Parity reflects every state through the origin: the new wavefunction at $x$ is the old one at $-x$. Doing it twice changes nothing, so its eigenvalues are $+1$ and $-1$, the even and odd functions. When the potential is the same on both sides, parity commutes with the Hamiltonian and each level can be given a definite parity — and then a simple symmetry argument says which matrix elements must vanish. Selection rules for light, one of the most useful facts in atomic physics, start here.

A face seen in a mirror is the same face reversed. A symmetric face is its own reflection; a face turned to one side becomes a face turned to the other. Parity sorts states the same way.

A lopsided state and its mirror image $\hat{\Pi}\psi(x)=\psi(-x)$. Their sum is even and their difference is odd, which is how every function splits into an even and an odd part.

$\hat{\Pi}$ reflects through the origin. It squares to the identity, it is both unitary and Hermitian, and it reverses position and momentum.

• The eigenvalues are $\pm1$: even functions have parity $+1$, odd functions $-1$. Every function is an even part plus an odd part.
• If $V(-x)=V(x)$ then : energy eigenstates can be chosen with definite parity, and parity is conserved.

Write $\hat{x}$ as minus its reflection, $-\hat{\Pi}\hat{x}\hat{\Pi}$, and let each act on the state beside it. The matrix element comes back as minus the product of the two parities times itself; with equal parities that is minus itself, so it is zero.

From $\hat{\Pi}\hat{x}\hat{\Pi}=-\hat{x}$.

Substitute.

$\hat{\Pi}$ is Hermitian with real eigenvalues, so it gives $\pi_{a}$ on the left and $\pi_{b}$ on the right.

Both parities are $+1$ or both are $-1$.

A number equal to minus itself is zero.

Reflecting twice gives back the original state, so parity squares to one and its eigenvalues are $+1$ and $-1$: the even and the odd functions.

Under parity both position and momentum change sign: a reflected particle is on the other side and moving the other way.

If $V(-x)=V(x)$, parity commutes with the Hamiltonian: the kinetic term has $\hat{p}^{2}$, which parity does not change. Energy eigenstates can then be chosen even or odd.

An odd operator such as $x$ has zero matrix elements between two states of the same parity. It connects only states of opposite parity.

Great work! You now know how to:

• find parity eigenvalues of wavefunctions in one and three dimensions
• prove that $x$ connects only states of opposite parity
• decide when parity is conserved