Symmetry in Quantum Mechanics

State Wigner's theorem and explain its significance for quantum symmetry transformations.

Represent symmetry transformations as unitary or antiunitary operators on Hilbert space.

Relate the symmetry group of a Hamiltonian to the degeneracy structure of its energy spectrum.

Derive a conservation law from a continuous symmetry of the Hamiltonian.

Apply symmetry principles (e.g., parity) to derive selection rules for transition matrix elements.

Describe the properties of the time-reversal operator for integer and half-integer spin systems.

Derive Kramers' degeneracy theorem from time-reversal symmetry.