Answer
See explanation.
Work Step by Step
For a standing wave inside the box, there is a node at each wall, and an integer number of half-wavelengths fit in the box: $n(\frac{\lambda}{2})=L$.
a. $n\lambda=2L$ and $E_n=\frac{p^2}{2m}$
$$E_n=\frac{(h/\lambda)^2}{2m}$$
$$E_n=\frac{n^2h^2}{8mL^2}$$
b. We substitute $L=a_0=0.5292\times10^{-10}m$ and n = 1 to find an energy of $2.15\times10^{-17}J=134 eV$.
This is a poor model for a 3D hydrogen atom.