These are lecture notes based on two semester courses taught at University of Hyderabad for several batches. In the first phase only the lecture notes have are being made aailable and these provide an introduction to the subject.
These have to be supplemented by a set of assignments which will be made available in the second phase.
TABLE OF CONTENTS
1. Classical Theories Revisited
Classical wave and particle concepts revisited;
Structure of classical theory;
Formulations of classical theory.
2. Basic Concepts in Vector Spaces
Groups, Fields and Vector Spaces;
Basis and Dimension; Linear Combination, Basis and Dimension;
Linear Functionals; Dual of a Vector Space;
Linear Operators-I; Algebra of Linear Operators; Commutator; Inverse of an Operator;
Eigenvalues and Eigenvectors; Basis and Matrix Representation; Matrix Representation;
Change of Basis
3. Inner Product Spaces
Inner Product in Vector Spaces; Cauchy Schwarz and Triangle Inequalities;
Orthogonality and Gram Schmidt Procedure;
Linear Operators in Inner Product Spaces;
Hermitian and Unitary Operators;
Complete o.n. Sets; Representation in o.n basis;
Dirac Bra and Ket Notation;
Spectral Theorem Functions of Operators.
4. Infinite Dimensional vector spaces
5. Thrity years that shook Physics
Black Body Radiation; Photoelectric Effect; Atomic Spectra; Alpha Decay;
Specific Heat of Solids and Gases; Planck's Theory of Black Body Radiation;
Einstein's Explanation of Photoelectric Effect; Bohr Model of H Atom;
Bohr Sommerfeld Quantization; Compton Effect; Correspondence and
Complementarity Principle; Franck Hertz Experiment; de Broglie Waves;
Davisson Germer Experiment; Discovery of Spin; Stern Gerlach Experiment;
Einstein's Theory of Specific Heat of Solids;
6 Postulates of Quantum Mechanics
Photon Polarisation Experiment; Double Slit Experiment;
Superposition principle for states
Measurement of position and momentum
Uncertainty Principle
New Concepts in Quantum Theory;
Appearance of Probability in Quantum Mechanics
Superposition principle and the first postulate of quantum mechanics
The Second Postulate of Quantum Mechanics
Examples of Operators in Quantum Mechanics;
The Third Postulate of Quantum Mechanics
Probabilities and Average Values;
Measurement in Quantum Mechanics.
7. Canonical Quantization
Canonical quantization rule;
Heisenberg microscope;
Uncertainty principle;
General form of uncertainty principle;
Simultaneous measurement of two dynamical variables
8. Eigenvalues Using Commutators
Harmonic oscillator;
Angular momentum;
Summary of important results
Time Development in quantum mechanics;
Schrodinger equation; Ehrenfest theorem; Time development of averages;
Constants of motion; Stationary states;
Schrodinger, Heisenberg and Dirac pictures
10. Working with Representations
Coordinate representation; Eigenfunctions of momentum;
Momentum representation; Other representations;
Change of representation Time Dependent
11. Time Dependent Schrodinger Equation
Schrodinger equation in coordinate representation;
Conservation of probability;
Propagator
12. Free Particle
Free particle in one dimension;
Free particle in two and three dimensions;
Periodic boundary condition and box normalization;
Particle in a box in one dimension;
Particle in a box in two and three dimensions
13. Energy Eigenvalue Problems in One Dimension
Square well; Square barrier; Square well problem ;
Delta function potential;
Harmonic oscillator;
General properties of motion in one dimension;
Periodic potential
14. Reflection and Transmission
Probability current and definition of reflection and transmission coefficients;
Reflection and transmission through a square barrier;
Bound states as poles in transmission coefficient in the complex energy plane.
15. Problems in Two Dimensions
Free particle in polar coordinates;
Harmonic oscillator in two dimensions ;
Charged particle in a uniform magnetic field.
16. Spherically Symmetric Potential Problems
Angular Momentum in coordinate representation;
Separation of variables in polar coordinates;
Solution of radial equation for a constant potential;
Free particle solution in polar coordinates ;
Square well and hard sphere;
General properties of solutions;
Isotropic oscillator in three dimension;
Hydrogen atom energy levels in Schr\"odinger mechanics;
Accidental degeneracy;
Energy levels of positronium and alkali atoms
17. Angular Momentum in Quantum Mechanics
Summary of angular momentum eigenvalues and eigenvectors;
Addition of angular momenta; Clebsch Gordon coefficients;
Relation with rotations; Tensor operators; Wigner Eckart theorem
18. Scattering in quantum mechanics
Definition scattering amplitude and cross section;
Boundary conditions for the wave function;
Green function; A simple example of integral equation;
Green function for Poisson equation
Integral equation for scattering;
Born approximation; Validity of Born approximation;
Square well potential.
19. Method of Partial Waves
Partial wave expansion of plane waves;
Statement of the problem; Radial wave function;
The phase shift; Relating cross to phase shifts;
Phase shifts for a square well potential.
20. Spin and Identical Particles
Spin as a Dynamical Variable;
Spin wave function and spin operators;
Identical particles in quantum mechanics;
Symmetrization postulate;
A first look at the He Spectrum;
21. Variation Method
Ritz variation method for ground state
22. WKB Approximation
Semi-Classical Expansion;
Validity of lowest order WKB approximation;
Connection Formulae; Continuity requirement; solution for a linear potential;
Bohr Sommerfeld quantization;
Tunnelling and through a barrier;
WKB approximation fo radial equation
23 Time Independent Perturbation Theory
Non degenerate case
First and Second order non degenerate perturbation theory
Degenerate case First order correction to energy levels and wave function
Second order correction to the energy levels.
24. Approximation Methods for Time Dependent Problems
Sudden approximation;
Adiabatic approximation;
Time dependent perturbation theory;
Transition amplitude; Perturbation theory ;
I-Transition to a set of discrete levels.
II-Periodic perturbation-- resonant absorption; Probability for esonance transitions; Details of Solution
III-Transitions to a final state in continuum; Fermi golden rule;
White light
25. Semi-Classical Theory of Radiation
Maxwell's equations; Scalar and vector potential;
Gauge invariance; Charged particle in electro magnetic fields;
Dipole approximation; Absorption and induced emission;
Spontaneous emission;
Einstein A, B Coefficients;
Absorption of a white light
Symmetries in Classical and Quantum Mechanics Wigner's theorem; Rotation group in quantum mechanics; Time reversal in quantum mechanics; Noether's theorem; Poincare groupMore Later
Foundations of Quantum MechanicsCopenhagen interpretation of quantum mechanics; Statistical interpretation of quantum mechanics; Bohr correspondence and complementarity principles; Many worlds interpretations of quantum mechanics; Hidden variables; EPR paradox and Bell's theorem; Entanglement; Measurement problem; Collapse of wave function; Schrodinger cat.
Quantum Information
Atomic Physics
Molecular Physics
Nuclear Physics
Elementary Particle Physics
Relativistic Quantum Mechanics
Second Quantization
Quantum Hamilton Jacobi Scheme
Applications of Group Theory in Quantum Mechanics
Formal Theory of Scattering