Lecture Notes from PHYS622 - Quantum Mechanics I, as taught by Dr. Jay Sau in Fall, 2015.

Lecture 1 (08/31/15): Why is Quantum Weird? [JS]
- Wave-Particle Duality: The Nature of Light
- Planck Radiation Spectrum: A Conundrum
- Einstein's Photoelectric Effect
- Interference (again)

Lecture 2 (09/02/15): Modernizing Young's Experiment [JS]
- Photon Interferometers
- Complex Amplitudes

Lecture 3 (09/04/15): Building Up to the Hilbert Space
- Beam Splitters
- Matrix Methods
- Testing Bombs
- Spin: Stern-Gerlach
- Orthogonality

Lecture 4 (09/09/15): Expanding on Stern-Gerlach
- Stern-Gerlach
- Bra-Ket Notation
- "Games with Stern-Gerlach"

Lecture 5 (09/11/15): Spectra & The Hilbert Space
- Atomic Spectra
- Hilbert Space
- Dual Vectors (bras)
- Operators
- Matrix Representation of Operators

Lecture 6 (09/14/15): Operators
- Operators/Matrices
- Observables

Lecture 7 (09/16/15): Unitary Operators
- Projection Operator/Measurement/Wavefunction Collapse
- Unitary Operators
- Diagonalization

Lecture 8 (09/18/15): Operator Relations
- Eigenvalues/Eigenvectors
- Hermitian Operators
- Gram-Schmidt Orthonormalization
- Degeneracy
- Compatible/Commuting Observables

Lecture 9 (09/21/15): Specialty Observables
- Commuting/Compatible Observables
- Position Operators
- Translation
- Generators

Lecture 10 (09/23/15): Translating x-hat to p-hat
- Commutation Relations
- Hamilton-Jacobi Identity

Lecture 11 (09/25/15): Campbell-Baker-Hausdorff & Wavefunctions
- Commutation Relations
- Campbell-Baker-Hausdorff Identities
- Wavefunctions
- Uncertainty Principle

Lecture 12 (09/28/15): The Uncertainty Principle
- Wave Packets
- Uncertainty Principle
- Cauchy-Schwartz Inequality

Lecture 13 (09/30/15): Quantum Cryptography
- Uncertainty
- Cryptography
- Composite Systems

Lecture 14 (10/02/15): Product & Entangled States
- Tensor Product Space
- Interaction & Entanglement
- EPR: Einstein-Podolsky-Rosen Paradox

Lecture 15 (10/05/15): Measurement & Entanglement
- Measuring From Entangled Perspective
- Density Matrices
- Reduced Density Matrix

Lecture 16 (10/07/15): The Density Matrix
- Density Matrix
- Bloch Spheres

Lecture 17 (10/09/15): Entropy & The Schrodinger Equation
- Entropy of Density Matrices
- Connection to Entanglement
- Schrodinger's Equation
- Working with the Schrodinger Equation

Lecture 18 (10/12/15): Dynamics & Heisenberg Representation
- Bloch Sphere Dynamics
- Quantum Zeno & Superoperators
- Heisenberg Representation

Lecture 19 (10/14/15): Heisenburg's Equation of Motion
- Heisenburg EOM
- Ehrenfest's Theorem
- Harmonic Oscillators

Lecture 20 (10/16/15): Schrodinger's Wave Equation
- Harmonic Oscillator
- Coherent States
- The Schrodinger Wave Equation
- The Time-Independent SWE

Lecture 21 (10/19/15): Classical Limits & Approximations
- SWE in the Classical Limit
- Bohmian Mechanics
- WKB Approximation

Lecture 22 (10/21/15): WKB Approximation
- Eikonal Approximation Derivation
- WKB "Connection" Formulas
- Bohr-Sommerfeld Quantization

Lecture 23 (10/23/15): Bound States & Propagators [JS]
- WKB Applications: Bound States
- Even/Odd Wedge Potential
- Tunneling 
- Propagators
- Propagators and the T.I.S.E. (Midterm - Problem 3)

Lecture 24 (10/26/15): Path Integrals
- Propagator as a Transition Amplitude
- Propagator as a Path Integral
- Dirac & Feynman: Connection to the Classical Action

Lecture 25 (10/28/15): Principle of Least Action
- Path Integral
- Getting W(Delta-t)
- Feynman Path Integral

Lecture 26 (10/30/15): Electromagnetism in Quantum Mechanics [JS]
- "Baby" Gauge Invariance
- Electromagnetic Interferometry 
- Gravitational Interferometry
- Hamiltonian/Schrodinger's Eq. in EM-Fields

Lecture 27 (11/02/15): Schrodinger's Equation in EM Fields [JS]
- Schrodinger's Equation in EM Fields
- Current/Conservation

Lecture 28 (11/04/15): Gauge Invariance
- Invariance of B & E
- Invariance of Mechanical Momentum
- Invariance of Schrodinger's Wave Equation
- Invariance of Current
- Action-Angle: The Semiclassical Approximation

Lecture 29 (11/06/15): Landau Levels & the Aharanov-Bohm Effect
- HW8 P2: Landau Levels
- HW8 P3: Chiral Edge States
- HW8 P1: Stoke's Theorem in Quantum Mechanics
- Aharanov-Bohm Effect

Lecture 30 (11/09/15): Midterm Hindsight & Angular Momentum 
- Midterm Q1(a)
- Midterm Q2(b)
- Midterm Q2(c)
- Midterm Q3(b)
- Angular Momentum
- Rotation Symmetry

Lecture 31 (11/11/15): Rotation
- Classical Rotation
- Quantum Rotation
- Generators
- Finite Rotations
- Simplest Example: Spin-1/2

Lecture 32 (11/13/15): Spin
- Spin Angular Momentum
- Spin Generator of Rotations
- Rotating States
- Groups & Lie Algebra

Lecture 33 (11/16/15): Ladder & Angular Momentum Operators
- Ladder Operators w.r.t. J
- Rotation Matrices D(R)
- Homework 9, Problem 1
- Orbital Angular Momentum

Lecture 34 (11/18/15): Orbital Angular Momentum
- Orbital Angular Momentum
- Spherical/Central Potential

Lecture 35 (11/20/15): Central Potential Problem
- HW9 P2: Generators of Body Axis Rotations
- Central Potential Problem
- Separable Wavefunctions
- Application to the Central Potential Problem

Lecture 36 (11/23/15): Solving Central Potentials [JS]
- Radial Schrodinger Equation
- Solving the l=0 [Central Potential] Case
- Wavefunctions and the Parity Operator
- Three Generally Solvable Cases
    1. The free particle & the infinite spherical well
    2. Hydrogen atom - the Coulomb potential
    3. 3D harmonic oscillator

Lecture 37 (11/25/15): 3D Simple Harmonic Oscillator
- The 3D Simple Harmonic Oscillator
- SU(3) Invariance
- The l=0 Case
- Vector Operators
- Scalar Operators & Rotational Invariance

Lecture 38 (11/30/15): Spherical Vectors & The 3D Harmonic Oscillator
- 3D Simple Harmonic Oscillator
- Spherical Vectors
- Applications to the 3D S.H.O. 

Lecture 39 (12/02/15): Angular Momentum Addition
- Addition of Angular Momentum
- Basis Options
- Clebsch-Gordan Coefficients 

Lecture 40 (12/04/15): Properties of Clebsch-Gordan Coefficients
- Bounds on Allowable m-Values
- Bounds on Allowable j-Values
- Recursion Relation for C-G Coefficients
- Schwinger's Oscillator
- Spherical Vector Operators
- Spherical Tensors

Lecture 41 (12/07/15): The Clebsch-Gordan Series
- The Clebsch-Gordan Series
- Spherical Tensors
- Wigner-Eckart Theorem

Lecture 42 (12/09/15): Wigner-Eckart Theorem
[Guest Lecturer: Victor Yakovenko]
- General Relation for Spherical Tensors
- Proving the Wigner-Eckart Theorem
- Recursive Relationship
- Example: Qualifier Problem on Wigner-Eckart