Orthonormality (5 min)

Consider the Hilbert space ${\cal H}$ of wave functions $\psi(x)$ of the infinite potential well on the interval $[0,L]$ with $\psi(0)=\psi(L)=0$. Show that the basis vectors

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

form an orthonormal system.