The Physics of Quantum Mechanics
James Binney
and
David Skinner

iv
Copyright c
circlecopyrt
2008 James Binney and David Skinner
Published by Capella Archive 2008

Contents
Preface
ix
1
Probability and probability amplitudes
1
1.1
The laws of probability
4
•
Expectation values 5
1.2
Probability amplitudes
6
•
Two-slit interference 8
•
Matter waves? 9
1.3
Quantum states
10
•
Quantum amplitudes and measurements 10
⊲
Complete sets of amplitudes 11
•
Dirac notation 12
•
Vector spaces and their adjoints 13
•
The energy rep-
resentation 16
•
Orientation of a spin-half particle 17
•
Polarisation of photons 18
1.4
Measurement
20
⊲
Problems 21
2
Operators, measurement and time evolution
23
2.1
Operators
24
⊲
Functions of operators 27
⊲
Commutators 28
2.2
Evolution in time
29
•
Evolution of expectation values 31
2.3
The position representation
32
•
Hamiltonian of a particle 35
•
Wavefunction for well
defined momentum 37
⊲
The uncertainty principle 38
•
Dynamics of a free particle 39
•
Back to two-slit in-
terference 41
•
Generalisation to three dimensions 43
⊲
The virial theorem 43
⊲
Problems 45
3
Harmonic oscillators and magnetic fields
49
3.1
Stationary states of a harmonic oscillator
50
3.2
Dynamics of oscillators
54
•
Anharmonic oscillators 55
3.3
Motion in a magnetic field
59
•
Gauge transformations 61
•
Landau Levels 63
⊲
Displacement of the gyrocentre 65
•
Aharonov-Bohm ef-
fect 67
⊲
Problems 70
4
Transformations & Observables
76
4.1
Transforming kets
77
•
Translating kets 77
•
Continuous transformations
and generators 80
•
The rotation operator 82
•
Discrete transformations 82

vi
Contents
4.2
Transformations of operators
84
4.3
Symmetries and conservation laws
89
4.4
The Heisenberg picture
91
4.5
What is the essence of quantum mechanics?
92
⊲
Problems 95
5
Motion in step potentials
97
5.1
Square potential well
97
•
Limiting cases 101
⊲
(a) Infinitely deep well
101
⊲
(b)
Infinitely narrow well 102
5.2
A pair of square wells
103
•
Ammonia 106
⊲
The ammonia maser 107
5.3
Scattering of free particles
110
•
Reflection off a potential well 110
•
Tunnelling through a
potential barrier 113
5.4
What we have learnt
115
⊲
Problems 117
6
Composite systems
122
6.1
Composite systems
123
•
Collapse of the wavefunction 128
•
Operators for com-
posite systems 129
•
Development of entanglement 131
•
Einstein–Podolski–Rosen experiment 132
⊲
Bell’s inequality 134
6.2
Quantum computing
138
6.3
The density operator
145
•
Reduced density operators 150
•
Shannon entropy 153
•
Thermodynamics 156
6.4
Measurement
159
⊲
Problems 161
7
Angular Momentum
166
7.1
Eigenvalues of
J
z
and
J
2
167
•
Rotation spectra of diatomic molecules 170
7.2
Orbital angular momentum
174
•
L
as the generator of circular translations 175
•
Spectra
of
L
2
and
L
z
177
•
Orbital angular momentum eigenfunc-
tions 177
•
Orbital angular momentum and parity 182
•
Orbital angular momentum and kinetic energy 183
•
Legendre polynomials 185
7.3
Three-dimensional harmonic oscillator
186
7.4
Spin angular momentum
192
•
Spin and orientation 193
•
Spin-half systems 195
⊲
The

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