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Quantum Mechanics
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PHYS 130B Lecture Notes
Chapter 1: Qubit
1.1 States and Observables
1.1.1 What is a Qubit
1.1.2 State and Representation
1.1.3 Hermitian Operators
1.2 Measurement
1.2.1 Measurement Postulate
1.2.2 Uncertainty and Incompatibility
1.2.3 Measurement Operators
1.3 Time Evolution
1.3.1 Unitary Evolution
1.3.2 Schrödinger Picture
1.3.3 Heisenberg Picture
Chapter 2: Identical Particles
2.1 Bosons and Fermions
2.1.1 Tensor Product
2.1.2 Symmetrization
2.1.3 Second Quantization
2.2 Angular Momentum
2.2.1 Angular Momentum Algebra
2.2.2 Spin Representations
2.2.3 Addition of Angular Momenta
2.3 Anyons
2.3.1 Exchange Statistics
2.3.2 Fractional Quantum Hall Anyons
2.3.3 Toric Code
Chapter 3: Path Integral
3.1 Quantization
3.1.1 Geometric Optics
3.1.2 Physical Optics
3.1.3 Particle-Wave Unification
3.2 Propagator
3.2.1 Path Integral Formulation
3.2.2 Schrödinger Equation
3.2.3 Free Particle Propagator
3.3 Stationary Phase
3.3.1 Stationary Phase Approximation
3.3.2 WKB Approximation
3.3.3 Bohr-Sommerfeld Quantization
3.4 Imaginary Time
3.4.1 Wick Rotation
3.4.2 Statistical Mechanics
3.4.3 Instantons
Chapter 4: Phase and Gauge
4.1 Gauge Field
4.1.1 Gauge Principle
4.1.2 Electromagnetic Coupling
4.1.3 Gauge Invariance
4.2 Berry Phase
4.2.1 Berry Phase
4.2.2 Aharonov-Bohm Effect
4.2.3 Flux Ring
4.3 Landau Level
4.3.1 Cyclotron Motion
4.3.2 Landau Quantization
4.3.3 Quantum Hall Effect
4.4 Spin and Monopole
4.4.1 Classical Spin
4.4.2 Dirac Monopole
4.4.3 Monopole Harmonics
Chapter 5: Perturbation Theory
5.1 Time-Independent Perturbation Theory
5.1.1 Toy Model
5.1.2 Non-Degenerate Perturbation Theory
5.1.3 Degenerate Perturbation Theory
5.2 Time-Dependent Perturbation Theory
5.2.1 Interaction Picture
5.2.2 Fermi’s Golden Rule
5.2.3 Applications
Chapter 6: Quantum Foundations
6.1 Density Matrix
6.1.1 Mixed States
6.1.2 Entropy
6.1.3 Quantum Statistics
6.2 Entanglement
6.2.1 Composite Systems
6.2.2 Entanglement Measures
6.2.3 Bell Inequality
6.3 Generalized Measurement
6.3.1 Projective Measurement
6.3.2 POVM
6.3.3 Quantum Channels
6.4 Open Quantum Systems
6.4.1 Decoherence
6.4.2 Lindblad Master Equation
6.4.3 Error Correction
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