THE FEYNMAN–HELLMANN THEOREM

Feynman and Hellmann proved a simple theorem about derivatives of energy levels with respect to parameters, which is often helpful in evaluating the expectation values required for computing first-order perturbation theory. Let H(λ) be a one parameter set of Hamiltonians, En(λ) one of its discrete eigenvalues and |Ψ_n(λ)〉 the corresponding normalized eigenstate.
Assume that we are working at a value of λ where the eigenstate is nondegenerate. The norm of | n(λ)〉 is independent of λ so

If we write

and use the previous equation, we find the Feynman–Hellmann result

This can be viewed as a generalization of the first-order perturbation theory formula to Hamiltonians that depend on nonlinear functions of λ.