Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 4 / 59 / b / v

With the supplied sum, the second-order approximation isFor an electron in the $1\mu \mathrm{m}$ box, $E_1^{(0)}=6.02\times 10^{-26}\mathrm{J}$ and the correction is $-4.61\times 10^{-27}\mathrm{J}$, giving $5.56\times 10^{-26}\mathrm{J}=3.47\times 10^{-7}\mathrm{eV}$.

For $10\mu \mathrm{m}$, the formal result is approximately $-4.61\times 10^{-23}\mathrm{J}=-2.88\times 10^{-4}\mathrm{eV}$. It is not a credible approximation: even the minimum electric potential energy in that box is only $-5\mu \mathrm{eV}$, whereas this answer is much lower. The ground-to-first-excited-state mixing estimate increases from $0.160$ to $160$ when the length grows tenfold. The shift grows as $a^4$ while the level spacing decreases as $a^{-2}$, so the larger box is far outside the perturbative regime.