A spatial inversion sends $$\mathbf r\rightarrow-\mathbf r.$$ The corresponding quantum parity operator $P$ acts on a wavefunction as $$(P\psi)(\mathbf r)=\psi(-\mathbf r).$$ Applying inversion twice returns the original state, so $$P^2=I.$$ Therefore a parity eigenstate can have only eigenvalue $+1$ or $-1$: $$P|\psi\rangle=\pm|\psi\rangle.$$ States with parity $+1$ are called even, while states with parity $-1$ are odd.

In one dimension, an even wavefunction satisfies $$\psi(-x)=\psi(x),$$ while an odd wavefunction satisfies $$\psi(-x)=-\psi(x).$$ For a symmetric potential $$V(-x)=V(x),$$ the Hamiltonian commutes with parity. Energy eigenstates can therefore be chosen to have definite parity, except that degeneracy can require an appropriate choice of basis inside the degenerate subspace.

Parity immediately constrains matrix elements. Position is odd under inversion: $$P,x,P^{-1}=-x.$$ If both initial and final states have the same parity, then the integrand in $$\langle f|x|i\rangle$$ is odd and its integral over symmetric space vanishes. An operator with odd parity therefore connects opposite-parity states but has zero matrix element between same-parity states.

This makes parity a powerful selection-rule tool: one can often determine that a transition amplitude vanishes without evaluating the full integral.

Parity is a discrete symmetry. Unlike continuous translations or rotations, it does not come with an infinitesimal parameter; its main role here is to classify quantum states and constrain which matrix elements and transitions are allowed.