Lectures on Quantum Mechanics Jean-Louis Basdevant Lectures on Quantum Mechanics Professor Jean-Louis Basdevant Physics Department École Polytechnique 91128 Palaiseau France jean-louis.basdevant@polytechnique.edu Cover illustration: Siné, Schrödinger’s cat Library of Congress Control Number: 2006936625 ISBN-10: 0-387-37742-5 ISBN-13: 978-0-387-37742-1 e-ISBN-10: 0-387-37744-1 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 written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights 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 Praise of physics 1.1 The interplay of the eye and the mind 1.2 Advanced technologies 1.3 The pillars of contemporary physics 1.3.1 Mysteries of light 1.3.2 Fundamental structure of matter 1.4 The infinitely complex 1.5 The Universe 12 A quantum phenomenon 2.1 Wave behavior of particles 2.1.1 Interferences 2.1.2 Wave behavior of matter 2.1.3 Analysis of the phenomenon 2.2 Probabilistic nature of quantum phenomena 2.2.1 Random behavior of particles 2.2.2 A nonclassical probabilistic phenomenon 2.3 Conclusions 2.4 Phenomenological description 13 16 16 17 18 20 20 20 21 23 Wave function, Schră odinger equation 3.1 Terminology and methodology 3.1.1 Terminology 3.1.2 Methodology 3.2 Principles of wave mechanics 3.2.1 The interference experiment 3.2.2 Wave function 3.2.3 Schră odinger equation 3.3 Superposition principle 25 25 25 26 27 27 27 29 30 viii Contents 3.4 Wave packets 3.4.1 Free wave packets 3.4.2 Fourier transformation 3.4.3 Shape of wave packets 3.5 Historical landmarks 3.6 Momentum probability law 3.6.1 Free particle 3.6.2 General case 3.7 Heisenberg uncertainty relations 3.7.1 Size and energy of a quantum system 3.7.2 Stability of matter 3.8 Controversies and paradoxes 3.8.1 The 1927 Solvay Congress 3.8.2 The EPR paradox 3.8.3 Hidden variables, Bell’s inequalities 3.8.4 The experimental test 31 31 32 33 33 35 35 36 36 37 38 40 40 41 41 42 Physical quantities 4.1 Statement of the problem 4.1.1 Physical quantities 4.1.2 Position and momentum 4.2 Observables 4.2.1 Position observable 4.2.2 Momentum observable 4.2.3 Correspondence principle 4.2.4 Historical landmarks 4.3 A counterexample of Einstein and its consequences 4.3.1 What we know after a measurement? 4.3.2 Eigenstates and eigenvalues of an observable 4.3.3 Wave packet reduction 4.4 The specific role of energy 4.4.1 The Hamiltonian 4.4.2 The Schră odinger equation, time and energy 4.4.3 Stationary states 4.4.4 Motion: Interference of stationary states 4.5 Schră odingers cat 4.5.1 The dreadful idea 4.5.2 The classical world 45 46 46 47 48 49 49 50 50 51 53 54 55 56 56 57 58 59 60 60 63 Energy quantization 5.1 Methodology 5.1.1 Bound states and scattering states 5.1.2 One-dimensional problems 5.2 The harmonic oscillator 5.2.1 Harmonic potential 65 65 66 67 67 67 Contents 5.2.2 Energy levels, eigenfunctions 5.3 Square well potentials 5.3.1 Square potentials 5.3.2 Symmetric square well 5.3.3 Infinite well, particle in a box 5.4 Double well, the ammonia molecule 5.4.1 The model 5.4.2 Stationary states, the tunnel effect 5.4.3 Energy levels 5.4.4 Wave functions 5.4.5 Inversion of the molecule 5.5 Illustrations and applications of the tunnel effect 5.5.1 Sensitivity to the parameters 5.5.2 Molecular structure 5.6 Tunneling microscopy, nanotechnologies 5.6.1 Nanotechnologies 5.6.2 Classical limit ix 68 69 69 70 73 74 74 75 76 78 79 81 81 82 84 84 85 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 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 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 ˆ 151 8.4.1 Operators a ˆ, a ˆ† , and N 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 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 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 ˆ L ˆ , and L ˆ z 182 10.2.2 Eigenfunctions common to H, 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 .. .Lectures on Quantum Mechanics Jean-Louis Basdevant Lectures on Quantum Mechanics Professor Jean-Louis Basdevant Physics Department... components are the optoelectronic components of the new generation where light is used instead of electricity Laser photons replace electrons They collect and transmit information directly on. .. How can one verify such an assumption? One way is to perform interference and diffraction experiments The first experimental confirmation is due to Davisson and Germer in 1927 It is a diffraction experiment