Quantum Mechanics Semester -1
Quantum Mechanics and Applications (36Hrs)
2.1. Experimental foundation of quantum mechanics: Elementary ideas of black body radiation, photoelectric effect and atomic spectra. Need of quantum mechanics. Concept of matter wave, de Broglie relation, uncertainty principle and its consequences.
2.2. Postulates of Quantum Mechanics: State function or wave function postulate: Born
interpretation of the wave function, well behaved functions, orthonormality of wave
functions. Operator postulate: Operator algebra, linear and nonlinear operators,
Laplacian operator, commuting and noncommuting operators, Hermitian operators and their properties, eigen functions and eigen values of an operator. Eigen value
postulate: eigen value equation, eigen functions of commuting operators .Expectation
value postulate. Postulate of time-dependent Schrödinger equation, conservative
systems and time-independent Schrödinger equation.
2.3. Translational motion: Free particle in one-dimension, particle in a one dimensional box with infinite potential walls, particle in a one-dimensional box with finite potential walls-tunneling, particle in a three dimensional box ,separation of variables, degeneracy.
2.4. Vibrational motion: One-dimensional harmonic oscillator (complete treatment), Hermite equation(solving by method of power series), Hermite polynomials, recursion relation, wave functions and energies-important features, harmonic oscillator model and molecular vibrations.
2.5. Rotational motion: Co-ordinate systems, cartesian, cylindrical polar and spherical polar coordinates and their relationships. The wave equation in spherical polar coordinates-particle on a ring, the phi equation and its solution, wave functions in the real form. Non-planar rigid rotor (or particle on a sphere),separation of variables, the phi and the theta equations and their solutions, Legendre and associated Legendre equations, Legendre and associated Legendre polynomials. Spherical harmonics (imaginary and real forms),polar diagrams of spherical harmonics.
2.6. Quantization of angular momentum, quantum mechanical operators corresponding to angular momenta (Lx, Ly, Lz and L2 ),commutation relations between these operators. Spherical harmonics as eigen functions of angular momentum operators Lz and L2 . Ladder operator method for angular momentum, space quantization.
2.7. Quantum Mechanics of Hydrogen-like Atoms:Potential energy of hydrogen-like systems. The wave equation in spherical polar coordinates: separation of variables-r, theta and phi equations and their solutions, wave functions and energies of hydrogenlike atoms. Orbitals:Radial functions, radial distribution functions, angular functions and their plots. Dirac's relativistic equation for hydrogen atom (Elementary idea only).
2.8. Spin orbitals:Construction of spin orbitals from orbitals and spin functions,spin
orbitals for many electron atoms, symmetric and antisymmetric wave functions.
Pauli's exclusion principle,slater determinants.