Text and reference books: J. Sakurai, Modern Quantum Mechanics; C. Cohen-Tannoudji, Quantum Mechanics (2 vols); Landau and Lifshitz, Quantum Mechanics (Course of Theoretical Physics, Vol.3); Schiff; and so on. Two classics are Dirac's monograph and the Feynman lectures (vol.3).
The first part is a review of basic QM:
Stern-Gerlach experiments, operator formulation of QM, Dirac kets/bras/operators, matrix representations, uncertainty relations, position/momentum operators, time evolution and the Schrodinger equation, Schrodinger/Heisenberg pictures;
Harmonic oscillator (creation-annihilation operators etc), position space representations and Schrodinger's equation. Perturbation theory (time-independent non-degenerate), fine structure etc in the Hydrogen atom; some calculational tricks in the H-atom. Perturbation theory, degenerate, more systematically. Angular momentum and addition, Clebsch-Gordan coefficients etc.
Time-dependent potentials and perturbation theory; variational methods.
The second part comprises two subtopics:
(i) Path integrals: transition amplitude: time slicing to sum over paths and the path integral; additional insertions, correlation functions; free particle; path integral for the harmonic oscillator, Fourier variables; (Polchinski, "String theory", vol.1, Appendix, first few pages; Srednicki, "Quantum field theory", chapters on path integrals in QM)
(ii) Quantum entanglement: EPR and spin singlet correlations; 2 spins (via angular momentum addition), entangled states --> reduced density matrix --> entanglement entropy.
Witten's lectures on Information theory (arxiv:1805.11965, upto sec.3.5; also Preskill's lectures) -- classical information theory (Shannon entropy, conditional entropy, mutual information, relative entropy); quantum information and entanglement (density matrices, purification, von Neumann entropy, mutual information, relative entropy, monotonicity of relative entropy).
Regular assignments, midsem, endsem