## ABSTRACT

If the Hamiltonian operator depends on a parameter λ, which may be a particle charge or mass, a field strength, or simply a dummy parameter artificially introduced to apply perturbation theory, then the eigenvalues and eigenfunctions of Hˆ will also depend on that parameter. If we differentiate < |(Hˆ − E)| >= 0 with respect to λ

〈 ∂

∂λ

∣∣∣(Hˆ − E)∣∣∣ 〉 + 〈

∣∣∣∣∣ ∂(Hˆ − E)

∂λ

∣∣∣∣∣ 〉 + 〈

∣∣∣(Hˆ − E)∣∣∣ ∂ ∂λ

〉 = 0 (3.5)

and take into account equation (3.2), then we obtain the Hellmann-Feynman theorem [25] ∂E

∂λ < | >=

〈

∣∣∣∣∣ ∂Hˆ

∂λ

∣∣∣∣∣ 〉 , (3.6)

or

∂E

∂λ = 〈 ∂Hˆ

∂λ

〉 (3.7)

if is normalized to unity. In what follows we show that the general results just derived facilitate the application of pertur-

bation theory.