Starting September 23, all regular lectures were shadowed online via Zoom and also recorded (both video and sound) on the UT cloud. Likewise, all the extra lectures (starting from the first on September3) were recorded. All these recording are available via Canvas: log in to Canvas, select this class, then follow the link to Zoom.

The first 8 regular lectures were not recorded; sorry. However, you can find the scans of my notes — or rather, the scans of what I wrote under the document camera — in this folder.

**Update 12/4 evening:** I added the scans of my hand-written notes for the lectures between 9/23 and 11/16.

**Update 12/10 2PM:** I have added the scans for the last 3 weeks of classes.

- August 26 (Thursday):
- 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 31 (Tuesday):
- Hilbert space formalism: definition; examples; normalizable and un-normalizable states; bases; position basis and momentum basis; overlaps and probabilities.
- September 2 (Thursday):
- Observables, bases, and operators:
observables and bases; expectation values; operators;
eigenstates and eigenvalues; matrices; Hermiticity;
position and momentum operators;
~~functions of operators~~. - Extra lecture on September 3 (Friday):
- Contour integrals:
complex contours and analytic functions; contour deformations; residues; handling
*z*=∞; Gaussian integrals.

Gaussian wave packets in coordinate and momentum spaces. - September 7 (Tuesday):
- Compatible observables:
functions of operators; translation operator;
compatibility and commuting operators; mutual spectra; diagonal and block-diagonal matrices;
resolving degeneracies; compatibility of complex observables;
~~functions of commuting operators~~. - September 9 (Thursday):
- Functions of several commuting opertors.

Incompatible observables and uncertainty relations: general uncertainty relation; Heisenberg's uncertainty principle for positions and momenta; Heisenberg's microsscope example; order of measurements; quantum uncertainty beyond the observer effect. - September 14 (Tuesday):
- Unitary operators.

Symmetries: general rules; translations of space.

Time evolution: unitary time evolution operator; Shrödinger equation. - September 16 (Thursday):
- Time evolution:
Hamiltonian operator and stationary states; non-stationary states;
Heisenberg–Dirac equation; Ehrenfest equation;
unitary equivalence;
~~Shrödinger and Heisenberg pictures of QM~~. - Extra lecture on September 17 (Friday):
- Classical mechanics and its canonical quantization: Lagrangian formalism; Hamiltonian formalism; charged particle in electric and magnetic fields; canonical positions, momenta, and commutation relations; Poisson brackets and commutator brackets; universality theorem; quantizing dependent variables.
- September 21 (Tuesday):
- Shrödinger and Heisenberg pictures of QM.

Conseration laws in quantum mechanics: conserved operators; application to diagonalizing the Hamiltonian; relation to symmetries; two-body example. - September 23 (Thursday):
- Harmonic oscillators:
general harmonic oscillator; â, â
^{†}, and n̂ operators; eigenvalues of the n̂; degeneracies and extra degrees of freedom. - September 28 (Tuesday):
- Finished harmonic oscillators: matrix elements; seeing oscillations in non-stationary states. 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 30 (Thursday):
- Coherent states of multiple oscillators:
review of coherent states; generalize to multiple oscillators; coherent states of the vibrating 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. - Extra lecture on October 1 (Friday):
- Identical bosons:
identical bosons; occupation numbers; creation and annihilation operators;
1-body and 2-body operators; creation and annihilations fields.

Bose–Einstein condensation: particles / fields duality; classical field for the condensate; coherent state of BEC and the classical field; Landau–Ginzburg model; ground state and μ>0; flowing BEC in the classical field language. - October 5 (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;
lasers.

Introduction to wave mechanics: Schrödinger equation and boundary conditions in 1 dimension; bound and unbound states. - October 7 (Thursday):
- Un-bound states in 1 dimensions: bound vs. unbound states; stationary states and wave packets; reflection and transmission; step potential example; continuity of the wave function and its first derivative.
- October 12 (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; began WKB for un-bound states. - October 14 (Thursday):
- Tunneling: general analysis; cathode example; centrifugal barrier for radial motion; tunneling in nuclear fusion.
- Extra lecture on October 15 (Friday):
- Saddle point method and Airy functions: integrals of exponentials with large exponents; Laplace method for real functions; saddle point method for complex integrals; Airy equation and its Laplace transform; Airy functions as contour integrals; asymptotic behavior of Airy functions; Airy functions and the WKB approxymation in QM.
- October 19 (Tuesday):
- More WKB:
clarifying a few points about tunneling and fusion;
limitations on WKB in 3 dimensions; WKB phases for interference;
neutron interference due to gravity.

Charged particle in EM fields: Ehrenfest equations; gauge transforms of the wave function; covariant derivatives and kinematic momenta; covariant Schrödinger equation. - October 21 (Thursday):
- Aharonov–Bohm effect:
schematics of the experiment; un-gauging
**A**; propagation amplitudes and their gauge dependence; interference and magnetic flux; AB effect and the chage quantization.

Magnetic monopoles: heuristics of magnetic charge quantization; Dirac monopoles; Dirac's charge quantization; monopoles in GUTs and in the string theory. - Extra lecture on October 22 (Friday):
- Introduction to path integrals: discretizing time; normalization; phase-space path integral; coordinate-space path integral; partition function; harmonic oscillator example.
- October 26 (Tuesday):
- Symmetries in quantum mechanics:
groups of symmetries; unitary operators for the symmetries; Schrödinger and Heisenberg pictures of symmetries;
continuous symmetries and their generators; translation symmetry as an example.

Rotations in 2 dimensions: generator J_{2d}; relation to the angular momentum, orbital, spin, or combined; spectrum of L_{2d}=L_{z}; spectrum of S_{2d}, real particles vs. anyons. - October 28 (Thursday):
- Rotation symmetry:
active and passive rotations; transformation lawa for scalars, vectors and tensors;
symmetric and antisymmetric tensors.

SO(3) rotation group: infinitesimal rotations; generators are angular momenta J_{x,y,z}; commutation relations for the generators. - November 2 (Tuesday):
- Continuous groups like SO(3) and their representations:
recovery of group product from the commutator algebra of the generators;
representations of groups and of the generators.

Spectrum and eigenstates of the angular momentum: Ĵ_{z}, Ĵ_{±}, and**Ĵ**^{2}operators; |*α,j,m*⟩ states and the spectrum of*j*and*m*; matrix elements of Ĵ_{x,y,z}in the |*α,j,m*⟩ basis; multiplets and representations of the angular momentum and of the rotation group. - November 4 (Thursday):
- Clarify representations vs. multiplets.
Central potential: orbital angular momentum
**L**; spherical harmonics; radial wave function; degeneracies of the energy spectrum (briefly).

Spin: non-trivial rotations by 2π; Spin(3)≅SU(2) symmetry group;~~spin-statistics theorem~~. - Extra lecture on November 5 (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 9 (Tuesday):
- Identical particles:
Permutation symmetry and its representations; bosons and fermions; spin–statistics theorem;
anyons in 2 space dimensions.

Degeneracy of the central potential problems: extra symmetries for the harmonic potential; extra symmetries for the Coulomb potential.

Started adding angular momenta: simultaneous rotations of separate degrees of freedom;**Ĵ**=**Ĵ**_{1}+**Ĵ**_{2}. - November 11 (Thursday):
- Adding angular momenta:
reorganizing (j
_{1})⊗(j_{2}) multiplet into multiplets of j; spindle diagrams; Clebbsch–Gordan coefficients; adding several angular momenta. - November 16 (Tuesday):
- Wigner–Eckard theorem: reduced matrix elements; the scalar case; the vector case; spherical tensors; Wigner–Eckard theorem for the tensor operators; projection theorem for vector operators; magnetic moment example.
- November 18 (Thursday):
- Perturbation theory:
perturbative expansion; leading order energy correction; example; corrections to degenerate energies;
fine structure example; Zeeman effect example;
~~linear Stark effect example~~. - Extra lecture on November 19 (Friday):
- Density operators: mixed states and probabilities; time evolution of density operators; entropy; ρ̂ in statistical mechanics.
- November 23 (Tuesday):
- Parity symmetry:
polar and axial vectors; states of definite parity; selection rules;
implications for the linear Stark effect.

Second-order perturbation theory: formal perturbation series; first-order δ|ψ⟩ and second-order δE.

Quadratic Stark effect: formal series and selection rules;*m*-dependence of δE;~~atomic polarizability; hydrogen atom example~~. - November 30 (Tuesday):
- Transitions:
transitions to a perturbation; transition probabilities;
transitions to continuum; density of final states;
*δE∼ℏ/t*; Fermi's golden rule for the transition*rate*. - December 2 (Thursday):
- Emission of photons by atoms: electric dipole approximation; EM matrix element for spontaneous emission of a photon; calculating the net transition rate; hydrogen 2p→1s example; spontaneous vs. stimulated emission; emission line broadening.

Last Modified: December 10, 2021. Vadim Kaplunovsky

vadim@physics.utexas.edu