This is a two-term graduate course that covers most of the essential topics of modern nonrelativistic quantum mechanics. The course is primarily intended for graduate students with background in Physics.

1. Addition of angular momentums and direct product space. Spin orbit interaction. Model problem: N non-interacting spin-1/2 particles and the Dicke states.

2. N-particle systems. Indistinguishable particles and Pauli exclusion principle. System of spin-1/2 particles and exchange interaction.

3. Introduction to second quantization methods. Operators on Hilbert space of Dirac states. Model problem: 1D chain of strongly-interacting spin-1/2 particles.

4. Symmetries in quantum mechanics. Invariance under unitary transformations and conservation laws. Space inversion symmetry and parity. Lattice symmetry: Bloch waves and energy bands. Time reversal symmetry and its consequences.

5. Approximation methods in quantum mechanics: variational methods, time-independent perturbation theory. Time-independent perturbation theory in case of degenerate states. Selection rules for orbital angular momentum.

6. Energy spectrum of the hydrogen atom revisited: fine structure and hyperfine splitting.

7. Hydrogen atom in static electric and magnetic fields: quadratic and linear Stark effects, Zeeman splitting and Paschen-Back effect.

8. Time-dependent perturbation theory. Interaction picture and Dyson series for the time-evolution operator. Transitions under time-dependent perturbations: adiabatic and sudden perturbations.

9. Harmonic perturbation and interaction of quantum particles with electromagnetic field. The Fermi’s golden rule. Stimulated emission and absorption of electro-magnetic waves by a quantum particle. Spontaneous emission and the Einstein coefficients. Exactly solvable time-dependent problem: two-level system approximation and the Rabi oscillations.

10. Introduction to the quantum electrodynamics (QED): quantization of electro-magnetic field. Operators for electric and magnetic fields. Photons and vacuum fluctuations of electro-magnetic field.

11. Interaction of a quantum particle with electromagnetic field revisited: beyond semi-classical description. Derivation of the rate of spontaneous emission. The Lamb shift and renormalization of electron mass.

12. Introduction to quantum statistical physics. Density matrix formalism and statistical ensembles. System of non-interacting quantum particles: the Boltzmann, Bose-Einstein and Fermi-Dirac distributions.

13. Description of open quantum systems. Dephasing. Density matrix approach and the master equation. Model problem: the spin-boson model and optical Bloch equations.

Students who take the course are expected to be familiar with general topics in Classical Mechanics, Electrodynamics and Calculus. This course requires a pass in A216 Quantum Mechanics I.