PHYS812
Quantum Mechanics II
Course description
This course will introduce advanced concepts and techniques of quantum mechanics and demonstrate their relevance to research applications.
Lecture 1
Syllabus and general course information.
Review: How does one solve the Schrödinger equation? Example: hydrogen-like atom.
Special hydrogenic systems: positronium, muonium, antihydrogen, muonic and hadronic atoms.
Lecture 2
Symmetries & conservation laws
Orbital angular momentum & spin
Quantum mechanics of angular momentum
Lecture 3
Addition of the angular momenta. Clebsch-Gordon coefficients, 3j and 6j symbols.
Graphical representation.
Wigner-Eckart theorem. Irreducible tensor operators.
Lectures 4 - 5
Graphical representation. Angular momentum diagrams.
Lectures 6 - 7
Research example: parity nonconservation in atoms.
Topics: conservation laws, perturbation theory, angular momentum addition, reduced matrix elements, Wigner-Eckart theorem, hyperfine states, summation over magnetic moments: use of angular diagrams.
Lectures 8 - 9
Identical particles. Bosons and fermions. Quarks and colors. Symmetric and antisymmetric wave functions.
Slater determinants. Many-particle operators. Rules for calculation of matrix elements of one-particle and two-particle operators. Example: energy levels of two-electron atoms and ions (He and He-like ions).
Practical application of perturbation theory and variational method.
Lectures 10 - 11
Second quantization (example: atomic electrons). Normal form of operator product.
Many-particle operators in second quantizations. Example: Coulomb two-particle matrix element.
Application: calculation of energy levels in He and He-like ions (general case: LS coupled states).
Lecture 12
Second quantization. Normal form of operator product and expectation values. Contractions.
Wick’s theorem. Second quantization: closed shell systems.
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Lectures 13 - 14
Self-consistent fields.
Hartree-Fock equations: He-like systems. Hartree-Fock equations: closed-shell systems.
Lectures 15 - 16
Scattering. Differential cross section. The Born approximation. Validity of the Born approximation.
Solving scattering problems: examples.
Lectures 17 - 18
The method of partial waves. Optical theorem. Calculation of phase shifts. Examples.
Scattering of two identical particles.
Lecture 19
Relativistic Quantum Mechanics. Klein-Gordon equation and the interpretation of the Klein-Gordon equation.
The Dirac equation.
Lecture 20
Plane wave solutions of the Dirac equation. Spherical spinors.
Hydrogen-like systems again (relativistic version). Dirac energy levels.
Lecture 21
A particle in electromagnetic field. Magnetic effects: The Aharonov-Bohn effect.
Flux quantization in superconductors. Josephson junctions. Superconducting devices.