Answer
$L = 0.426~nm$
Work Step by Step
We can find the energy of the emitted photon:
$E = \frac{h~c}{\lambda}$
$E = \frac{(6.626\times 10^{-34}~J~s)(3.0\times 10^8~m/s)}{200\times 10^{-9}~m}$
$E = 9.939\times 10^{-19}~J$
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:
$\Delta E = \frac{3~h^2}{8mL^2} = 9.939\times 10^{-19}~J$
We can find the length of the box:
$\frac{3~h^2}{8mL^2} = 9.939\times 10^{-19}~J$
$L^2 = \frac{3~h^2}{(8m)(9.939\times 10^{-19}~J)}$
$L = \sqrt{\frac{3~h^2}{(8m)(9.939\times 10^{-19}~J)}}$
$L = \sqrt{\frac{(3)~(6.626\times 10^{-34}~J~s)^2}{(8)(9.109\times 10^{-31}~kg)(9.939\times 10^{-19}~J)}}$
$L = 4.26\times 10^{-10}~m$
$L = 0.426~nm$
