Quantum Mechanics: Lecture Log

This is the lecture log for the graduate Quantum Mechanics class PHY 389 K, taught in Fall 2023 by professor Vadim Kaplunovsky, unique=57825.

Most lectures should be video recorded and the records available on Canvas. For the few lectures that did not get recorded because of technical glitches, I shall scan the notes I have used in class and links the scans to this page.

August 22 (Tuesday):

Syllabus and admin: course content, textbook, prerequisites, homework, exams, grades, etc.
Quantum states: Stern–Gerlach experiment; spin states; analogy to light polarization; 2D Hilbert space and its bases.

August 24 (Thursday):

Hilbert space formalism: definition; examples; normalizable and un-normalizable states; bases; position basis and momentum basis; overlaps and probabilities; observables and bases; expectation values; operators; eigenstates and eigenvalues.

Likbez lecture on August 25 (Friday):

Contour integrals: complex contours and analytic functions; contour deformations; residues; handling z=∞; Gaussian integrals.
Gaussian wave packets in coordinate and momentum spaces.

August 29 (Tuesday):

Operators and matrices: matrices pf operators; kernels; Hermiticity; eigenvalues and eigenstates; functions of operators; translation operators; compatible observables and commuting operators; diagonal and block-diagonal matrices.

August 31 (Thursday):

Finish compatible observables: Co-diagonalization of commuting operators; basis building; complex observables; functions of several operators.
Heisenberg's uncertainty principle: Δx·Δp≥ℏ/2; observer effect and the microscope example; quantum uncertainty beyond the observer effect; uncertainty relation for the De Broglie waves.
Uncertainty for general incompatible observers: ΔA\middot;½ΔB≥|⟨[A,B]⟩|; special case of position and momentum.

September 5 (Tuesday):

Unitary operators.
Symmetries: general rules; translations of space.
Time evolution: unitary time evolution operator; Schrödinger equation.

September 7 (Thursday):

Time evolution: Hamiltonian operator and stationary states; non-stationary states; Heisenberg–Dirac equation; Ehrenfest equation.
Unitary equivalence.

Extra lecture on September 8 (Friday):

Classical mechanics and its canonical quantization: Lagrangian formalism; Hamiltonian formalism; charged particle in electric and magnetic fields; canonical positions, momenta, and commutation relations; quantizing dependent variables; Poisson brackets and commutator brackets; bracket universality theorem.

September 12 (Tuesday):

Schrödinger and Heisenberg pictures of QM.
Conservation laws in quantum mechanics: conserved operators; application to diagonalizing the Hamiltonian; relation to symmetries; two-body example.
Begin Harmonic oscillators: general harmonic oscillator; â, â^†, and n̂ operators.

September 14 (Thursday):

Harmonic oscillators: spectrum of the n̂ operator; degeneracies and extra degrees of freedom; matrix elements; ⟨Q⟩, ⟨Q⟩, ΔQ, and ΔP in stationary states; seeing harmonic oscillations in non-stationary states; coherent states (briefly).

September 19 (Tuesday):

Coherent states of a harmonic oscillator.
Multiple oscillators and quanta: multiple harmonic oscillators; quantizing waves on a string; modes and quanta; quanta as 1d phonons; multi-phonon states; phonons are identical bosons.

September 21 (Thursday):

Coherent states of the vibrating string: mixing states of with different N_phonons; coherent states of multiple oscillators; coherent states and semi-classical waves on the string.
Electromagnetic waves and photons: modes in a cavity; polarizations; energy and Maxwell equations for the modes; quantum operators for the modes; lowering and raising operators; diagonalizing the Hamiltonain and reorganizing the eigenstates by the total number of quanta; quanta as photons; multiphoton states and identical bosons; coherent states and the classical EM waves; coherent states in lasers.

Extra lecture on September 22 (Friday):

Identical bosons: identical bosons; occupation numbers; creation and annihilation operators; wave-function language vs. Fock-space language; one-body operators; two-body operators; non-relativistic quantum fields; “second quantization”; field-particle duality.

September 26 (Tuesday):

Emission and absorbtion of light: Fermi's Golden rule for transitions; atom-photon interaction and its matrix elements; absorbtion, spontaneous emission, and stimilated emission of EM waves; thermal equilibrium of atoms and EM waves.
Lasers: optical cavity; pumping and population inversion; 3-level, 4-level and multi-level lasers; ruby laser and helium-neon laser examples; masers.

September 28 (Thursday):

Wave mechanics in 1 dimension: Schrödinger equation in 1d and asymptotic boundary conditions at x=±∞; bound and unbound states; stationary states and wave packets; reflection and transmission; step potential example; continuity rules for wave functions; total reflection.

October 3 (Tuesday):

Wentzel–Kramers–Brillouin (WKB) approximation: ψ(x) in the classically allowed region and the Hamilton–Jacobi equation for its phase; phase and magnitude of a WKB wave-functions; limits of applicability; classically forbidden regions; boundaries and Airy functions; Bohr–Sommerfeld quantization rule for the bound states.
Introduction to the saddle point method: the real case.

October 5 (Thursday):

Airy functions and the saddle point method: Saddle point method for the complex case; Airy equation and the Laplace method; Airy functions as contour integrals; asymptotic behavior of the Airy functions; relation to WKB.
Begin tunneling: WKB approximation for the tunneling; bounce action of the reversed potential barrier.

Extra lecture on October 6 (Friday):

Bose–Einstein condensation and superfluidity: coherent BEC states; Landau–Ginzburg theory; density and velocity of a superfluid; beyond the coherent state approximation; quasiparticle spectrum: phonons, rotons, knocked out atoms, and anything in between; origin of superfluidity.

October 10 (Tuesday):

Tunneling examples: electrons escaping a cathode; centrifugal barrier for radial motion; nuclear fusion.
More WKB: limitations on WKB in 3 dimensions; WKB phases for interference; neutron interference due to gravity.

October 12 (Thursday):

Charged particle in EM fields: Ehrenfest equations; gauge transform of the wave function; covariant derivatives and kinematic momenta; covariant Schrödinger equation.
Aharonov–Bohm effect: schematics of the experiment; un-gauging A; propagation amplitudes and their gauge dependence; interference and magnetic flux; AB effect and charge quantization.
Intro to magnetic monopoles: AB effect and heuristics of magnetic charge quantization.

October 17 (Tuesday):

Magnetic monopoles: Dirac monopoles; Dirac's charge quantization; monopoles in GUTs and in the string theory.
Rotation symmetries in 2 dimensions: Overview of symmetries; rotation group in 2D; angular momentum generates the rotations; spectrum of L_z and S_z.

October 19 (Thursday):

Rotation symmetries in 3 dimensions: rotaions by 2π and 4π; 3D rotation group SO(3); components of J generate all rotations; commutation relations of angular momenta; scalars, vectors, tensors, and their commutation relations.
Lie groups and Lie algebras: Lie groups and their generators; Lie algebras of commutators; Baker–Campbell–Hausdorff formula for the finite group elements.

Extra lecture on October 20 (Friday):

Introduction to path integrals: discretizing time; normalization; phase-space path integral; coordinate-space path integral; partition function; harmonic oscillator example.

October 24 (Tuesday):

Groups, Algebras, Representations, and multiplets: Baker–Campbell–Hausdorff formula for the finite group elements; representations of general Lie groups and Lie algebras; SO(3) examples; multiplets of quantum states.
Representations of the angular momentum algebra: J², J_z, and J_± operators; cnains of |α,j,m⟩ states; spectrum of j and m; matrix elements of angular momenta; multiplets and representations of angular momenta; representations of finite rotations; examples for j=0,½,1.

October 26 (Thursday):

SO(3) versus Spin(3): rotaions by 2π; Spin(3) as a double cover of the SO(3); Spin(3)≅SU(2); geometries of the SO(2), SO(3), and Spin(3) group manifolds.
Orbital angular momentum and Spin: particle in a central potential; |n,ℓ,m⟩ states and their energies; 3D harmonic potential: extra degeneracy due to SO(3) symmetry enhanced to SU(3); spins of bosons and fermions; neutron interference example.

October 31 (Tuesday):

Identical particles: Permutation symmetry and its representations; bosons and fermions; spin-statistics theorem.
Adding angular momenta: simultaneous rotations of separate degrees of freedom, and Ĵ=Ĵ₁+Ĵ₂; reorganizing (j₁)⊗(j₂) multiplet into multiplets of j; spindle diagrams; triangle rule.

November 2 (Thursday):

Finish adding angular momenta: Clebbsch–Gordan coefficients; symmetry and antisymmetry for j₁=j₂. LS and JJ couplings; 3j symbols.
Started Wigner–Eckard theorem: introduction and the scalar case.

Extra lecture on November 3 (Friday):

Brief intro to quantum information: qubits; entanglement; quantum information cannot be copied; information loss.
Mixed states and density operators: lost entangled particle; density matrix and density operator ρ̂; eigenvalues of ρ̂ and probabilities; pure and mixed states.

November 7 (Tuesday):

Wigner–Eckard theorem: the vector case; spherical tensors; Wigner–Eckard theorem for tensor operators; projection theorem for vector operators; magnetic moment example.
Parity symmetry: polar and axial vectors; states of definite parity; selection rules.

November 9 (Thursday):

Perturbation theory: perturbative expansion; leading order energy correction; examples; corrections to degenerate energies; fine structure example; Zeeman effect example.

November 14 (Tuesday):

Examples of degenerate perturbation theory: Zeeman effect; linear Stark effect.
Formal perturbation theory: recursive calculation of δ^(k)E_n and δ^(k)|n⟩ to each order k.

November 16 (Thursday):

Second-order perturbation theory: general formula for the δ⁽²⁾E_n; quadratic Stark effect as an example; atomic polarizability.
Transitions: transitions due to a perturbation; transition probabilities.

Extra lecture on November 17 (Friday):

Density operators: mixed states and probabilities; time evolution of density operators; entropy; ρ̂ in statistical mechanics.

November 21 and 23:

No classes: Thanksgiving holiday.

November 28 (Tuesday):

Fermi's golden rule: time versus energy difference in transition probabilities, δE≲ℏ/δt; rransitions to continuum and density of final states; transition rate and Fermi's golden rule; net and partial transition rates; exponential decay.
Emission of photons by atoms: electric dipole approximation; EM matrix element for spontaneous emission of a photon; photon density of states; calculating the emission rate.

November 30 (Thursday):

Finished emission of photons by atoms: net and partial emission rates; hydrogen 2p→1s example.
Absorbtion of photons: transition rate, incoming flux and cross-section; line broadening and line strength; hydrogen 1s→2p example; relation between emission rates and absorbtion line strengths.
Ionization by photons: density of states for a free electron; ionization cross-section; hydrogen example.