 Parity

For a quantum state that is an eigenstate of the parity operator, parity is its eigenvalue $+1$ or $-1$. In a scalar position representation in quantum mechanics, the parity operator acts by $(P\psi )(\mathbf{r})=\psi (-\mathbf{r})$. Since $P^2=I$, these are its only possible eigenvalues: even wavefunctions have parity $+1$ and odd wavefunctions have parity $-1$.

If the Hamiltonian commutes with the parity operator, its energy eigenstates can be chosen to have definite parity. In the quantum harmonic oscillator, parity alternates as $(-1{)}^n$. This helps establish selection rules: the position operator is odd under spatial reflection, so its matrix element between two states of the same parity vanishes.