राष्ट्रीय विज्ञान शिक्षा एवंअनुसंधान संस्थान

National Institute of Science Education and Research

National Institute of Science Education and Research

PG Core Course

- Hydrogen atom including 1.s coupling and hyperfine interaction.
- helium atom introduction to exchange and correlation; variational calculation of ground and exited-states.
- Introduction to the idea of effective potentials for electrons in many-electron atoms (Hartree theory and idea of self-consistency); use of Clementi-Roetti wave-function.

- HIlbert space (states, operators, evolution)
- One dimensional problems & Harmonic oscillator, delta & periodic pots
- Bound states vs scattering states
- The central force problem
- The hydrogen atom, hard and soft sphere
- Time-independent perturbation theory, WKB approximation, variational method
- Time-dependent perturbation theory, Heisenberg and interaction represtations
- Dirac equation

- Review of thermodynamics, thermodynamic potentials, thermodynamic equilibrium and stability
- Gibbs distribution: Ensembles, classical and quantum free particles, systems with continuous and discrete spectrum, degenerate Fermi systems, Bose-Einstein condensation.
- Interacting system: Cluster an Virial expansions, radial distribution function.
- Introduction to response, fluctuation and noise, Einstein formula.

- Electrostatics in vacuum, force, field, potentials and enegry.(4 hrs)
- Electrostatic boundary conditions and conductors. (2 hrs)
- Solution of Laplace's equation in one, two and three dimensions, uniqueness theorem, methods of images, separation of variables, multipole expansion.(12 hrs)
- Dielectrics(4 hrs)
- Current distributions, magnetic fields and magnetostatic boundary conditions(4 hrs)
- Motion of charges in E & B fields, energy and momentum of electromagnetic fields(8 hrs)

- Vectors and Tensors (index notation,vector analysis in curvilinear coordinates. Cartesian tensors and four vectors, General tensors).
- Review of Linear Algebra with emphasis on applications to physical problems (linear transformations + Matrix representations, Eigen values + Eigen Vectors, Innner product spaces).
- Review of complex analysis with applications (Cauchy-Riemann equations, Complex integration, Cauchy theorems, Contour integration, Branch points and branch cuts, Applications to integrals, series etc.)

- Two-body Central force problem (reduced mass), planet orbits,Virial theorem.
- Collisions and scattering, CM and lab frames, scattering cross section.
- Motion in non-inertial frames, Coriolis force.
- principle of virtual work, constraints, D Alemberts principle.
- Generalized coordinates, velocities and momenta, Lagranges formulation.
- principle of list action, fomulation by Maupertuls, Euler, Hamilton, Liouvilles theorem.
- Hamilton's equations, poisson brackets.