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.
