Categories: Physics, Quantum mechanics.

# Hellmann-Feynman theorem

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

\begin{aligned} \hat{H}(\lambda) \ket{\psi_n(\lambda)} = E_n(\lambda) \ket{\psi_n(\lambda)} \end{aligned}

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

\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 $\lambda$, which could be a scalar or a vector. This yields:

\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 $\inprod{\psi_m}{\psi_n} = \delta_{mn}$:

\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 $\inprod{\nabla_\lambda \psi_m}{\psi_n} = - \inprod{\psi_m}{\nabla_\lambda \psi_n}$. Using this result to replace $\inprod{\nabla_\lambda \psi_m}{\psi_n}$ in the previous equation leads to:

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

\begin{aligned} \boxed{ \nabla_\lambda E_n = \matrixel{\psi_n}{\nabla_\lambda \hat{H}}{\psi_n} } \end{aligned}

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

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