The parity

For symmetric potentials $V(x)=V(-x)$, the Schrödinger equation has an important property: If $\psi(x)$ is a solution of $\hat{H}\psi(x)=E \psi(x)$, then also $\psi(-x)$ is a solution with the same $E$, i.e. $\hat{H}\psi(-x)=E \psi(-x)$ (replace $-x \to x$ and note that $\partial_x^2 = \partial_{-x}^2$. Since $\hat{H}$ is linear, also linear combinations of solutions with the same eigenvalue $E$ are solutions with eigenvalue $E$, in particular the symmetric (even) and anti symmetric (odd) linear combinations

$$\psi_e(x):=\frac{1}{\sqrt{2}}[\psi(x)+\psi(-x)],\quad \psi_o(x):=\frac{1}{\sqrt{2}}[\psi(x)-\psi(-x)].$$ (107)

These are the solutions with even (e) and odd (o) parity, respectively.