Quantum Mechanics 2

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Physics

Synopsis:

Symmetry in quantum mechanics, invariance and conservation laws, approximate methods, scattering theory, systems of identical particles

Requirements:

Knowledge of the basic course of physics and subject 02KVAN - Quantum mechanics

Syllabus of lectures:

1. Symmetry: general formalism, continuous and discrete transformations, generators. Translation, rotation.

2. Parity, time inversion. Gauge transformation, particle in an electromagnetic field.

3. Addition of angular momenta: Clebsch-Gordan coefficients , 6j-symbols, Irreducible tensor operators, Wigner-Eckart theorem.

4. Elementary theory of representations: Energy, coordinate and momentum representations, General properties of solutions of Schroedinger equation, Free particle solution, decomposition of the plane wave into partial waves.

5. Time evolution and propagators: Schroedinger, Heisenberg and Dirac pictures, Resolvent, stationary Green function, Propagator, retarded a advanced Green operator, Lippmann-Schwinger equation and perturbative solution for the evolution operator

6. Approximate methods: Variational method, helium atom. WKB method, connection formulas, tunneling.

7. Time-dependent perturbation theory, various perturbations, Fermi golden rule. Transitions between discrete levels and into continuum, particle scattered by an external field.

8. Particle in e.m. field: Pauli equation, photoeffect.

9. Introduction into scattering theory: From time-dependent to time-independent description, Wave operators, S-matrix and T-matrix, Stationary scattering states, Lippmann-Schwinger equation, scattering amplitude and cross section.

10. Born series, partial waves, phase shifts. Solutions in coordinate and momentum representations.

11. Systems of identical particles: Pauli principle, (anti)symmetrization of wave functions. One-particle basis, Slater determinants,

12. Fock space, creation and annihilation operators, one- and two-particle operators, Hartree-Fock method.

Syllabus of tutorials:

1. Symmetry:

general formalism, continuous and discrete transformations, generators. Translation, rotation.

2. Parity, time inversion.

Gauge transformation, particle in an electromagnetic field.

3. Addition of angular momenta: Clebsch-Gordan coefficients , 6j-symbols, irreducible tensor operators, Wigner-Eckart theorem.

4. Elementary theory of representations:

Energy, coordinate and momentum representations, General properties of solutions of Schroedinger equation, Free particle solution, decomposition of the plane wave into partial waves.

5. Time evolution and propagators: Schroedinger, Heisenberg and Dirac pictures, Resolvent, stationary Green function, Propagator, retarded a advanced Green operator, Lippmann-Schwinger equation and perturbative solution for the evolution operator

6. Approximate methods:

Variational method, helium atom. WKB method, connection formulas, tunneling.

7. Time-dependent perturbation theory, various perturbations, Fermi golden rule. Transitions between discrete levels and into continuum, particle scattered by an external field.

8. Particle in e.m. field: Pauli equation, photoeffect.

9. Introduction into scattering theory:

From time-dependent to time-independent description, Wave operators, S-matrix and T-matrix, Stationary scattering states, Lippmann-Schwinger equation, scattering amplitude and cross section.

10. Born series, partial waves, phase shifts. Solutions in coordinate and momentum representations.

11. Systems of identical particles:

Pauli principle, (anti)symmetrization of wave functions.

One-particle basis, Slater determinants,

12. Fock space, creation and annihilation operators, one- and two-particle operators, Hartree-Fock method.

Study Objective:

Knowledge:

Advanced quantum-mechanical methods, perturbative formulation and second quantization

Abilities:

Application of quantum description and various (in particular perturbative) methods of solution on realistic microscopic systems

Study materials:

Key references:

[1] D.J. Griffiths: Introduction to Quantum Mechanics, Prentice Hall, 2nd edition, 2004

[2] J. Formánek: Introduction in Quentum mechanics I,II, Academia, 2004 (in Czech)

Recommended references:

[3] J.R. Taylor: Scattering Theory, J. Wiley and Sons, 1972

[4] E. Merzbacher: Quantum Mechanics, 3rd edition, John Wiley, 1998

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: