QUANTUM MECHANICS

States, Amplitude and Operators: Hermitian operators and their properties, Unitary operators, Dirac’s Bra and Ket notation: Normalization and orthogonality conditions; Orthonormality; 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.

 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.

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;

Two State Systems: 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.

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.

 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.

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. Space inversion, Time Reversal.

Angular momentum; Components of angular momentum operator in Cartesian and spherical polar coordinates, Commutation relations.

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).

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.