Categories: Physics, Quantum mechanics.

Hellmann-Feynman theorem

Consider the time-independent Schrödinger equation, where the Hamiltonian H^\hat{H} depends on some parameter λ\lambda whose meaning we will not specify:

H^(λ)∣ψn(λ)⟩=En(λ)∣ψn(λ)⟩\begin{aligned} \hat{H}(\lambda) \ket{\psi_n(\lambda)} = E_n(\lambda) \ket{\psi_n(\lambda)} \end{aligned}

Assuming all eigenstates ∣ψn⟩\ket{\psi_n} are normalized, this gives us the following basic relation:

⟨ψm∣H^∣ψn⟩=En⟨ψm∣ψn⟩=δmnEn\begin{aligned} \matrixel{\psi_m}{\hat{H}}{\psi_n} = E_n \inprod{\psi_m}{\psi_n} = \delta_{mn} E_n \end{aligned}

We differentiate this with respect to λ\lambda, which could be a scalar or a vector. This yields:

δmn∇λEn=∇λ⟨ψm∣H^∣ψn⟩=⟨∇λψm∣H^∣ψn⟩+⟨ψm∣∇λH^∣ψn⟩+⟨ψm∣H^∣∇λψn⟩=Em⟨ψm∣∇λψn⟩+En⟨∇λψm∣ψn⟩+⟨ψm∣∇λH^∣ψn⟩\begin{aligned} \delta_{mn} \nabla_\lambda E_n &= \nabla_\lambda \matrixel{\psi_m}{\hat{H}}{\psi_n} \\ &= \matrixel{\nabla_\lambda \psi_m}{\hat{H}}{\psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n} + \matrixel{\psi_m}{\hat{H}}{\nabla_\lambda \psi_n} \\ &= E_m \inprod{\psi_m}{\nabla_\lambda \psi_n} + E_n \inprod{\nabla_\lambda \psi_m}{\psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n} \end{aligned}

In order to simplify this, we differentiate the orthogonality relation ⟨ψm∣ψn⟩=δmn\inprod{\psi_m}{\psi_n} = \delta_{mn}:

0=∇λδmn=∇λ⟨ψm∣ψn⟩=⟨∇λψm∣ψn⟩+⟨ψm∣∇λψn⟩\begin{aligned} 0 = \nabla_\lambda \delta_{mn} = \nabla_\lambda \inprod{\psi_m}{\psi_n} = \inprod{\nabla_\lambda \psi_m}{\psi_n} + \inprod{\psi_m}{\nabla_\lambda \psi_n} \end{aligned}

Meaning that ⟨∇λψm∣ψn⟩=−⟨ψm∣∇λψn⟩\inprod{\nabla_\lambda \psi_m}{\psi_n} = - \inprod{\psi_m}{\nabla_\lambda \psi_n}. Using this result to replace ⟨∇λψm∣ψn⟩\inprod{\nabla_\lambda \psi_m}{\psi_n} in the previous equation leads to:

δmn∇λEn=(Em−En)⟨ψm∣∇λψn⟩+⟨ψm∣∇λH^∣ψn⟩\begin{aligned} \delta_{mn} \nabla_\lambda E_n &= (E_m - E_n) \inprod{\psi_m}{\nabla_\lambda \psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n} \end{aligned}

For m=nm = n, we therefore arrive at the Hellmann-Feynman theorem, which is useful when doing numerical calculations that often involve minimizing energies with respect to λ\lambda:

∇λEn=⟨ψn∣∇λH^∣ψn⟩\begin{aligned} \boxed{ \nabla_\lambda E_n = \matrixel{\psi_n}{\nabla_\lambda \hat{H}}{\psi_n} } \end{aligned}

While for m≠nm \neq n, we get the Epstein generalization of the Hellmann-Feynman theorem, which is for example relevant for the Berry phase:

(En−Em)⟨ψm∣∇λψn⟩=⟨ψm∣∇λH^∣ψn⟩\begin{aligned} \boxed{ (E_n - E_m) \inprod{\psi_m}{\nabla_\lambda \psi_n} = \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n} } \end{aligned}

References

  1. G. Grosso, G.P. Parravicini, Solid state physics, 2nd edition, Elsevier.