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QUANTUM MECHANICS - I
Course Outline
Historical background, wavefunction, superposition principle, wave packets, Schrodinger Equation, probability and current densities, expectation value, Ehrenfest's theorem.
General formalism : linear vectors and operators in Hilbert space, observables, commuting operators, momentum representation and uncertainty principle, unitary transformation.
Schrodinger and Heisenberg representation, equation of motion.
Applications : One dimensional problems, linear harmonic oscillator, polynomial solution, creation and annihilation operators.
Central forces, angular momentum, spherical harmonics, spin, addition of angular momentum. Motion in a well, free and bound states in Coulomb potential.
Texts/References :
R. Shankar, Principles of Quantum Mechanics, 2nd edn., Plenum 1994.
L. Landau and E. Lifsitz, Quantum Mechanics, Pergamon Press, 1965.
J. J. Sakurai, Modern Quantum Mechanics, Addison wesley, 1994. **
L. I. Schiff, quantum Mechanics, 3rd edn., McGraw Hill, 1968.
B.H. Bransden and C. J. Jochain, Introduction to Quantum Mechanics, ELBS Longman, 1989.
** Main Text for the Course
Lecture 0: Old quantum theory and problems with classical physics.
Lecture 1: Wave-particle duality and birth of quantum mechanics, old quantum theory and problems with classical physics.
Lecture 2: Postulates of Quantum Mechanics, Copenhagen Interpretation, Linear Vector Space, Operators, Hermitian Operators, Dirac Bra-Ket notation.
Lecture 3: Measurement in Quantum Mechanics, Compatible and incompatible observables, Stern-Gerlach experiment, Spin operators in ket-bra basis.
Lecture 4: Operators in Hilbert Space, Uncertainty Product, Basis in Hilbert space, Unitary Operators.
One dimensional Potential Problems: Particle in a box, Square well Potential, Potential Barrier and Tunnelling, Attractive Delta Function Potential.
Probability Current & Equation of Continuity
Harmonic Oscillator- I Series solution, Hermite polynomials, zero point energy, Properties of one dimensional potential
Harmonic Oscillator Potential (Dirac Representation) Harmonic Oscillator - II
Lecture 5: Position and Momentum Representation of a quantum state
Schrodinger and Heisenberg Pictures
Angular Momentum 1 : Angular momentum as generators of rotation, Properties of angular momentum, Matrix representation for j=1/2 and 1.
Angular Momentum 2 : Angular momentum for Spin- 1/2, Pauli spinors
Angular Momentum 3 : Eigenstates of orbital angular momentum, Legendre polynomials, Spherical harmonics
Hydrogen Atom - Series solution, separation of variables, Laguerre Polynomials
Landau Levels and Quantum Hall Effect - Electrons in a magnetic field, the classical Hamiltonian, solution of quantum mechanical problem, the ground state, Landau levels and their degeneracies, Hall effect - classical and quantum, qualitative explanation of Integer Quantum Hall effect.
EPR and Bell’s Inequalities - EPR Paradox, Entanglement, Bell’s Theorem, CHSH Inequalities, Aspect Experiment.
Problem Sets and Tests and Quizzes
1. Set 1
2. Set 2
3. Quiz 1
4 Set 3
5. Mid-semsester Test (2018)