Infinite Square Well

Setup

For $0<x<a$ and infinite walls at the boundaries, the wavefunction satisfies

$$\psi(0)=\psi(a)=0.$$

Solutions

The normalized stationary states are

$$\psi_n(x)=\sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right),$$

with energies

$$E_n=\frac{n^2\pi^2\hbar^2}{2ma^2}.$$