qb - The Physics of Quantum Mechanics James Binney and David Skinner iv This book is a consequence of the vision and municence of Walter of Merton who

qb - The Physics of Quantum Mechanics James Binney and...

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The Physics of Quantum Mechanics James Binney and David Skinner
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iv This book is a consequence of the vision and munificence of Walter of Merton, who in 1264 launched something good Copyright c circlecopyrt 2008–2013 James Binney and David Skinner Published by Cappella Archive 2008; revised printings 2009, 2010, 2011
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Contents Preface x 1 Probability and probability amplitudes 1 1.1 The laws of probability 3 Expectation values 4 1.2 Probability amplitudes 5 Two-slit interference 6 Matter waves? 7 1.3 Quantum states 7 Quantum amplitudes and measurements 7 Complete sets of amplitudes 8 Dirac notation 9 Vector spaces and their adjoints 9 The energy rep- resentation 12 Orientation of a spin-half particle 12 Polarisation of photons 14 1.4 Measurement 15 Problems 15 2 Operators, measurement and time evolution 17 2.1 Operators 17 Functions of operators 20 Commutators 20 2.2 Evolution in time 21 Evolution of expectation values 23 2.3 The position representation 24 Hamiltonian of a particle 26 Wavefunction for well defined momentum 27 The uncertainty principle 28 Dynamics of a free particle 29 Back to two-slit in- terference 31 Generalisation to three dimensions 31 Probability current 32 The virial theorem 33 Problems 34 3 Harmonic oscillators and magnetic fields 37 3.1 Stationary states of a harmonic oscillator 37 3.2 Dynamics of oscillators 41 Anharmonic oscillators 42 3.3 Motion in a magnetic field 45 Gauge transformations 46 Landau Levels 47 Displacement of the gyrocentre 49 Aharonov-Bohm ef- fect 51 Problems 52 4 Transformations & Observables 58 4.1 Transforming kets 58 Translating kets 59 Continuous transformations
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vi Contents and generators 60 The rotation operator 62 Discrete transformations 62 (a) The parity operator 62 Mirror operators 63 4.2 Transformations of operators 64 The parity operator 66 Mirror operators 68 4.3 Symmetries and conservation laws 68 4.4 The Heisenberg picture 70 4.5 What is the essence of quantum mechanics? 71 Problems 73 5 Motion in step potentials 75 5.1 Square potential well 75 Limiting cases 78 (a) Infinitely deep well 78 (b) Infinitely narrow well 78 5.2 A pair of square wells 79 Ammonia 81 The ammonia maser 83 5.3 Scattering of free particles 84 The scattering cross section 86 Tunnelling through a potential barrier 87 Scattering by a classically allowed region 88 Resonant scattering 89 The Breit–Wigner cross section 92 5.4 How applicable are our results? 95 5.5 Summary 98 Problems 99 6 Composite systems 104 6.1 Composite systems 105 Collapse of the wavefunction 108 Operators for com- posite systems 109 Development of entanglement 110 Einstein–Podolski–Rosen experiment 111 Bell’s inequality 113 6.2 Quantum computing 116 6.3 The density operator 121 Reduced density operators 125 Shannon entropy 127 6.4 Thermodynamics 129 6.5 Measurement 132 Problems 135 7 Angular Momentum 139 7.1 Eigenvalues of J z and J 2 139 Rotation spectra of diatomic molecules 142 7.2 Orbital angular momentum 145 L
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