: ¤³¤ÎÊ¸½ñ¤Ë¤Ä¤¤¤Æ... : lecture_10 : The Born-Oppenheimer Approximation

# Proof of the Hellman-Feynman Theorem

Consider a system with a Hamiltonian that depends on some parameters . Let be an eigenvector of with eigenvalue

 (16)

We further assume that is normalized so that
 (17)

The Hellman-Feynman theorem states that
 (18)

The proof of the Hellman-Feynman theorem is straightforward. We begin with the fact that
 (19)

Differentiating both sides yields
 (20)

Since is an eigenvector of , this can be written as

However, since is normalized, we have, from the normalization condition:

Hence, the term in square brackets vanishes, and we have
 (21)

which is just the Hellman-Feynman theorem.

: ¤³¤ÎÊ¸½ñ¤Ë¤Ä¤¤¤Æ... : lecture_10 : The Born-Oppenheimer Approximation
Mark Tuckerman Ê¿À®17Ç¯3·î8Æü