Answer
$E_1 = 2.05~MeV$
$E_2 = 8.22~MeV$
$E_3 = 18.5~MeV$
Work Step by Step
We can write a general expression for the energy levels:
$E_n = \frac{n^2~h^2}{8~m~L^2}$
We can find the energy when $n = 1$:
$E_1 = \frac{1^2~h^2}{8~m~L^2}$
$E_1 = \frac{(1)^2~(6.626\times 10^{-34}~J~s)^2}{(8)~(1.67\times 10^{-27}~kg)~(10\times 10^{-15}~m)^2}$
$E_1 = 3.286\times 10^{-13}~J$
$E_1 = (3.286\times 10^{-13}~J)(\frac{1~eV}{1.6\times 10^{-19}~J})$
$E_1 = 2.05\times 10^6~eV$
$E_1 = 2.05~MeV$
We can find the energy when $n = 2$:
$E_2 = \frac{2^2~h^2}{8~m~L^2}$
$E_2 = \frac{(2)^2~(6.626\times 10^{-34}~J~s)^2}{(8)~(1.67\times 10^{-27}~kg)~(10\times 10^{-15}~m)^2}$
$E_2 = 1.3145\times 10^{-12}~J$
$E_2 = (1.3145\times 10^{-12}~J)(\frac{1~eV}{1.6\times 10^{-19}~J})$
$E_2 = 8.22\times 10^6~eV$
$E_2 = 8.22~MeV$
We can find the energy when $n = 3$:
$E_3 = \frac{3^2~h^2}{8~m~L^2}$
$E_3 = \frac{(3)^2~(6.626\times 10^{-34}~J~s)^2}{(8)~(1.67\times 10^{-27}~kg)~(10\times 10^{-15}~m)^2}$
$E_3 = 2.9576\times 10^{-12}~J$
$E_3 = (2.9576\times 10^{-12}~J)(\frac{1~eV}{1.6\times 10^{-19}~J})$
$E_3 = 1.85\times 10^7~eV$
$E_3 = 18.5~MeV$
