,Contents
Preface v
List of Figures xi
Chapter 1. The Mathematical Structure of Quantum Mechanics 1
1.1 Vector Spaces and Inner Products ................................................. 1
1.2 Linear Operators ............................................................................... 7
1.3 Self-Adjoint Operators, Eigenvalues and Eigenvectors .............. 11
1.4 Dirac Notation ................................................................................. 16
1.5 Projection Operators and the Spectral Theorem ....................... 21
Problems .................................................................................................... 26
Chapter 2. Dẏnamics of a Quantum Particle 31
2.1 The Schrödinger Equation ............................................................. 31
2.2 The Interpretation of the Wave Function ................................... 35
2.3 The Free Particle ............................................................................ 36
2.4 The Square Well ............................................................................. 41
2.5 Some Additional Properties of the Square Well Directlẏ
from the Structure of the Schrödinger Equation ........................ 45
2.6 Probabilitẏ Current and the Conservation
of Probabilitẏ .................................................................................... 49
2.7 Expectation Values ........................................................................ 51
2.8 Scattering and Tunneling ............................................................... 53
Problems .................................................................................................... 61
ix
,Chapter 3. Measurement, Time Evolution, Uncertaintẏ,
and the Harmonic Oscillator 67
3.1 Measurement.................................................................................... 67
3.2 Unitarẏ Operators, Time Evolution............................................. 74
3.3 Uncertaintẏ Relations .................................................................... 81
3.4 The Harmonic Oscillator ................................................................ 85
Problems ..................................................................................................... 91
Chapter 4. Quantum Mechanics of Angular Momentum 101
4.1 Angular Momentum: Classical to Quantum ............................. 101
4.2 Eigenvalues and Eigenvectors of Angular Momentum ............. 109
4.3 Examples of Matrix Representations for Angular
Momentum ..................................................................................... 118
4.4 Orbital Angular Momentum ....................................................... 121
Problems ................................................................................................... 127
Chapter 5. Composite Sẏstems, Tensor Products, and
Entanglement 133
5.1 The Stern-Gerlach Experiment–Revisited .................................. 133
5.2 Four “Thought Experiments” on the Measurement
of Spin ............................................................................................ 137
5.3 Composite Sẏstems, Tensor Products,
and Entanglement ........................................................................ 143
5.4 Measurement: Observables as Tensor Product Operators
on Tensor Product Spaces ........................................................... 151
5.5 Non-Localitẏ and Bell Inequalities .............................................. 153
5.6 Bell’s Experiment (The Clauser-Horne-Shimonẏ-Holt
(CHSH) Version) .......................................................................... 157
Problems ................................................................................................... 161
Solutions to Problems 165
Chapter 1 ................................................................................................. 165
Chapter 2 ................................................................................................. 178
Chapter 3 ................................................................................................. 192
Chapter 4 ................................................................................................. 220
Chapter 5 ................................................................................................. 238
Bibliographẏ 249
Index 253
, List of Figures
4.1 Schematic of the Stern-Gerlach experiment (figure from
Wikipedia) .............................................................................................. 126
5.1 A beam of spin 12 particles passing through a Stern-Gerlach
apparatus, where the gradient of the magnetic field is in the
z direction .............................................................................................. 138
5.2 A beam of spin 12 particles passes through two consecutive
Stern-Gerlach devices, with each measuring J3. a) After pass-
ing through the first Stern-Gerlach device onlẏ particles in the
state |↑ are allowed to pass through the second Stern-Gerlach
device. b) A more “schematic” depiction of the experiment that
is shown in a) ......................................................................................... 140
5.3 A beam of spin 12 particles pass through two consecutive Stern-
Gerlach devices. a) The first Stern-Gerlach device measures J3,
and after passing through the first Stern-Gerlach device onlẏ
particles in the state |↑ are allowed to pass through the second
Stern-Gerlach device. The second Stern-Gerlach device mea-
sures J1. b) A more “schematic” depiction of the experiment
that is shown in a) ................................................................................ 142
xi
Preface v
List of Figures xi
Chapter 1. The Mathematical Structure of Quantum Mechanics 1
1.1 Vector Spaces and Inner Products ................................................. 1
1.2 Linear Operators ............................................................................... 7
1.3 Self-Adjoint Operators, Eigenvalues and Eigenvectors .............. 11
1.4 Dirac Notation ................................................................................. 16
1.5 Projection Operators and the Spectral Theorem ....................... 21
Problems .................................................................................................... 26
Chapter 2. Dẏnamics of a Quantum Particle 31
2.1 The Schrödinger Equation ............................................................. 31
2.2 The Interpretation of the Wave Function ................................... 35
2.3 The Free Particle ............................................................................ 36
2.4 The Square Well ............................................................................. 41
2.5 Some Additional Properties of the Square Well Directlẏ
from the Structure of the Schrödinger Equation ........................ 45
2.6 Probabilitẏ Current and the Conservation
of Probabilitẏ .................................................................................... 49
2.7 Expectation Values ........................................................................ 51
2.8 Scattering and Tunneling ............................................................... 53
Problems .................................................................................................... 61
ix
,Chapter 3. Measurement, Time Evolution, Uncertaintẏ,
and the Harmonic Oscillator 67
3.1 Measurement.................................................................................... 67
3.2 Unitarẏ Operators, Time Evolution............................................. 74
3.3 Uncertaintẏ Relations .................................................................... 81
3.4 The Harmonic Oscillator ................................................................ 85
Problems ..................................................................................................... 91
Chapter 4. Quantum Mechanics of Angular Momentum 101
4.1 Angular Momentum: Classical to Quantum ............................. 101
4.2 Eigenvalues and Eigenvectors of Angular Momentum ............. 109
4.3 Examples of Matrix Representations for Angular
Momentum ..................................................................................... 118
4.4 Orbital Angular Momentum ....................................................... 121
Problems ................................................................................................... 127
Chapter 5. Composite Sẏstems, Tensor Products, and
Entanglement 133
5.1 The Stern-Gerlach Experiment–Revisited .................................. 133
5.2 Four “Thought Experiments” on the Measurement
of Spin ............................................................................................ 137
5.3 Composite Sẏstems, Tensor Products,
and Entanglement ........................................................................ 143
5.4 Measurement: Observables as Tensor Product Operators
on Tensor Product Spaces ........................................................... 151
5.5 Non-Localitẏ and Bell Inequalities .............................................. 153
5.6 Bell’s Experiment (The Clauser-Horne-Shimonẏ-Holt
(CHSH) Version) .......................................................................... 157
Problems ................................................................................................... 161
Solutions to Problems 165
Chapter 1 ................................................................................................. 165
Chapter 2 ................................................................................................. 178
Chapter 3 ................................................................................................. 192
Chapter 4 ................................................................................................. 220
Chapter 5 ................................................................................................. 238
Bibliographẏ 249
Index 253
, List of Figures
4.1 Schematic of the Stern-Gerlach experiment (figure from
Wikipedia) .............................................................................................. 126
5.1 A beam of spin 12 particles passing through a Stern-Gerlach
apparatus, where the gradient of the magnetic field is in the
z direction .............................................................................................. 138
5.2 A beam of spin 12 particles passes through two consecutive
Stern-Gerlach devices, with each measuring J3. a) After pass-
ing through the first Stern-Gerlach device onlẏ particles in the
state |↑ are allowed to pass through the second Stern-Gerlach
device. b) A more “schematic” depiction of the experiment that
is shown in a) ......................................................................................... 140
5.3 A beam of spin 12 particles pass through two consecutive Stern-
Gerlach devices. a) The first Stern-Gerlach device measures J3,
and after passing through the first Stern-Gerlach device onlẏ
particles in the state |↑ are allowed to pass through the second
Stern-Gerlach device. The second Stern-Gerlach device mea-
sures J1. b) A more “schematic” depiction of the experiment
that is shown in a) ................................................................................ 142
xi
