The Physics of Quantum Mechanics

407 Pages·2013·4.4 MB·English

by  James Binney

THE PHYSICS OF QUANTUM MECHANICS

The Physics of Quantum Mechanics
JAMES BINNEY
RudolfPeierlsCentreforTheoreticalPhysicsandMertonCollege,
UniversityofOxford
DAVID SKINNER
DepartmentofAppliedMathematicsandTheoreticalPhysics,
UniversityofCambridge
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Contents
Preface xii
1 Introduction 1
1.1 Origins 2
1.2 Measurements 6
• Measurement involves disturbance 7
⊲ Heisenberg microscope 7 • Ideal measurements 8
• Summary 10
1.3 Probability amplitudes 11
• Two-slit interference 12
1.4 Quantum states 14
• Observables 14 ⊲ Complete sets of amplitudes 14
• Vector spaces and their duals 16 • The energy represen-
tation 19 • Polarisation of photons 20
1.5 Summary 23
Problems 24
2 Operators, measurement and time evolution 25
2.1 Operators 25
⊲ Functions of operators 29 ⊲ Commutators 29
2.2 Evolution in time 31
• Evolution of expectation values 32
2.3 The position representation 34
• Hamiltonian of a particle 37 • Wavefunction for well-
defined momentum 38 ⊲ The uncertainty principle 39
• Dynamics of a free particle 41 • Back to two-slit in-
terference 43 • Generalisation to three dimensions 44
⊲ Probability current 45 ⊲ The virial theorem 46
2.4 Summary 47
Problems 48
viii Contents
3 Oscillators 52
3.1 Stationary states of a harmonic oscillator 52
3.2 Dynamics of oscillators 56
• Anharmonic oscillators 58
Problems 61
4 Transformations and observables 66
4.1 Transforming kets 66
• Translating kets 67 • Continuous transformations and
generators 69 • The rotation operator 71 • Discrete
transformations 72 ⊲ The parity operator 72 ⊲ Mirror
operators 73
4.2 Transformations of operators 74
⊲ The parity operator 76 ⊲ Mirror operators 78
4.3 Symmetries and conservation laws 79
4.4 The Heisenberg picture 81
4.5 What is the essence of quantum mechanics? 83
Problems 85
5 Motion in step potentials 88
5.1 Square potential well 88
• Limiting cases 91 ⊲ Infinitely deep well 91 ⊲ Infinitely
narrow well 92
5.2 A pair of square wells 93
• Ammonia 96 ⊲ The ammonia maser 97
5.3 Scattering of free particles 99
⊲ The scattering cross-section 102 • Tunnelling through
a potential barrier 103 • Scattering by a classically
allowed region 104 • Resonant scattering 106
⊲ The Breit–Wigner cross-section 109
5.4 How applicable are our results? 113
5.5 Summary 115
Problems 117
6 Composite systems 123
6.1 Composite systems 124
• 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 137
Contents ix
6.3 The density operator 144
• Reduced density operators 148 • Shannon entropy 151
6.4 Thermodynamics 154
6.5 Measurement 158
Problems 162
7 Angular momentum 167
2
7.1 Eigenvalues of Jz and J 168
• Rotation spectra of diatomic molecules 171
7.2 Spin and orbital angular momentum 174
• Orbital angular momentum 174 ⊲ L as the generator of
2
circular translations 176 ⊲ Spectra of L and Lz 177
• Spin angular momentum 177
7.3 Physics of spin 180
• Spin-half matrices 181 • Spin-one matrices 182
• The Stern–Gerlach experiment 183 ⊲ Stern–
Gerlach experiment with spin-one atoms 186
• Precession in a magnetic field 187 • The classical
limit 188
7.4 Orbital angular-momentum eigenfunctions 191
• Orbital angular momentum and parity 196 • Orbital
angular momentum and kinetic energy 196 • Legendre
polynomials 198
7.5 Three-dimensional harmonic oscillator 199
7.6 Addition of angular momenta 205
• Case of two spin-half systems 209 • Case of spin-one
and spin-half 211 • The classical limit 212
Problems 213
8 Hydrogen 219
8.1 Gross structure of hydrogen 220
• Emission-line spectra 226 • Radial eigenfunctions 226
• Shielding 231 • Expectation values for r k 234
−
8.2 Fine structure and beyond 236
• Spin–orbit coupling 236 • Hyperfine structure 241
Problems 243
9 Motion in a magnetic field 247
9.1 Hamiltonian for motion in a magnetic field 248
• Classical equations of motion 248
9.2 Gauge transformations 250
• Probability current 251