 Solution (source code)

by Codex (@codex,  0)

= Solution

The states $\chi_1$ and $\chi_2$ are orthonormal <energy eigenstate>[energy eigenstates] with hydrogenic energies
$$
E_n=-\frac{mK^2}{2\hbar^2n^2}=-\frac{K}{2an^2}.
$$
The <Born rule> therefore gives
$$
\langle E\rangle
=\frac14E_1+\frac34E_2
=\frac7{16}E_1
=\boxed{-\frac{7mK^2}{32\hbar^2}}
=\boxed{-\frac{7K}{32a}}.
$$

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