Course Details
Subject {L-T-P / C} : PH4005 : Quantum Mechanics - I { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Prof. Sasmita Mishra
Syllabus
Linear Vector Space and matrix formulation of Quantum Mechanics: Dirac ket representation and mathematical definition of a vector space, scalar product, definition of dual vectors, Cauchy-Schwartz inequality, linear operators and their algebra, hermitian operator, unitary operator, projection operator. Linear independence of vectors, dimension of a space, basis and span, completeness of basis, degenerate case, Gram-Schmidt orthogonalization method. Measurement process in quantum mechanics and its relation with expectation value, compatible and incompatible observables, complete set of commutating observables. Matrix representation of a ket, outer product and matrix representation of an operator, basis transformation, similarity transformation, eigenvalues and eigenvectors of a matrix. Infinite-dimensional vector spaces the space of square-integrable (L2) functions and its definition, weight functions, continuous basis, Hilbert space, position eigenket, position space wave function, translation operator, momentum-space wave function, self adjoint extension of momentum operator for finite interval. Quantum Dynamics: Schrödinger picture, time evolution operator and time dependence of expectation value, spin precession. Heisenberg picture, Heisenberg equation of motion. Time evolution of base kets and transition amplitude, density operator and density matrix, pure and mixed ensemble, ensemble averages and density operator, time evolution of ensembles. Theory of angular momentum: Introducing angular momentum as generator of rotation. Eigenvalues, eigenstates and matrix elements of orbital angular momentum operator. Euler rotation and representations of rotation operator. Stern-Gerlach experiment and introduction to the idea of spin, representation of spin operator in bra-ket notation, matrix representation and Pauli matrices, quantum dynamics and spin precession. Rotation operator for spin-1/2 and its matrix representation. Addition of angular momentum, Clebsch-Gordan (CG) coefficients. CG coefficients and rotation matrices. Tensor operators, Wigner-Eckart theorem, projection theorem. Discrete Symmetries in Quantum Mechanics: Parity symmetry and operator, wave functions under parity, parity selection rule. Discrete time reversal symmetry, time reversal operator, wave functions under time reversal, time reversal for a spin-1/2 system and Kramers degeneracy.
Course Objectives
- Get familiar with formulation of Quantum Mechanics through linear algebra and matrix algebra.
- Solving problems in Quantum Mechanics and getting a clear idea of eigenenergies and eigenfunctions.
- Solving problems in more than one dimension and multi-particle systems.
Course Outcomes
The students will get familiar with solving problems in Quantum Mechanics through Dirac's bra and ket notations. The students will learn the spin and angular momentum algebra to be applied in the advanced courses.
Essential Reading
- D. J. Griffith, Introduction to Quantum Mechanics, Pearson, 2nd Edition (2007)
- J. J. Sakurai, Jim Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University press
Supplementary Reading
- C. Cohen-Tannoudji, Quantum Mechanics, Volume 1: Basic Concepts, Tools, and Applications, 2nd ed., Wiley
- R. Shankar, Principles of Quantum Mechanics, Plenum Publishers, 2nd Edition (1994)