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Lectures On Quantum Mechanics

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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

9. Time Development

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

More  Later 

Symmetries in Classical and Quantum Mechanics Wigner's theorem; Rotation group in quantum mechanics; Time reversal in quantum mechanics; Noether's theorem; Poincare group
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


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