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
circlecopyrt
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
7.1
Eigenvalues of
J
z
and
J
2
139
•
Rotation spectra of diatomic molecules 142
7.2
Orbital angular momentum
145
•
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