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:**J**^{2}, 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**=**Ĵ**_{1}+**Ĵ**_{2}; reorganizing (j_{1})⊗(j_{2}) multiplet into multiplets of j; spindle diagrams; triangle rule. - November 2 (Thursday):
- Finish adding angular momenta:
Clebbsch–Gordan coefficients;
symmetry and antisymmetry for j
_{1}=j_{2}. 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.

Last Modified: November 30, 2023. Vadim Kaplunovsky

vadim@physics.utexas.edu