Quantum Mechanics

I(a)        Hermitian operators and their properties, Unitary operators, Dirac’s Bra and Ket notation: Normalization and orthogonality conditions; Orthnormality; Eigen states and eigen values of an operator; Degeneracy.

            States of a quantum mechanical system, Representation of quantum-mechanical states, Properties of quantum mechanical amplitudes, Operators and change of state, a complete set of basis states, product of linear operators.

I(b)      Process of measurement; Expectation values; Time dependence of quantum mechanical amplitudes; Observables with no classical analogue: spin; Dependence of quantum mechanical amplitude on position: the wave functions; Super position of amplitudes: interference.

II(a)    Hamiltonian matrix and the time evolution of  Quantum mechanical States; Hermiticity of the

           Hamiltonian matrix;

II(b)    Time independent perturbation of an nondegenerate system; Harmonic Oscillator and simple matrix

           examples of time independent perturbation, Degenerate Perturbation Theory.

III(a)  Energy eigen states of a two state system; Diagonalizing the energy matrix, Time independent perturbation of a two state system, the perturbation solution: weak field and strong field cases; General description of a two state system: Pauli matrices; Ammonia molecule as an example of two state system.

III(b)   Transitions in a two state system; Time dependent perturbations: The Golden Rule, Adiabatic and Sudden Perturbation, Phase space, Energy width of quasi stationary states.

IV(a)    Compatible observables and constants of motion; Symmetry transformation and conservation laws; Invariance of the Hamiltonian; Invariance under space and time translations and space rotation and conservation of momentum, energy and angular momentum. Space inversion, Time Reversal.

IV(b)    Coordinate representation of operators: position, momentum and angular momentum, Time Dependence of expectation values, Components of angular momentum operator in Cartesian and spherical polar coordinates, Commutation relations.

V(a)      Angular momentum operators and their eigen values; Matrix representation of the angular momentum operators and their eigen states; Coordinate representation of the orbital angular momentum operators and their eigen states (Spherical Harmonics).

V(b)     Composition of angular momenta: Clebsch-Gordon Coefficients; Recursion relations; Construction procedure; C.G. Coefficients for simple cases (j1 = ½ , j2 = ½ ;  j1=1, j2 = ½;  j1=1, j2=1), Irreducible spherical tensor operators,  Wigner-Eckart theorem.