UNIT 1:Introduction to Quantum mechanics: Wave function, expectation values, Schrodinger equation for free particles, Bound state problems.Linear Vector Spaces: Basics, Inner Product Spaces, Dual spaces and the Dirac Notation, Subspaces, Linear Operators, Matrix elements of linear operators, Active and Passive transformations, The Eigenvalue problem, Functions of Operators and related concepts, Generalization to infinite dimensionsUNIT 2:The Postulates, Basic postulates of quantum mechanics, Observables and operators, Measurements in quantum mechanics, Time evolution of the systems state, Symmetries and conservation laws. Connecting quantum mechanics and classical mechanics.Properties of One-Dimensional Motion: Bound, Unbound, and Mixed States, Symmetric potentials and parity, free particle, Potential step, Potential barrier and Well, Infinite square well potential, Finite square well potential.UNIT 3:Review of the Classical Oscillator, Quantization of the Oscillator (Coordinate Basis), The Oscillator in the Energy Basis, Passage from the Energy Basis to the position Basis. Matrix Representation of Various Opera- tors, Expectation Values of Various Operators. General expression for uncertainty relations.UNIT 4:Introduction, Orbital Angular Momentum, General Formalism, Matrix Representation, Geometrical Representation, Spin Angular Momentum, Experimental Evidence, theory of Spin, Spin 1/2 and Pauli Matrices. Eigen functions of orbital angular momentum: The Eigen value Problem of L2 and Lz, Properties of the Spherical Harmonics.UNIT 5:Rotations in Quantum Mechanics: Infinitesimal and Finite Rotations, Properties of the Rotation Operator, Euler Rotations, Rotation Matrices. Addition of Angular Momenta: Addition of two Angular Momenta: General formalism, Calculation of the ClebschGordan Coefficients, Addition of more than two angular momenta, Coupling of Orbital and Spin Angular Momenta, Rotation matrices for coupling two angular momenta, Scalar, Vector, and Tensor Operators.
