Answer
$6.1\times10^{-10}m$.
Work Step by Step
The energy levels in a 1D box are given by the equation $E_n=\frac{n^2 h^2}{8mL^2}$.
The ground state has n=1, and the first excited level has n = 2.
$\Delta E = E_2-E_1=\frac{2^2 h^2}{8mL^2}-\frac{1^2 h^2}{8mL^2}=\frac{3 h^2}{8mL^2}$
We are told that this energy difference is $ 3.0 eV=4.8\times10^{-19}J$. Solve for L.
$$L=h\frac{\sqrt{3}}{\sqrt{8m \Delta E}}=6.1\times10^{-10}m $$
