Answer
$\lambda = 13.7~nm$
Work Step by Step
We can find an expression for energy of an electron in this potential well:
$E_n = (\frac{h^2}{8m~L^2})~n^2$
$E_n = \frac{h^2}{8m~L^2}~n^2$
$E_n = \frac{(6.626\times 10^{-34}~J~s)^2}{(8)(9.109\times 10^{-31}~kg)~(250\times 10^{-12}~m)^2}~n^2$
$E_n = (9.640\times 10^{-19}~J)~n^2$
$E_n = (9.640\times 10^{-19}~J)(\frac{1~eV}{1.6\times 10^{-19}~J})~n^2$
$E_n = (6.02~eV)~n^2$
Note that light with a shorter wavelength has more energy than light with a longer wavelength.
The shortest wavelength will be emitted when the electron jumps between states with the largest energy difference.
We can find the energy in the third excited state where $n=4$:
$E_4 = (6.02~eV)~(4)^2 = 96.3~eV$
We can find the energy in the ground state where $n=1$:
$E_1 = (6.02~eV)~(1)^2 = 6.02~eV$
We can find the difference in energy:
$E_4-E_1 = (96.3~eV) - (6.02~eV) = 90.28~eV$
We can find the wavelength associated with this energy:
$\lambda = \frac{hc}{E}$
$\lambda = \frac{(4.136\times 10^{-15}~eV~s)(3.0\times 10^8~m/s)}{90.28~eV}$
$\lambda = 13.7\times 10^{-9}~m$
$\lambda = 13.7~nm$
