**[Cr:4, Lc:3, Tt:1, Lb:0]**

- Classical limit of Quantum mechanics: Semi-classical quantization,
WKB. Coherent states as “best approximants” to classical behaviour.
Squeezed states.
This topic will explore solutions of the Schroedinger equation using approximate methods, mainly based on the saddle point method. Relationships to Bohr-Sommerfeld methods and to the path integral will lead to approximate wavefunctions, energy levels, and to classical mechanics.

- Perturbation theory: time dependent and independent, Standard material including degenerate cases. Borel resummation, Diagrammatics, Fermi golden rule Non perturbative effects. Instantons.
- Quantum systems in classical fields: Aharonov-Bohm, Landau levels etc. Studying quantum systems coupled to classical electric and magnetic fields. Phases in quantum mechanics. Hall effect, Hofstadter problem. Problems with Semi-classical theory of radiation (Bohr-Rosenfeld analysis).
- Scattering theory: 1-d, 2-d and 3-d. Poles of the scattering matrix. Analyticity properties. Reference: e.g., Sakurai
- Symmetry in Quantum mechanics: Ordinary and supersymmetry. Conserved quantum numbers. Degeneracy and splitting. Wigner-Eckart theorem Representations of symmetry groups. Galilean invariance in quantum mechanics
- Matrix Quantum Mechanics (and quantum gravity): This topic will explore the quantum mechanics of systems with large numbers of degrees of freedom. Large N limit, Nuclear energy levels, Thomas-Fermi model, and a relation to quantum gravity are possible sidelights.
- Quantum Light: Quantum description of optical fields. classical and non-classical light. Photon statistics, sub-Poisson light, squeezed light.

- J. J. Sakurai, Modern Quantum Mechanics, Addison Wesley (1993).
- S. Coleman, Aspects of symmetry: Selected Erice lectures, Cambridge University Press (1988).
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press (1995).