Syllabus (PHYSICS)

Course Type: MAJ-12

Semester: 7

Course Code: BPHSMAJ12C

Course Title: Quantum Mechanics and Applications

(L-P-Tu): 4-2-0

Credit: 6

Practical/Theory: Combined

Course Objective: This course aims to provide a comprehensive foundation in quantum mechanics and its applications to atomic systems. It introduces the time-dependent and time-independent Schrödinger equations, emphasizing their role in describing quantum states, wave functions, and uncertainty principles. Students will analyze bound-state problems such as the particle in a potential well and the quantum harmonic oscillator, and extend the study to hydrogen-like atoms through angular momentum formalism and radial wavefunctions. The course explores atomic behavior in external electric and magnetic fields, covering spin, Zeeman and Stark effects, and their physical significance. Many-electron atoms are studied using Pauli’s exclusion principle, coupling schemes, Hund’s rules, and term symbols, with applications to atomic spectra. Practical sessions focus on solving the radial Schrödinger equation numerically for hydrogen-like systems and simple molecular vibrations, strengthening computational and problem-solving skills.

Learning Outcome: This course aims to provide a comprehensive foundation in quantum mechanics and its applications to atomic systems. It introduces the time-dependent and time-independent Schrödinger equations, emphasizing their role in describing quantum states, wave functions, and uncertainty principles. Students will analyze bound-state problems such as the particle in a potential well and the quantum harmonic oscillator, and extend the study to hydrogen-like atoms through angular momentum formalism and radial wavefunctions. The course explores atomic behavior in external electric and magnetic fields, covering spin, Zeeman and Stark effects, and their physical significance. Many-electron atoms are studied using Pauli’s exclusion principle, coupling schemes, Hund’s rules, and term symbols, with applications to atomic spectra. Practical sessions focus on solving the radial Schrödinger equation numerically for hydrogen-like systems and simple molecular vibrations, strengthening computational and problem-solving skills.

Theory (4 Credits)

Schrödinger Equation

Time dependent Schrödinger equation: Time dependent Schrödinger equation and dynamical evolution of a quantum state; Properties of Wave Function. Interpretation of Wave Function Probability and probability current densities in three dimensions; Conditions for Physical Acceptability of Wave Functions. Normalization. Linearity and Superposition Principles. Eigenvalues and Eigenfunctions. Position, momentum and Energy operators; commutator of position and momentum operators; Expectation values of position and momentum. Wave Function of a Free Particle. (8 Lectures)

Time independent Schrödinger equation-Hamiltonian, stationary states and energy eigenvalues; expansion of an arbitrary wavefunction as a linear combination of energy eigen functions; General solution of the time dependent Schrödinger equation in terms of linear combinations of stationary states; Application to spread of Gaussian wave-packet for a free particle in one dimension; wave packets, Fourier transforms and momentum space wave function; Position-momentum uncertainty principle. (10 Lectures)

General discussion of bound states in an arbitrary potential

Continuity of wave function, boundary condition and emergence of discrete energy levels; application to one-dimensional problem-square well potential; Quantum mechanics of simple harmonic oscillator-energy levels and energy eigenfunctions using Frobenius method; Hermite polynomials; ground state, zero point energy and uncertainty principle. (10 Lectures)

Quantum theory of hydrogen-like atoms

Time independent Schrödinger equation in spherical polar coordinates; separation of variables for second order partial differential equation; angular momentum operator & quantum numbers; Radial wavefunctions from Frobenius method; shapes of the probability densities for ground & first excited states; Orbital angular momentum quantum numbers l and m; s, p, d, shells.

(10 Lectures)

Atoms in Electric and Magnetic Fields
Electron angular momentum. Space quantization. Electron Spin and Spin Angular Momentum. Larmor’s Theorem. Spin Magnetic Moment. Stern-Gerlach Experiment. Zeeman Effect: Electron Magnetic Moment and Magnetic Energy, Gyromagnetic Ratio and Bohr Magneton.

(8 Lectures)

Atoms in External Magnetic Fields
Normal and Anomalous Zeeman Effect. Paschen Back and Stark Effect (Qualitative Discussion only). (4 Lectures)

Many Electron Atoms
Pauli’s Exclusion Principle. Symmetric and Antisymmetric Wave Functions. Periodic table. Fine structure. Spin orbit coupling. Spectral notations for Atomic States. Total angular momentum. Vector Model. Spin-orbit coupling in atoms- L-S and J-J couplings. Hund’s Rule. Term symbols. Spectra of Hydrogen and Alkali Atoms (Na etc.). (10 Lectures)

Practical (2 Credits)

1. Solve the s-wave Schrodinger equation for the ground state and the first excited state of the

hydrogen atom
2. Solve the s -wave radial Schrodinger equation for an atom
3. Solve the s -wave radial Schrodinger equation for a particle of mass m
4. Solve the s -wave radial Schrodinger equation for the vibrations of hydrogen molecule

Reading References

Theory

1. Introduction to Quantum Mechanics, D. J. Griffiths, 3^rd Edn, Cambridge University Press
2. A Text book of Quantum Mechanics, P.M. Mathews and K. Venkatesan, McGraw Hill
3. Quantum Mechanics: Concepts and Applications, N. Zettili, 3^rd Edition, Wiley
4. Quantum Mechanics, R. Eisberg and R. Resnick, 2^nd Edn., Wiley.
5. Quantum Physics, H.C. Verma, 2^nd Edn., Surya Publication
6. Physics of Atoms and Molecules, B.H. Bransden and C.J. Joachain, Pearson, 2^nd Edition
7. Fundamentals of Molecular Spectroscopy, C.B. Banwell and McCash, 6^th Edn., Midtech.
8. Foundation of Quantum Mechanics, A B Gupta, Books & Allied Pvt Ltd

Practical

1. An introduction to computational Physics, T. Pang, 2nd Edn., Cambridge Univ. Press
2. Simulation of ODE/PDE Models with MATLAB®, OCTAVE and SCILAB: Scientific &

Engineering Applications: A. Vande Wouwer, P. Saucez, C. V. Fernández, Springer.