Answer
$L = 18~fm$
Work Step by Step
We can write a general equation for the energy:
$E_n = \frac{n^2~h^2}{8mL^2}$
The energy difference between the $n = 2$ state and the $n = 1$ state is:
$\Delta E = \frac{3~h^2}{8mL^2} = 2.0~MeV$
We can find the length of the box:
$\frac{3~h^2}{8mL^2} = 2.0~MeV$
$L^2 = \frac{3~h^2}{(8m)(2.0~MeV)}$
$L = \sqrt{\frac{3~h^2}{(8m)(2.0~MeV)}}$
$L = \sqrt{\frac{(3)~(6.626\times 10^{-34}~J~s)^2}{(8)(1.67\times 10^{-27}~kg)(2.0\times 10^6~eV)(1.6\times 10^{-19}~J/eV)}}$
$L = 1.8\times 10^{-14}~m$
$L = 18~fm$
