PHY 511:

1. Introduction (1)
Experimental motivations of quantum mechanics. Basic concepts of wave mechanics; its insufficiency.

 2. Basic formalism (8)
Quantum states. Hilbert space. Bra and ket vectors. Linear superpositions. Scalar (inner) product. Linear operators, commutator and anticommutator. Identity, adjoint and self-adjoint (Hermitian) operators. Orthonormal sets and matrix formalism. Eigenvalues and eigenstates. Outer products and projection operators. Change of basis. Matrix diagonalization. Measurements and “measurements" (preliminary discussion). Compatible and incompatible observables. Heisenberg's uncertainty relation. Spin operator; Stern-Gerlach experiment and its description. Hamiltonian operator, Schrödinger and Heisenberg pictures of quantum dynamics; spin precession. Discrete and continuous spectra.

 3. 1D quantum particle (9)
Coordinate operator, reduction to wave mechanics. Operator of momentum; plane waves, wave packets. Ehrenfest theorem, continuity equation, probability current. Reflection from potential step. Tunneling through delta-functional and rectangular barriers. 1D scattering and transfer matrices. Resonant tunneling. 1D rectangular quantum well. Harmonic oscillator, Fock (stationary) states, creation and annihilation operators, Glauber (coherent) states, squeezed states. WKB approximation, classical turning points, Bohr-Sommerfeld quantization rule. Metastable states and their decay. Quantum oscillations in double well potentials. Motion in periodic potential; Bloch theorem, energy bands and gaps. Propagator, Feynman path integral.

 4. 2D and 3D motion (8)
Generalization to higher dimensions. Two-slit interference description. Motion in EM field; Aharonov-Bohm effect. 2D and 3D scattering characterization. Born approximation, optical theorem, eikonal approximation.

PHY 512:

Variable separation. Partial quantum confinement, Landau levels. 2D and 3D rectangular quantum wells and harmonic oscillators. Symmetry at rotation, angular momentum; 2D and 3D rotators and spherical quantum wells. Partial phase method, hard sphere scattering, resonant scattering. Bohr’s atom.

 5. Perturbation theories (4)
Constant perturbation in non-degenerate and degenerate systems; anharmonic oscillator, Stark effect. Spin addition to orbital momentum; Clebsh-Gordan coefficients; Zeeman effect. Time-dependent perturbation theory; Rabi oscillations. Transitions in continuous spectrum, "Fermi" Golden Rule.

 6. Open systems, quantum statistics, and quantum measurements (4)
Pure and mixed quantum states. Density matrix. Classical mixture in thermal equilibrium. Wigner function. Density matrix dynamics without and with interaction with environment, dephasing. Quantum measurements, QND. Bayes theorem. Bell’s inequalities and the local reality problem.

 7. Identical particles (7)
Permutation symmetry, indistinguishability principle, bosons and fermions. Two electron systems, singlet and triplet states, helium atom, covalent (chemical) bond. Atoms, periodic table of elements. Second quantization for bosons and fermions, Fermi gas of interacting electrons. Hartri and Hartri-Fock approximations.

 8. Quantum theory of EM field (5)
Electromagnetic field modes and their quantization. Casimir effect. The notion of photon; its energy, momentum, and angular momentum. EM field statistics, coherence, 2nd order correlation functions, photon bunching and antibunching. Quantum EM field interaction with charged particles. Spontaneous and induced transitions, rate of electric dipole transitions, Einstein coefficients.

 9. Relativistic theory of particles with finite mass (5)
Relativistic Schrödinger equation, particles and antiparticles. Dirac equation, introduction of spin. Relativistic Fermi particles in EM field, spin-orbit interaction, application to atomic spectra. Relativistic theory of hydrogen atom.

 10. Quantum field theory preview (1)

Wave mechanics reminder
Bra-ket formalism
Matrix language
Eigenstates and eigenvalues
Quantum measurements

---

C. OTHER COURSE MATERIALS

Gaussian wave packets: 1, 2, 3
Resonant tunneling through: 2 barrier system,  N barrier system
Energy bands
Propagators:  free particle,  harmonic oscillator
Legendre polynomials

---

D. HOMEWORKS AND EXAMS

PHY 511:
 

Homeworks: 1, 2, 3 (with solutions), 4, 5, 6, 7, 8, 9,10
Exams (with solutions): Midterm 1, Midterm 2,  Final

PHY 512:
 

Homeworks: 1, 2,  3, 4, 5,  6 (with solution), 7,  8, 9, 10,  11
Exams (with solutions): Midterm 1, Midterm 2, Final 

---