Syllabus

Module 1: ( 7 hours)
Wave-particle duality and de Broglie’s hypothesis, Quantum description of a particle and concept of wave-packet, wave-packet of a free particle and natural emergence of Heisenberg uncertainty relation. Time evolution of a free wave-packet, concept of group velocity and its relation with particle velocity, spread of Gaussian wave-packet for a free particle in one dimension.

Module 2: (5 hours)
Young’s double slit and polarized photon experiments – observations, interpretation and necessity of quantum mechanical description. Physical interpretation of momentum space wave function and Plancherel theorem. Time dependent and independent Schrödinger equation, properties of wave functions, physical acceptability of wave functions, linearity requirement and superposition principles, statistical interpretation of wave function.

Module 3: (5 hours)
Time independent Schrödinger equation and stationary states, energy eigenvalues, completeness of energy eigenfunctions, probability current and continuity of wave function, Eigenvalues and eigenfunctions of Hermitian operator. Hamiltonian operator, position, momentum and energy operators, commutator of position and momentum operators

Module 4: (4 hours)
Expectation values of position, momentum, and arbitrary quantum mechanical operator. Interpretation of expectation value from experimental point of view. Ehrenfest’s theorem.
Discussion on bound states and scattering states in an arbitrary potential: boundary condition and emergence of discrete energy levels.

Module 5: (15 hours)
Applications: One-dimensional problems, energy eigenvalues, eigenfunctions of infinite and finite square well potential and delta function potential. Quantum mechanical scattering and tunneling in one dimension across a step potential and rectangular potential barrier. Simple harmonic oscillator-energy levels and energy eigenfunctions ground state, zero point energy and uncertainty principle, three-dimensional quantum harmonic oscillator and degenerate eigenstates. Hydrogen atom: complete solution of radial and angular equations, wave functions and energy spectrum.

To understand wave-particle duality and how a free particle can be described by a wave packet. Understanding the emergence of the Schrodinger equation from the notion of wave packet, the properties and interpretation of wave function.

To learn the connection between position and momentum space wave functions. To obtain time-independent Schrödinger equation from time-dependent Schrödinger equation.

To learn the time-independent Schrodinger equation as an eigenvalue equation, the role of Hermitian operators in quantum mechanics, and the physical interpretation of commuting and non-commuting operators.

To understand the difference between bound states and scattering states and the role of boundary conditions in giving rise to discrete energy levels.

5. Solving time independent Schrödinger equation for different potentials.

At the end of the course, students will be able to:
CO1: On successful completion of this course, the students should be able to appreciate the dual nature of objects and when it behaves like a particle and when like a wave.

CO2: They should be able to appreciate the physical interpretation of the wave function and the properties it should satisfy.

CO3: They should be able to explain how classical mechanics can be retrieved from quantum mechanics using the expectation values.

CO4: After completing the course the student should be able to appreciate the difference between bound state and scattering state problems from the nature of the potential.

CO5: They should be able to solve independent Schrödinger equation for standard potentials, like the finite and infinite potential well and potential barrier, the harmonic oscillator and the Coulomb potential.

N. Zettili, Quantum Mechanics: Concepts and Applications, Wiley , 2nd Edition (2009).

D. J. Griffith and D. F. Schroeter, Introduction to Quantum Mechanics, Cambridge University Press , 3rd Edition (2018).

C. Cohen-Tannoudji, Bernard Diu, Frank Laloe., Quantum Mechanics, Vol-I, Wiley VCH , 2nd Edition (2019).

R. Shankar, Principles of Quantum Mechanics, Springer , 2nd Edition (2014).