Course Identification

Two semesters of undergraduate QM

1. Motion in external electromagnetic field. Gauge invariance. Symmetries in the presence of gauge fields. The gauge principle - guessing interactions by gauging symmetries. Uniform magnetic field. Electromagnetic field in different gauges. Landau levels.
2. Quantization of electromagnetic field. Review of the canonical quantization of a field using elastic string as an example. Casting Maxwell equations into canonical Hamiltonian form. Wave functionals. Photons. Number, coherent and squeezed states. Elements of quantum optics. Casimir effect. Interaction with (non relativistic) matter. Multiple expansion. Dipole and higher order transitions. Selection rules. Sum rules.
3. Second quantization. Review of the many body wave functions for identical particles. Treating Schroedinger equation as a field and its quantization. Equivalence of two approaches. Fock space. Examples of using the second quantization - Hartree-Fock equations for fermions, Gross-Pitaevski equation for bosons. Elementary excitations. Bogoliubov spectrum. Thomas-Fermi method.
4. The semiclassical approximation. Expansion around stable minima. Multidimensional systems. The WKB approximation. Connection formulae. Bohr-Sommerfeld quantization. Tunneling.
5. The scattering theory. Scattering amplitude and cross section. The partial wave decomposition, phase shifts. Formal scattering theory - the Lippman-Schwinger equation. The Born approximation. The WKB approximation. Inelastic scatering and reactions.
6. Time dependent problems and methods of approximation. Adiabatic theory. The Berry phase. Interacting fast and slow systems-the Born-Oppenheimer approach.
7. Pure and Mixed Systems in Quantum Mechanics. The Density Matrix. Wigner transform.

The learning process is supported and tested by weekly tutorial sessions, homework assignments and final written exam. As a result of studying this course.

* The course’s content might be modified to adjust it  to the audiences’ background

Upon successful completion of this course students should be able to:

1. Demonstrate good proficiency in topics in advanced quantum mechanics such as the density matrix, physics of decoherence, quantum mechanics of the motion in magnetic field, canonical quantization of simple fields, quantization of EM field and its interaction with non-relativistic matter, second quantization viewed as a quantization of the Schrodinger field and some basic many body phenomena, adiabatic and semiclassical theory.
2. Continue with further advanced studies of detailed quantum mechanical description of complex atomic, nuclear, quantum optical phenomena as well as basic solid state physics.

G. Baym QM, Landau and Lifshits QM, Messiah, QM