Quantum Mechanics Demystified

Preface ix
Acknowledgments xi

CHAPTER 1 Historical Review 1
Blackbody Radiation and Planck’s Formula 1
The Photoelectric Effect 6
The Bohr Theory of the Atom 7
de Broglie’s Hypothesis 10
Quiz 11

CHAPTER 2 Basic Developments 13
The Schrödinger Equation 13
Solving the Schrödinger Equation 18
The Probability Interpretation and Normalization 24
Expansion of the Wavefunction and Finding
Coefficients 35
The Phase of a Wavefunction 44
Operators in Quantum Mechanics 46
Momentum and the Uncertainty Principle 54
The Conservation of Probability 59
Quiz 63

CHAPTER 3 The Time Independent Schrödinger Equation 65
The Free Particle 66
Bound States and 1-D Scattering 74
Parity 88
Ehrenfest Theorem 95
Quiz 96

CHAPTER 4 An Introduction to State Space 99
Basic Definitions 99
Hilbert Space Definitions 100
Quiz 110

CHAPTER 5 The Mathematical Structure of Quantum
Mechanics I 111
Linear Vector Spaces 111
Basis Vectors 122
Expanding a Vector in Terms of a Basis 124
Orthonormal Sets and the Gram-Schmidt
Procedure 124
Dirac Algebra with Bras and Kets 125
Finding the Expansion Coefficients in the
Representation of Bras and Kets 127
Quiz 129

CHAPTER 6 The Mathematical Structure of Quantum
Mechanics II 131
The Representation of an Operator 133
Eigenvalues and Eigenvectors 142
The Hermitian Conjugate of an Operator 152
The Commutator 167
Quiz 172

CHAPTER 7 The Mathematical Structure of Quantum
Mechanics III 175
Change of Basis and Unitary Transformations 175
The Generalized Uncertainty Relation 185
Projection Operators 188
Functions of Operators 193
Generalization to Continuous Spaces 194
Quiz 203

CHAPTER 8 The Foundations of Quantum Mechanics 205
The Postulates of Quantum Mechanics 205
Spectral Decomposition 209
Projective Measurements 211
The Completeness Relation 212
Completely Specifying a State with a CSCO 220
The Heisenberg versus Schrödinger Pictures 221
Describing Composite Systems in Quantum
Mechanics 222
The Matrix Representation of a Tensor Product 223
The Tensor Product of State Vectors 224
The Density Operator 226
The Density Operator for a Completely
Mixed State 229
A Brief Introduction to the Bloch Vector 237
Quiz 239

CHAPTER 9 The Harmonic Oscillator 241
The Solution of the Harmonic Oscillator in the
Position Representation 241
The Operator Method for the Harmonic Oscillator 250
Number States of the Harmonic Oscillator 253
More on the Action of the Raising and Lowering
Operators 256
Quiz 258

CHAPTER 10 Angular Momentum 259
The Commutation Relations of
Angular Momentum 260
The Uncertainty Relations for
Angular Momentum 262
Generalized Angular Momentum and
the Ladder Operators 262
Matrix Representations of Angular Momentum 272
Coordinate Representation of Orbital Angular
Momentum and the Spherical Harmonics 283
Quiz 293

CHAPTER 11 Spin-1/2 Systems 295
The Stern-Gerlach Experiment 296
The Basis States for Spin-1/2 Systems 298
Using the Ladder Operators to Construct Sx, Sy 300
Unitary Transformations for Spin-1/2 Systems 308
The Outer Product Representation of the Spin
Operators 310
The Pauli Matrices 312
The Time Evolution of Spin-1/2 States 317
The Density Operator for Spin-1/2 Systems 328
Quiz 329

CHAPTER 12 Quantum Mechanics in Three Dimensions 331
The 2-D Square Well 332
An Overview of a Particle in a Central Potential 341
An Overview of the Hydrogen Atom 342
Quiz 356

Final Exam 357
Answers to Quiz and Exam Questions 363
References 385
Index 387




Quantum Field Theory Demystified


CHAPTER 1 Particle Physics and Special Relativity 1
CHAPTER 2 Lagrangian Field Theory 23
CHAPTER 3 An Introduction to Group Theory 49
CHAPTER 4 Discrete Symmetries and Quantum Numbers 71
CHAPTER 5 The Dirac Equation 85
CHAPTER 6 Scalar Fields 109
CHAPTER 7 The Feynman Rules 139
CHAPTER 8 Quantum Electrodynamics 163
CHAPTER 9 Spontaneous Symmetry Breaking and the Higgs Mechanism 187
CHAPTER 10 Electroweak Theory 209
CHAPTER 11 Path Integrals 233
CHAPTER 12 Supersymmetry 245
Final Exam 263
Solutions to Quizzes and Final Exam 281
References 289
Index 291