,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. Dynamics 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 Directly
from the Structure of the Schrödinger Equation . . . . . . 45
2.6 Probability Current and the Conservation
of Probability . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.7 Expectation Values . . . . . . . . . . . . . . . . . . . . . . 51
2.8 Scattering and Tunneling . . . . . . . . . . . . . . . . . . 53
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
ix
,x Elementary Quantum Mechanics
Chapter 3. Measurement, Time Evolution, Uncertainty,
and the Harmonic Oscillator 67
3.1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Unitary Operators, Time Evolution . . . . . . . . . . . . . 74
3.3 Uncertainty 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 Systems, 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 Systems, Tensor Products,
and Entanglement . . . . . . . . . . . . . . . . . . . . . . 143
5.4 Measurement: Observables as Tensor Product Operators
on Tensor Product Spaces . . . . . . . . . . . . . . . . . . 151
5.5 Non-Locality and Bell Inequalities . . . . . . . . . . . . . 153
5.6 Bell’s Experiment (The Clauser-Horne-Shimony-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
Bibliography 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 only 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 only
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. Dynamics 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 Directly
from the Structure of the Schrödinger Equation . . . . . . 45
2.6 Probability Current and the Conservation
of Probability . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.7 Expectation Values . . . . . . . . . . . . . . . . . . . . . . 51
2.8 Scattering and Tunneling . . . . . . . . . . . . . . . . . . 53
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
ix
,x Elementary Quantum Mechanics
Chapter 3. Measurement, Time Evolution, Uncertainty,
and the Harmonic Oscillator 67
3.1 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Unitary Operators, Time Evolution . . . . . . . . . . . . . 74
3.3 Uncertainty 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 Systems, 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 Systems, Tensor Products,
and Entanglement . . . . . . . . . . . . . . . . . . . . . . 143
5.4 Measurement: Observables as Tensor Product Operators
on Tensor Product Spaces . . . . . . . . . . . . . . . . . . 151
5.5 Non-Locality and Bell Inequalities . . . . . . . . . . . . . 153
5.6 Bell’s Experiment (The Clauser-Horne-Shimony-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
Bibliography 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 only 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 only
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
