= Solution
The <infinite square well> levels for width $L$ are $E_n=n^2h^2/(8m_eL^2)$. Starting in the <ground state>, the smallest excitation is $1\to2$, and its electric-dipole matrix element is nonzero. It therefore sets the longest absorption wavelength:
$$
\frac{hc}{\lambda_{\max}}=E_2-E_1=\frac{3h^2}{8m_eL^2},\qquad
\boxed{L=\sqrt{\frac{3h\lambda_{\max}}{8m_ec}}\simeq0.302\,\mathrm{nm}.}
$$
The corresponding <energy> separation is about $12.4\,\mathrm{eV}$. Elastic scattering of light is a different process from the stipulated absorption into an excited well state.
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