Hellmann-Feynman theorem

The Hellmann-Feynman theorem is a theorem in quantum mechanics , which relates the energy eigenvalues ​​of a time-independent Hamilton operator to the parameters it contains. It is named after its discoverers Hans Hellmann (1936) and Richard Feynman (1939).

In general, the theorem says:

${\displaystyle {\frac {\partial {E_{n}}}{\partial {\lambda }}}=\int {\psi _{n}^{*}{\frac {\partial {\hat {H}}}{\partial {\lambda }}}\psi _{n}d\tau }}$

${\displaystyle {\hat {H}}}$is the parameterized Hamilton operator ,

${\displaystyle E_{n}}$ is the nth eigenvalue of the Hamilton operator,

${\displaystyle \psi _{n}}$ is the nth eigenvector of the Hamilton operator,

${\displaystyle \lambda }$ is the parameter that interests

and means a complete integration over the entire domain of definition of the eigenvectors. ${\displaystyle d\tau }$

The proof

The proof, if you proceed in a purely formal manner, is quite simple. In Dirac's Bra-Ket notation, the following can be written:

${\displaystyle {\begin{aligned}{\frac {\partial E_{\lambda }}{\partial \lambda }}&={\frac {\partial }{\partial \lambda }}\langle \psi (\lambda )|{\hat {H}}_{\lambda }|\psi (\lambda )\rangle \\&=\langle {\frac {\partial \psi (\lambda )}{\partial \lambda }}|{\hat {H}}_{\lambda }|\psi (\lambda )\rangle +\langle \psi (\lambda )|{\hat {H}}_{\lambda }|{\frac {\partial \psi (\lambda )}{\partial \lambda }}\rangle +\langle \psi (\lambda )|{\frac {\partial {\hat {H}}_{\lambda }}{\partial \lambda }}|\psi (\lambda )\rangle \\&=E_{\lambda }\langle {\frac {\partial \psi (\lambda )}{\partial \lambda }}|\psi (\lambda )\rangle +E_{\lambda }\langle \psi (\lambda )|{\frac {\partial \psi (\lambda )}{\partial \lambda }}\rangle +\langle \psi (\lambda )|{\frac {\partial {\hat {H}}_{\lambda }}{\partial \lambda }}|\psi (\lambda )\rangle \\&=E_{\lambda }{\frac {\partial }{\partial \lambda }}\langle \psi (\lambda )|\psi (\lambda )\rangle +\langle \psi (\lambda )|{\frac {\partial {\hat {H}}_{\lambda }}{\partial \lambda }}|\psi (\lambda )\rangle \\&=\langle \psi (\lambda )|{\frac {\partial {\hat {H}}_{\lambda }}{\partial \lambda }}|\psi (\lambda )\rangle .\end{aligned}}}$

there applies:

${\displaystyle {\hat {H}}_{\lambda }|\psi (\lambda )\rangle =E_{\lambda }|\psi (\lambda )\rangle ,}$

${\displaystyle \langle \psi (\lambda )|\psi (\lambda )\rangle =1\Rightarrow {\frac {\partial }{\partial \lambda }}\langle \psi (\lambda )|\psi (\lambda )\rangle =0.}$

For a critical, mathematical consideration of this proof, see.

Individual evidence

1. ↑ Hellmann Introduction to Quantum Chemistry , Deuticke, Leipzig and Vienna 1937 (translation from Russian)
2. ↑ Richard Feynman Forces in Molecules , Physical Review, Volume 56, 1939, pp. 340-343
3. ↑ David Carfì: The pointwise Hellmann-Feynman theorem . In: AAPP Physical, Mathematical, and Natural Sciences . 88, No. 1, 2010, ISSN  1825-1242 . doi : 10.1478 / C1A1001004 .