Quantum Physics II

What this course covers

• The Schrödinger Equation
• Wave Function Must Be Complex
• First-Order Time Evolution
• Linearity and Superposition
• Probability Density and the Born Rule
• Conservation of Normalization
• Probability Current and Continuity Equation
• Energy Eigenstates and the Time-Independent Schrödinger Equation
• The Spectrum and Degeneracy
• Orthonormality of Energy Eigenstates
• Physical Equivalence of Wave Functions Differing by a Phase
• Bound states in one dimension
• Time-independent Schrödinger equation scaling
• No degeneracy theorem for bound states
• Energy eigenstates can be chosen real
• Bound-state solutions are real up to a global phase
• Even potentials and parity of eigenstates
• Bound-state parity is strictly even or odd
• Wave functions as vectors
• Position operator X̂ in matrix form
• Expectation value of position
• Eigenstates of the position operator
• Momentum operator P̂ in position representation
• Canonical commutation relation [X̂, P̂] = iℏ
• Momentum operator as a matrix
• Eigenstates of the momentum operator
• Stern-Gerlach experiment and spin-½ quantization
• Two-dimensional complex vector space as spin state space
• Eigenstates and eigenvalues of the Ŝ_z operator
• Basis states and linear superposition
• Column-vector representation of spin states
• Orthonormality of basis states
• Bra vectors and the inner product
• Eigenvalues from matrix polynomial equations
• Trace and eigenvalue sum relationship
• Pauli matrix commutation algebra
• Anti-commutator and Pauli matrix anti-commutation
• General Pauli matrix product formula
• Vector of operators and dot product notation (A·σ)
• (A·σ)(B·σ) product formula
• Spin operator along arbitrary direction (Ŝₙ)
• Cross product of operator triplets and S×S
• Vector spaces: definition and axioms
• Linear maps (operators) definition
• Operators on a vector space
• Examples of linear operators
• Space of operators as a vector space
• Operator multiplication (composition)
• Non-commutativity of operators
• The commutator of two operators
• T-invariant subspaces
• Eigenvalue equation (abstract form)
• Spectrum of an operator
• Non-invertibility and eigenvalues
• Linear independence of eigenvectors with distinct eigenvalues
• Characteristic polynomial and eigenvalue existence
• Degenerate eigenvalues
• Inner product definition and motivation
• Properties of the dot product as inner product axioms
• Linear Functionals on Vector Spaces
• Representing Linear Functionals via Inner Products
• Proof of the Representation Theorem
• Definition of the Adjoint (Hermitian Conjugate) Operator
• Linearity of the Adjoint Operator
• Hermitian operators and real expectation values
• Eigenvalues of Hermitian operators are real
• Orthogonality of eigenvectors for distinct eigenvalues
• Degeneracy of eigenvalues
• Unitary operators and norm preservation
• Invertibility of unitary operators
• S†S = I for unitary operators
• Unitary operators preserve inner products
• Unitary transformations map orthonormal bases to orthonormal bases
• Quantum uncertainty of an operator
• Variance formula for uncertainty
• Computing uncertainty: worked spin example
• Statement of the general uncertainty principle
• Hermiticity of the commutator times 1/i
• Position-momentum uncertainty principle
• Proof via the Cauchy-Schwarz inequality
• Energy-time uncertainty and state overlap
• Uncertainty principle lower bound on ground state energy
• Expectation values of X and P in symmetric bound states
• Variance identity linking ΔP and ⟨P²⟩
• Expectation value inequality ⟨A²⟩ ≥ ⟨A⟩²
• Minimizing the bounding function to extract the energy bound
• Diagonalization of linear operators
• Unitary diagonalizability and orthonormal eigenbases
• Quantum harmonic oscillator definition
• Canonical commutation relation [X, P] = iħ
• Hilbert space of square-integrable functions
• Factorization of the Hamiltonian
• Ground state energy lower bound from factorization
• Creation and annihilation operators A† and A
• Number operator N = A†A
• Ladder operator commutators with H
• Algebraic construction of the energy spectrum
• Unitary time evolution operator
• Time-independent Hamiltonian: solving for U
• Functions of operators on eigenstates
• Time-dependent commuting Hamiltonian: solving for U
• Time-ordered exponential for general H(T)
• Heisenberg picture of quantum mechanics
• Heisenberg Picture Operators
• Time-Dependent Creation and Annihilation Operators
• Motivation for Coherent States
• Unitary Translation Operator
• Action of Translation Operator on X̂ and P̂
• Definition of Coherent States
• Expectation Values in Coherent States
• Classical Time Evolution of Coherent States
• Minimum Uncertainty and Shape Preservation
• Coherent States in the Energy Basis
• Coherent states review
• Complex alpha plane representation
• Photon number uncertainty in coherent states
• Phase-number uncertainty relation
• Position and momentum uncertainties of coherent states
• Squeeze states via Hamiltonian change
• Bogoliubov transformation
• Squeezed uncertainties
• Photon states as quantum field theory
• Energy of the electromagnetic field
• Single-mode electromagnetic field configuration
• Electromagnetic energy as a harmonic oscillator
• Massless harmonic oscillator and unit analysis
• Quantization of the EM field via operator promotion
• Photon number states and energy quantization
• Electric field as a quantum operator
• Expectation value of electric field in photon number states
• General two-state Hamiltonian review
• Ammonia molecule as a two-state system
• Double-well potential and tunneling intuition
• Constructing the ammonia Hamiltonian
• Energy eigenvalues and the 2Δ energy gap
• Energy eigenstates as symmetric and antisymmetric superpositions
• Spin-direction picture for eigenstates
• Time evolution of the N-up state (Larmor precession)
• NMR spin Hamiltonian setup
• Non-commuting time-dependent Hamiltonians
• Rotating reference frame strategy
• Unitary transformation and rotated wave function
• Hamiltonian from a unitary operator
• Transformed Hamiltonian formula
• Similarity transformation of spin operators
• Time-independent rotating-frame Hamiltonian
• Tensor product vector space
• Zero vector in the tensor product space
• Inner product on the tensor product space
• Normalizing a two-particle state
• Entangled vs. separable states
• Determinant criterion for entanglement
• Singlet state properties
• Probability of joint spin measurements on entangled state
• Two-spin state overlap formula
• Physical realisation of entangled spin states
• Einstein–Podolsky–Rosen (EPR) argument
• Local realism: measurement corresponds to reality
• Local realism: locality (no distant influence)
• Hidden variables as implementation of local realism
• Vectors under rotations definition
• Scalars and vectors from dot and cross products
• Angular momentum commutation relations
• L_i commutes with L²
• General angular momentum algebra with J
• Vector identity for vectors under rotations
• L² in terms of R and P (operator identity)
• L² as an angular differential operator
• Hamiltonian for central potentials
• Spherical Harmonics as Angular Wavefunctions
• Differential Operator Forms of L², L_z, and L±
• Completeness and Orthogonality of Spherical Harmonics
• Constructing Spherical Harmonics via Ladder Operators
• Hamiltonian in Spherical Coordinates
• Separation of Variables for the 3D Schrödinger Equation
• The Radial Equation and the Substitution u = rF
• Central potential and radial wave function
• Effective potential and centrifugal barrier
• Energy spectrum diagram for central potentials
• Spherical Bessel functions for the free particle
• Infinite spherical well setup
• Bound states in 3D vs 1D potentials
• L=0 infinite spherical well solution
• Energy quantization and natural energy scale
• Zeros of spherical Bessel functions and general quantization
• Hellmann-Feynman Theorem
• First-Order Energy Correction via Perturbation
• Addition of Angular Momentum — Motivation
• Angular Momentum Algebra on a Vector Space
• Tensor Product of Angular Momentum Spaces
• Total Angular Momentum as a Sum
• Proof that Total Angular Momentum Satisfies the Algebra
• Spin-orbit coupling and fine structure
• Hyperfine structure from proton spin
• Magnetic dipole moments and g-factors
• Magnetic dipole field of the proton
• Hyperfine Hamiltonian construction
• Symmetry argument for angular integrals
• Ground-state hyperfine energy shift
• Addition of two spin-1/2 particles
• Addition of Angular Momentum
• Hydrogen Atom Energy Spectrum
• Runge-Lenz Vector Definition
• Conservation of the Runge-Lenz Vector
• R̂² in Terms of H and L²
• Orthogonality of R̂ and L̂
• Vector Operators Under Rotations

Syllabus

1. Lectures

   1. 1. Wave Mechanics
   2. 2. Wave Mechanics (continued)
   3. 3. Wave Mechanics (continued) and Stern-Gerlach Experiment
   4. 4. Spin One-half, Bras, Kets, and Operators
   5. 5. Linear Algebra: Vector Spaces and Operators
   6. 6. Linear Algebra: Vector Spaces and Operators (continued)
   7. 7. Linear Algebra: Vector Spaces and Operators (continued)
   8. 8. Linear Algebra: Vector Spaces and Operators (continued)
   9. 9. Dirac's Bra and Ket Notation
   10. 10. Uncertainty Principle and Compatible Observables
   11. 11. Uncertainty Principle and Compatible Observables (continued)
   12. 12. Quantum Dynamics
   13. 13. Quantum Dynamics (continued) | Heisenberg Picture
   14. 14. Quantum Dynamics (continued)
   15. 15. Quantum Dynamics (continued)
   16. 16. Quantum Dynamics (continued) and Two State Systems
   17. 17. Two State Systems (continued)
   18. 18. Two State Systems (continued), Multiparticle States and Tensor Products
   19. 19. Multiparticle States and Tensor Products (continued)
   20. 20. Multiparticle States and Tensor Products (continued) and Angular Momentum
   21. 21. Angular Momentum (continued)
   22. 22. Angular Momentum (continued)
   23. 23. Angular Momentum (continued)
   24. 24. Addition of Angular Momentum
   25. 25. Addition of Angular Momentum (continued)
   26. 26. Addition of Angular Momentum (continued)