Symmetries and Conservation Laws

Define a quantum symmetry based on the preservation of transition probabilities and state Wigner's theorem.

Outline the key steps in the proof of Wigner's theorem, showing why symmetries must be represented by either unitary or anti-unitary operators.

Construct the spatial translation operator and identify its infinitesimal generator as the momentum operator.

Prove that translational invariance of the Hamiltonian implies the conservation of momentum.

Construct the rotation operator for an infinitesimal rotation and identify the angular momentum operator as its generator.

Show that rotational invariance of the Hamiltonian implies conservation of angular momentum.

Analyze parity as a discrete symmetry, determine its eigenstates, and derive parity selection rules for matrix elements.