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9

The theory behind the trick is based on the Hellmann-Feynman (HF) theorem $$ \frac{dE_{\lambda}}{d\lambda}~=~\langle \psi_{\lambda} | \frac{d\hat{H}_{\lambda}}{d\lambda}| \psi_{\lambda} \rangle,\tag{A}$$ which works with a single derivative, but not with a square of a derivative, cf. OP's failed calculation (5) for the expectation value $\langle\frac{1}{r^2}...


3

That the trick works once is lucky coincidence, it is actually non-sense as presented. Consider, as an extreme counterexample, an operator $O$ independent from $e$, and a state $\lvert \psi\rangle$ whose eigenvalue w.r.t. $O$ is $e$. Then, by the same logic we have $$ 0 = \langle \psi \vert 0 \vert \psi \rangle = \langle \psi \vert \frac{\partial}{\partial ...


1

Electrons and nuclei both have spin. A spinning charged particle has a magnetic dipole moment. When a magnetic dipole is in a magnetic field, it experiences a force. This oversimplified description gives some brief intuition on the origin of the fine and hyperfine splittings. Fine structure is due to the interaction of the electron's spin with the ...



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