Quantum Mechanics In my class we are talking about the quant

[Quantum Mechanics] In my class we are talking about the quantum harmonic oscillator and I have a question.

Does parity have a generator? If so, what is it? If not, how is the condition of parity symmetry expressed in terms of the operators?

Solution

The parity transformation, P, is a unitary operator, acting on a state as follows: P(r) = e^i/2(r). Also, P²(r) = eⁱ(r), since an overall phase is unobservable. The operator P², which reverses the parity of a state twice, leaves the spacetime invariant, and so is an internal symmetry which rotates its eigenstates by phases eⁱ. If P² is an element e^iQ of a continuous U(1) symmetry group of phase rotations, then e^iQ/2 is part of this U(1) and so is also a symmetry. We can define P = Pe^iQ/2, which is also a symmetry, and so we can choose to call P our parity operator, instead of P. Note that P² = 1 and so P has eigenvalues ±1. However, when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than ±1.