Harmonic Oscillator: Parity

The harmonic oscillator functions have the symmetry property

$$\psi_n(x)=(-1)^n\psi_n(-x),$$ (253)

that is, they are symmetric or antisymmetric with respect to the symmetry point $x=0$ of the potential $V(x)$, in alternating order with the quantum number $n$. The lowest eigenstate $\psi_0(x)$ is symmetric and has no node, the next eigenstate $\psi_1(x)$ is anti-symmetric and has one node, $\psi_2(x)$ is again symmetric and has two zeros, and so on.