This is a course in quantum mechancis for first year graduate (M.Sc.) students
The course will meet from 9:10 to 10:25 am on Mondays and Wednesdays in the seminar hall at CMI.
The course will cover the following topics. Review of classical mecahncis: Newton's equation, Lagrangian formulation, action principle, Hamilton's equations and variational principle, Poisson brackets, symmetries, canonical transformations, Hamilton-Jacobi equation, separation of variables. Relation of Hamilton-Jacobi equation to Schrodinger equation, postulates of quantum mechanics, proof of Heisenberg uncertainty inequality, local conservation of probability density and probability current, Ehrenfest theorem and virial theorem. Rotations, Lie algebras and representations, invariant subspace, irreducibility, unitary equivalence, unitary irreducible representations of Lie algebra of infinitesimal rotations, matrix representation of angular momentum operators in spherial harmonic basis, spin angular momentum, spin half Pauli matrices and spin wave functions. Addition of angular momenta: tensor product of angular momentum representations and decomposition into irreducible representations, addition of two or more angular momenta. Simple harmonic oscillator and spectrum via representation of Lie algebra of canonical commutation relations, ladder operators. Linear momentum and angular momentum generate translations and rotations of wave functions. Particle in spherically symmetric potential, separation of variables, radial wave functions for free particle, particle in spherical well and hydrogen atom. Variational principles and approximations: Laplacian on a Riemannian manifold from electrostatic variational principle, Rayleigh-Ritz variational principle for Schrodinger eigenvalue problem, Lagrange multipliers, variational approximations for ground state of harmonic and anharmonic oscillators. Perturbation theory for stationary states of time-independent hamiltonians, two-state system, first order non-degenerate perturbation theory, second order correction to energy, anharmonic oscillator, first order degenerate perturbation theory. Pure and mixed ensembles in classical and quantum mechanics, basic properties of density matrices, time evolution, von Neumann entropy. Potential scattering in one dimension, S-matrix, time-independent scattering in three-dimensions, differential scattering cross section, partial wave expansion, phase shifts, optical theorem, infinitely hard sphere scattering. Time-dependent perturbation theory, first order time-dependent perturbation theory, harmonic perturbation, Fermi's golden rule. Born series: integral form of Schrodinger eigenvalue equation, Green's function of Helmholtz operator, first Born approimation for scattering amplitude, scattering amplitude for Coulomb potential, Rutherford differential cross section.The mid-term exam is scheduled for 9:30 am on Thursday 27 September 2012 in the seminar hall. Notes/books/discussions are not permitted. Answers must be written with a blue pen.
The final exam is scheduled for 1:30 pm on Monday 26 November 2012 at the seminar hall. Notes/books/discussions are not permitted. Answers must be written with a blue pen.