nLab Hellman-Feynman theorem

Idea

Hellman-Feynman theorem relates the time derivative of energy in quantum mechanical system with the expectation value of the time derivative of the Hamiltonian operator.

It is a consequence of the variational principle, hence it is valid not only to true eigenstates but also for some other wave functions (e.g. Hartree-Fock) steming from the variational principle.

References

• wikipedia: Hellman-Feynman theorem

Historical references in physics are

• H. Hellmann, Einfuehrung in die Quantenchemie (Franz Deuticke, Leipzig, 1937)

• R. P. Feynman, Forces in molecules, Phys. Rev. 56, 340 (1939)

A strong version with careful functional analysis can be found in

• David Carfì, The pointwise Hellman-Feynman theorem, AAPP Physical, Mathematical, and Natural Sciences 88 (1). no. C1A1001004 doi

The theorem also holds for the WKB approximations:

• K. Banerjee, W. K. B. approximation and scaling, Proc. Royal Soc. London A 363, No. 1712 (Nov. 1, 1978) 147-151 jstor