The Physics of Quantum Mechanics
James Binney
and
David Skinner
,iv
This book is a consequence of the vision and munificence of
Walter of Merton, who in 1264 launched something good
Copyright c 2008–2013 James Binney and David Skinner
Published by Cappella Archive 2008; revised printings 2009, 2010, 2011
,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
, 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
2
7.1 Eigenvalues of Jz and J 139
• Rotation spectra of diatomic molecules 142
7.2 Orbital angular momentum 145
• L as the generator of circular translations 146 • Spectra
of L2 and Lz 147 • Orbital angular momentum eigenfunc-
tions 147 • Orbital angular momentum and parity 151
• Orbital angular momentum and kinetic energy 151
• Legendre polynomials 153
7.3 Three-dimensional harmonic oscillator 154
7.4 Spin angular momentum 158
• Spin and orientation 159 • Spin-half systems 161 ⊲ The
Stern–Gerlach experiment 161 • Spin-one systems 164
• The classical limit 165 • Precession in a magnetic field 168
James Binney
and
David Skinner
,iv
This book is a consequence of the vision and munificence of
Walter of Merton, who in 1264 launched something good
Copyright c 2008–2013 James Binney and David Skinner
Published by Cappella Archive 2008; revised printings 2009, 2010, 2011
,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
, 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
2
7.1 Eigenvalues of Jz and J 139
• Rotation spectra of diatomic molecules 142
7.2 Orbital angular momentum 145
• L as the generator of circular translations 146 • Spectra
of L2 and Lz 147 • Orbital angular momentum eigenfunc-
tions 147 • Orbital angular momentum and parity 151
• Orbital angular momentum and kinetic energy 151
• Legendre polynomials 153
7.3 Three-dimensional harmonic oscillator 154
7.4 Spin angular momentum 158
• Spin and orientation 159 • Spin-half systems 161 ⊲ The
Stern–Gerlach experiment 161 • Spin-one systems 164
• The classical limit 165 • Precession in a magnetic field 168
