Consider the time-independent Schrödinger equation,
where the Hamiltonian depends on some parameter
whose meaning we will not specify:
Assuming all eigenstates are normalized,
this gives us the following basic relation:
We differentiate this with respect to ,
which could be a scalar or a vector.
In order to simplify this,
we differentiate the orthogonality relation
Meaning that .
Using this result to replace
in the previous equation leads to:
For , we therefore arrive at the Hellmann-Feynman theorem,
which is useful when doing numerical calculations
that often involve minimizing energies with respect to :
While for , we get the Epstein generalization
of the Hellmann-Feynman theorem, which is for example relevant for
the Berry phase:
- G. Grosso, G.P. Parravicini,
Solid state physics,
2nd edition, Elsevier.