Quantum Mechanics

I(a)    Operator formalism in Quantum Mechanics:  Hermitian operators and their properties, Unitary operator; Angular momentum; Components of angular momentum operator in Cartesian and spherical polar coordinates; Commutation relations.

Dirac’s Bra and Ket notation; Normalization and orthogonality conditions; Orthonormality; Eigen states and eigen values of an operator; Degeneracy;

I (b)   States, Amplitude and Operators: 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, Postulates, essential definitions and relations in Quantum mechanics.

II (a)  Observables and Description of Quantum Systems: 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; Identical particles.

II(b)   Stationary States of a Quantum System: Hamiltonian matrix and the time evolution of Quantum mechanical States; Hermiticity of the Hamiltonian matrix; Time independent perturbation of an arbitrary system; Harmonic Oscillator and simple matrix examples of time independent perturbation; 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(a)  Transition between Stationary States: Transitions in a two state system; Time dependent perturbations: The Golden Rule; Phase space, Emission and absorption of radiation; Induced dipole transition and spontaneous emission of radiation energy; Energy width of quasi stationary states.

III(b)  The co-ordinate Representation: Compatible observables; Quantum conditions and uncertainty relation; Coordinate representation of operators: position, momentum and angular momentum; Time dependence of expectation values:  Ehrenfest Theorem

IV (a) Symmetries: 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.

IV(b)  Irreducible tensor operators; Product of tensor operators; Combination of operator and eigen state; Wigner-Eckart theorem; Space inversion;

V(a)    Angular momentum : 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).