Building Blocks of Quantum Mechanics: Theory and Applications

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Notations
Formulas in SI units and Gaussian units
Table of fundamental constants
Chapter 1: Introduction
1.1. Brief history of quantum mechanics
1.2. Schrödinger equation
1.3. Probability interpretation of wave function
1.4. Stationary Schrödinger equation
1.5. Conservation of probability
1.6. Quantum superposition
1.6.1. No cloning theorem
1.6.2. Schrödinger cat
1.7. Operators
1.8. Quantum measurement
1.8.1. Stern-Gerlach experiment
1.9. Expectation values
1.10. Problems
Chapter 2: One-dimensional Eigen-problem
2.1. Symmetric potential and parity
2.2. Free particle
2.3. Delta-function normalization
2.4. Infinite square well potential
2.5. Finite square well potential
2.5.1. Bound states −V < E ≤ 0
2.5.2. Scattering states E > 0
2.6. Quantum tunneling
2.7. Delta-function potential
2.7.1. Bound state (α < 0 and E < 0)
2.7.2. Scattering state (E > 0)
2.8. The WKB approximation
2.8.1. Solution around a turning point
2.8.2. The connection formulae
2.8.3. Quantization of energy levels
2.9. Problems
Chapter 3: Representation theory of quantum states
3.1. Representation
3.1.1. Dirac bracket notations
3.1.2. Representation of quantum states
3.1.3. Hermitian operators
3.1.4. Eigenstates of Hermitian operators
3.1.5. Representation of operators
3.1.6. Representation of Schrödinger equation
3.1.7. Feynman-Hellmann theorem
3.2. Basis transformation
3.2.1. Example: From real to momentum space representation
3.3. Commutators
3.3.1. Properties of commutable operators
3.3.2. Properties of noncommutable operators
3.4. Schrödinger picture
3.4.1. Virial theorem
3.4.2. Ehrenfest theorem
3.5. Heisenberg picture
3.6. Uncertainty principle
3.7. The time-energy uncertainty principle
3.8. Problems
Chapter 4: Harmonic Oscillators
4.1. One-dimensional harmonic oscillator
4.1.1. Ladder operators
4.1.2. Eigen-spectrum
4.1.3. Eigenfunction
4.1.4. Occupation representation
4.2. Coherent state
4.2.1. Minimum uncertainty state
4.2.2. Wave function of the coherent state
4.3. Charged particles in an electromagnetic field
4.3.1. Minimal coupling
4.3.2. Gauge invariance
4.3.3. Probability current
4.3.4. Aharonov-Bohm effect
4.4. Landau levels
4.4.1. Landau gauge
4.4.2. Degeneracy of Landau levels
4.4.3. Symmetric gauge
4.4.4. Lowest Landau level
4.5. Problems
Chapter 5: Angular Momentum
5.1. Orbital angular momentum
5.2. General angular momentum
5.2.1. Matrix representation of angular momentum operators
5.3. Eigenfunctions of orbital angular momentum
5.4. Spin angular momentum
5.4.1. Pauli matrices
5.4.2. Eigenstates of S = 1/2
5.4.3. Qubit and Bloch sphere
5.5. Addition of two angular momenta
5.5.1. Clebsch-Gordan coefficients
5.5.2. Addition of two S = 1/2. spins
5.6. Wigner-Eckart theorem∗
5.6.1. Proof of the Wigner-Eckart theorem
5.7. Problems
Chapter 6: Central potential
6.1. Three-dimensional potential with spherical symmetry
6.2. Hydrogenic atom
6.2.1. Hamiltonian in the center-of-mass framework
6.2.2. Bound state solutions
6.2.3. Solutions by series expansion
6.2.4. Radial wave function
6.2.5. Rydberg formula
6.3. Partial wave method
6.3.1. Partial wave expansion
6.3.2. Scattering amplitude
6.3.3. Scattering cross section
6.3.4. Hard-sphere scattering
6.4. Supersymmetric quantum mechanics approach∗
6.4.1. Supersymmetric solution of the hydrogenic atom
6.5. Problems
Chapter 7: Identical Particles
7.1. Permutation symmetry
7.2. Bose-Einstein and Fermi-Dirac statistics
7.2.1. Exchange degeneracy
7.2.2. Anti-symmetrized wave functions
7.2.3. Symmetrized wave functions
7.2.4. Two identical particles
7.2.5. Exchange force
7.3. Free fermion gas
7.3.1. Particle in a periodic box
7.3.2. Fermi surface
7.3.3. Degeneracy pressure
7.4. Hydrogen molecule
7.5. Entanglement
7.5.1. Density matrix
7.5.2. Entangled state
7.5.3. Entanglement entropy
7.5.4. Bell bases
7.5.5. EPR paradox
7.6. Problems
Chapter 8: Symmetry and Conservation Law
8.1. Spatial translation invariance and momentum conservation
8.1.1. Translation operator
8.1.2. Generator of translations
8.1.3. Momentum conservation
8.1.4. Finite translation
8.1.5. Stone theorem
8.2. Galilean invariance
8.3. Noether theorem
8.4. Rotation and angular momentum conservation
8.4.1. Rotation in two dimensions
8.4.2. Rotation in three dimensions
8.4.3. Angular momentum conservation
8.5. Time translation invariance and energy conservation
8.6. Time-reversal symmetry
8.6.1. Time-reversal transformation
8.6.2. Even and odd operators
8.6.3. Nondegenerate energy eigenstate
8.6.4. Kramers degeneracy
8.7. Problems
Chapter 9: Approximate methods
9.1. Ground state wave function is node free
9.2. Variational ansatz
9.2.1. Half-harmonic oscillator
9.2.2. Ground state of helium
9.3. Stationary perturbation theory
9.3.1. First order correction
9.3.2. Second order correction
9.3.3. Anharmonic oscillator
9.4. Degenerate perturbation theory
9.4.1. Linear Stark effect
9.5. Problems
Chapter 10: Quantum transition
10.1. Time-dependent Schr¨odinger equation
10.1.1. Perturbation expansion
10.1.2. First order correction
10.1.3. Two-level systems
10.2. Monochromatic perturbation
10.2.1. Interaction of atoms with electromagnetic wave
10.2.2. Absorption and stimulated emission
10.2.3. Fermi’s golden rule
10.2.4. Selection rules∗
10.2.5. Constant perturbation
10.3. Einstein’s theory of radiation
10.4. Problems
Chapter 11: Adiabatic and diabatic Evolution
11.1. Adiabatic versus sudden approximations
11.1.1. The adiabatic theorem
11.1.2. Sudden approximation
11.1.3. Quantum Zeno effect
11.2. Landau-Zener transition
11.2.1. Rabi oscillation
11.2.2. Landau-Zener model
11.2.3. Landau-Zener transition
11.2.4. Derivation of the Landau-Zener formulas∗
11.3. Berry’s phase
11.3.1. Fictitious gauge vector
11.3.2. Fictitious magnetic field
11.3.3. Quantization of fictitious magnetic flux
11.3.4. A spin-1/2. particle in a magnetic flux
11.3.5. Charged particle moving around a magnetic flux
11.4. Problems
Chapter 12: Relativistic Quantum Mechanics
12.1. Relativistic covariance
12.2. Klein-Gordon equation
12.2.1. Current and density operators
12.2.2. Interpretation of the density operator
12.2.3. Negative energy
12.2.4. Nonrelativistic limit
12.3. Dirac equation
12.3.1. “Derivation” of the Dirac equation
12.3.2. Charge conservation
12.3.3. Covariant form and gamma matrices
12.3.4. Coupled with electromagnetic fields
12.3.5. Free-particle solutions
12.3.6. Spin: rotational symmetry
12.3.7. Antiparticles: charge conjugate
12.3.8. Negative energy: Dirac sea
12.4. Nonrelativistic limit of the Dirac equation
12.4.1. First order approximation
12.4.2. Second order approximation
12.4.3. Normalization
12.4.4. Effective Hamiltonian
12.5. Problems
Index