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Przedmiot: Symmetries and conservation laws in physics Prowadzący: dr hab. Piotr Magierski

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Przedmiot: Symmetries and conservation laws in physics

Prowadzący: dr hab. Piotr Magierski

Semestr: 8

Liczba godzin: 30


Liczba pkt. kredytowych

Preceding courses: classical mechanics, classical electrodynamics, statistical physics, quantum mechanics, group theory.

Forma zaliczenia: exam
  1. Program:

This lecture aims at presenting the role of symmetries in physics and their implications for conservation laws. In particular the following aspects will be covered: the relation between quantum numbers and symmetries, quantum state degeneracies, selection rules and spontaneous symmetry breaking.

  1. Symmetries and conservation laws in classical mechanics.

  2. Canonical transformations. Motion as a continuous canonical transformation.

  3. Invariants of canonical transformations.

  4. Action-angle variables, integrable systems.

  5. Bohr-Sommerfeld quantization rules for classical systems.

  6. Finite group theory (main results).

  7. Selected aspects of Lie group theory.

Generators and Lie algebras. Representations. Irreducible representations. Casimir operators. Irreducible operators and the Wigner-Eckhart theorem. Direct product representations, Clebsch-Gordan coefficients.

  1. Examples: one-dimensional rotation group, three-dimensional rotation group.

  2. Application: partial wave decomposition.

  3. Symmetries in quantum mechanics. Quantum state degeneracies. Selection rules.

  4. Examples: angular momentum algebra in quantum mechanics, two-dimensional harmonic oscillator, multipole radiation of the electromagnetic field.

  5. Time-reversal invariance. Kramers degeneracy.

  6. The Lorentz group. Generators and the Lie algebra. Irreducible representations.

  7. Examples of field theories in spaces of irreducible representations of the Lorentz group.

  8. Symmetries and conservation laws in field theories. Gauge symmetries and gauge fields. Spontaneous symmetry breaking, phase transitions.

Basic literature:

A. Goldstein, “Classical Mechanics”

V.I. Arnold, “Metody matematyczne mechaniki klasycznej”

J.P. Elliot, P.G. Dawber, “Symmetry in Physics”

M. Hamermesh, “Teoria grup w zastosowaniu do zagadnień fizycznych”

D.H. Perkins, “Wstęp do fizyki wysokich energii”

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