Energies and Eigenstates I (10-20 min)

Consider the motion of a particle of mass $m$ within the interval $[x_1,x_2]=[0,L], L>0$ between the infinitely high walls of the potential

$$V(x)=\left\{ \begin{array}{cc} \infty, & -\infty<x\le 0 \\ 0, & 0<x\le L \\ \infty & L<x< \infty \end{array} \right.$$ (7)

Show that the normalized energy eigenstate wave functions and energies are

$$\psi_n(x) = \sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right ),\quad E= E_n= \frac{n^2 \hbar^2 \pi^2}{2mL^2},\quad n=1,2,3,...$$ (8)