: この文書について...
: lecture_10
: The BornOppenheimer Approximation
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 HellmanFeynman theorem states that

(18) 
The proof of the HellmanFeynman 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 HellmanFeynman theorem.
: この文書について...
: lecture_10
: The BornOppenheimer Approximation
Mark Tuckerman
平成17年3月8日