Modern Physics

Preface to the Second Edition
Preface
Contents
1 Classical Statistical Mechanics
	1.1 Kinetic Theory of Gases
		1.1.1 Maxwell Distribution of Velocities
		1.1.2 Maxwell-Boltzmann Distribution of Energies
		1.1.3 Single-Particle Density of States
	1.2 Statistical Ensembles of Gibbs
		1.2.1 Microcanonical Ensemble
		1.2.2 Canonical Ensemble
		1.2.3 Grand Canonical Ensemble
		1.2.4 Many-Particle Density of States
	1.3 Heat Capacity of Gases and Solids
2 Special and General Relativity
	2.1 Electromagnetic Waves
		2.1.1 Lorentz Invariance of D'Alembert Operator
	2.2 Lorentz Transformations
		2.2.1 Thought Experiment with Light Bulb
	2.3 Einstein Postulates
	2.4 Relativistic Kinematics
		2.4.1 Length Contraction
		2.4.2 Time Dilation
		2.4.3 Transformation of Velocities
	2.5 Relativistic Dynamics
		2.5.1 Mechanical Work and Relativistic Energy
		2.5.2 Relativistic Energy and Linear Momentum
		2.5.3 Non-Relativistic Limit of the Energy
	2.6 Basic Concepts of General Relativity
		2.6.1 Spacetime Interval
		2.6.2 Curved Manifolds
		2.6.3 Equivalence Principle and Einstein Equations
		2.6.4 Non-Relativistic Limit of General Relativity
		2.6.5 Predictions of General Relativity
3 Quantum Properties of Light
	3.1 Black-Body Radiation
		3.1.1 Derivation of Planck's Law
	3.2 Photoelectric Effect
		3.2.1 Theoretical Explanation
	3.3 Energy and Linear Momentum of a Photon
	3.4 Compton Effect
		3.4.1 Theoretical Explanation
	3.5 Pair Production
4 Quantum Properties of Matter
	4.1 Heat Capacity of Solids: Einstein vs Debye
	4.2 Energy Spectra of Atoms
		4.2.1 Energy Spectrum of Hydrogen Atom
	4.3 Bohr's Model of Hydrogen Atom
		4.3.1 Derivation of Bohr's Formula
	4.4 Energy Levels and Photons
	4.5 Electromagnetic Transitions
	4.6 Einstein Coefficients
	4.7 Life-Time of an Atomic State
	4.8 Natural Linewidth
		4.8.1 Collisional Broadening
		4.8.2 Doppler Broadening
	4.9 Old Quantum Mechanics of Bohr, Wilson and Sommerfeld
	Further Reading
5 The Schrödinger Equation
	5.1 De Broglie Wavelength
		5.1.1 Explaining the Bohr Quantization
	5.2 Wave Mechanics of Schrödinger
		5.2.1 Derivation of Schrödinger's Equation
	5.3 Double-Slit Experiment with Electrons
	5.4 Formal Quantization Rules
		5.4.1 Schrödinger Equation for a Free Particle
		5.4.2 Schrödinger Equation for a Particle in an External Potential
	5.5 Madelung Transformation
	5.6 Stationary Schrödinger Equation
		5.6.1 Properties of the Hamiltonian Operator
		5.6.2 The Energy Levels are Real Numbers
		5.6.3 Orthogonality of Eigenfunctions
		5.6.4 Lebesgue Space of Square-Integrable Wave Functions
6 Solvable Problems
	6.1 One-Dimensional Square-Well Potential
	6.2 One-Dimensional Harmonic Potential
		6.2.1 Factorization Method
		6.2.2 Properties of Number Operator
	6.3 One-Dimensional Scattering
		6.3.1 Step Potential Barrier
		6.3.2 Rectangular Potential Barrier
	6.4 One-Dimensional Double-Well Potential
		6.4.1 One-Dimensional Double-Square-Well Potential
	6.5 WKB Method
		6.5.1 Quantum Tunneling
		6.5.2 Energy Spectrum
	6.6 Three-Dimensional Separable Potential
		6.6.1 Three-Dimensional Harmonic Potential
7 Axiomatization
	7.1 Matrix Mechanics and Commutation Rules
		7.1.1 Momentum Representation
	7.2 Time Evolution Operator
	7.3 Axioms of Quantum Mechanics
	7.4 Heisenberg Uncertainty Principle
		7.4.1 Uncertainty Principle for Non-Commuting Operators
	7.5 Time-Independent Perturbation Theory
	7.6 Time-Dependent Perturbation Theory and Fermi Golden Rule
	7.7 Variational Principle
8 Electrons and Atoms
	8.1 Electron in the Hydrogen Atom
		8.1.1 Schrödinger Equation in Spherical Polar Coordinates
		8.1.2 Selection Rules
	8.2 Pauli Exclusion Principle and the Spin
		8.2.1 Semi-Integer and Integer Spin: Fermions and Bosons
	8.3 The Dirac Equation
		8.3.1 The Pauli Equation and the Spin
		8.3.2 Dirac Equation with a Central Potential
		8.3.3 Relativistic Hydrogen Atom and Fine Splitting
		8.3.4 Relativistic Corrections to the Schrödinger Hamiltonian
	8.4 Spin Properties in a Magnetic Field
	8.5 Stark Effect
	8.6 Zeeman Effect
		8.6.1 Strong-Field Zeeman Effect
		8.6.2 Weak-Field Zeeman Effect
9 Many-Body Systems
	9.1 Identical Quantum Particles
		9.1.1 Spin-Statistics Theorem
	9.2 Non-Interacting Identical Particles
		9.2.1 Atomic Shell Structure and the Periodic Table of the Elements
	9.3 Interacting Identical Particles
		9.3.1 Electrons in Atoms and Molecules
	9.4 The Hartree-Fock Method
		9.4.1 Hartree for Bosons
		9.4.2 Hartree-Fock for Fermions
10 Quantum Statistical Mechanics
	10.1 Quantum Statistical Ensembles
		10.1.1 Quantum Microcanonical Ensemble
		10.1.2 Quantum Canonical Ensemble
		10.1.3 Quantum Grand Canonical Ensemble
	10.2 Bosons and Fermions at Finite Temperature
		10.2.1 Gas of Photons at Thermal Equilibrium
		10.2.2 Gas of Massive Bosons at Thermal Equilibrium
		10.2.3 Gas of Non-Interacting Fermions at Zero Temperature
	10.3 Connection Between Quantum Statistical Ensembles
		10.3.1 Grand Potential and Average Number Density
		10.3.2 Canonical Ensemble and Number Density
11 Quantum Information
	11.1 The Qubit
		11.1.1 Qubits and Measurements
		11.1.2 Single Qubit Quantum Gates
		11.1.3 Multiple Qubits
		11.1.4 Quantum Gates for 2-Qubits
	11.2 Entanglement
		11.2.1 Examples of Separable and Entangled states
		11.2.2 CNOT Gate and Entanglement
		11.2.3 Pure and Mixed States
		11.2.4 Von Neumann Entropy
		11.2.5 State of a System: Partial Trace
		11.2.6 Entanglement Entropy
		11.2.7 No-Cloning Theorem
		11.2.8 Quantum Teleportation
	11.3 Quantum Computing
		11.3.1 Deutsch-Jozsa Algorithm
		11.3.2 A Note on Quantum Hardware
12 Path Integral Formulation
	12.1 Path Integral Construction of the Propagator
		12.1.1 Wick Rotation and Thermodynamics
	12.2 Saddle-Point and Stationary Phase
		12.2.1 Gaussian Integrals
		12.2.2 Free Particle
		12.2.3 Particle in a Harmonic Potential
	12.3 Quantum Tunneling
	12.4 Path Integral for the Spin
Appendix A Dirac Delta Function
A.1  The Heaviside Step Function
A.2  The Strange Delta Function of Dirac
A.2.1  Dirac Function and the Integrals
A.3  Dirac Function in D Spatial Dimensions
Appendix B Complex Numbers
B.1  Set of Complex Numbers
B.2  Gauss Plane
B.2.1  Polar Representation
B.3  Euler Formula
B.3.1  Proof of the Euler Formula
B.3.2  De Moivre Formula
B.4  Fundamental Theorem of Algebra
B.5  Complex Functions
B.6  Gaussian and Fresnel integrals
Appendix C Fourier Transform
C.1  Geometric and Taylor Series
C.2  Fourier Series
C.2.1  Complex Representation of the Fourier Series
C.3  Fourier Integral
C.3.1  Properties of the Fourier Transform
C.3.2  Fourier Transform and Uncertainty Theorem
C.4  Fourier Transform of Space-Time Functions
Appendix D Differential Equations
D.1  First-Order ODE
D.1.1  Separation of Variables
D.2  Second-Order ODE
D.3  Newton Law as a Second-Order ODE
D.4  Partial Differential Equations
D.4.1  Wave Equation
D.4.2  Diffusion Equation
Appendix E Scattering Theory
E.1  Schrödinger Problem of Two Colliding Particles
E.2  Scattering Amplitude and Born Approximation
E.3  Cross-Section and S-Wave Scattering Length
*-20ptReferences
Index