7.8 Neutrino oscillations .....................................134
7.8.1 Lepton families ...................................134
7.8.2 Mechanism of the oscillations; reactor neutrinos .......135
7.8.3 Successive hermaphroditism of neutrinos .............138
8 Algebra of observables.....................................143
8.1 Commutation of observables ..............................143
8.1.1 Fundamental commutation relation ..................143
8.1.2 Other commutation relations .......................144
8.1.3 Dirac in the summer of 1925........................145
8.2 Uncertainty relations.....................................146
8.3 Evolution of physical quantities ...........................147
8.3.1 Evolution of an expectation value ...................147
8.3.2 Particle in a potential, classical limit.................148
8.3.3 Conservation laws .................................149
8.4 Algebraic resolution of the harmonic oscillator ..............150
8.4.1 Operators aˆ, aˆ†, and Nˆ ............................151
8.4.2 Determination of the eigenvalues ....................151
8.4.3 Eigenstates .......................................152
8.5 Commuting observables ..................................154
8.5.1 Theorem .........................................154
8.5.2 Example .........................................155
Contents xi
8.5.3 Tensor structure of quantum mechanics ..............155
8.5.4 Complete set of commuting observables (CSCO) ......156
8.5.5 Completely prepared quantum state .................157
8.6 Sunday, September 20, 1925 ..............................158
9 Angular momentum .......................................161
9.1 Fundamental commutation relation ........................162
9.1.1 Classical angular momentum........................162
9.1.2 Definition of an angular momentum observable........162
9.1.3 Results of the quantization .........................163
9.2 Proof of the quantization .................................163
9.2.1 Statement of the problem ..........................163
9.2.2 Vectors |j,m> and eigenvalues j and m .............164
9.2.3 Operators Jˆ± =Jˆx±iJˆy ...........................165
9.2.4 Quantization......................................166
9.3 Orbital angular momenta.................................168
9.3.1 Formulae in spherical coordinates ...................168
9.3.2 Integer values of m and (cid:1)...........................168
9.3.3 Spherical harmonics ...............................169
9.4 Rotation energy of a diatomic molecule.....................170
9.4.1 Diatomic molecule.................................171
9.4.2 The CO molecule..................................172
9.5 Angular momentum and magnetic moment .................173
9.5.1 Classical model ...................................173
9.5.2 Quantum transposition.............................175
9.5.3 Experimental consequences .........................175
9.5.4 Larmor precession .................................176
9.5.5 What about half-integer values of j and m?...........177
10 The Hydrogen Atom.......................................179
10.1 Two-body problem; relative motion........................180
10.2 Motion in a central potential..............................182
10.2.1 Spherical coordinates, CSCO .......................182
10.2.2 Eigenfunctions common to Hˆ, Lˆ2, and Lˆ ............182
z
10.2.3 Quantum numbers.................................183
10.3 The hydrogen atom......................................186
10.3.1 Atomic units; fine structure constant.................186
10.3.2 The dimensionless radial equation ...................188
10.3.3 Spectrum of hydrogen..............................191
10.3.4 Stationary states of the hydrogen atom...............191
10.3.5 Dimensions and orders of magnitude.................193
10.3.6 Historical landmarks...............................194
10.4 Muonic atoms...........................................195