Syllabus
Path integral approach: Derivation of path integral approach (phase space path integral and position
space path integral), Free particle and Harmonic oscillator in the path integral approach, Instanton contribution to path integral
Non-perturbative Quantum Mechanics: Limitations of perturbation theory, the asymptotic nature of the perturbation expansion, WKB approximation and Tunneling, Instantons in Quantum Mechanics, Instanton in double-well potential, Instanton in periodic potential, Decay of False vacuum and first-order phase transition.
Resurgence and resummation of perturbation theory: Non-convergence of perturbation theory,
Borel summability of perturbation series, application of these techniques to anharmonic oscillator with x^4 potential.
Berry Phase and Topological Phases in Quantum Mechanics, Berry connection and curvature of
Berry connections.
Quantization of constrained systems: Example of constrained systems, First class and Second class constraints, Dirac Bracket and Quantization, Examples like particles on a surface.
Fermionic Quantum Mechanics: Majorana Quantum Mechanics, Fermionic parity and its realisation on the Hilbert space, Path integral approach to Fermionic Quantum Mechanics and Grassmann variables, Kitaev’s mod 2 and mod 8 classification.
Other topics:
Supersymmetric Quantum Mechanics: What is supersymmetry in Quantum Mechanics, Witten index and topological invariance.
Geometry of Quantum States: Projective Hilbert space and Complex projective space (analogue of Bloch sphere for higher dimensional systems), Quantum probability amplitude and Fubini-Study metric.
Bootstrap method in Quantum Mechanics to find wave function and energy eigenstates, numerical techniques to implement bootstrap method.