---
title: "Hellmann-Feynman theorem"
date: 2021-11-29
categories:
- Physics
- Quantum mechanics
layout: "concept"
---

Consider the time-independent Schrödinger equation,
where the Hamiltonian $\hat{H}$ depends on a general parameter $\lambda$,
whose meaning or type 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
$\Inprod{\psi_m}{\psi_n} = \delta_{mn}$,
which ends up telling us that
$\Inprod{\nabla_\lambda \psi_m}{\psi_n} = - \Inprod{\psi_m}{\nabla_\lambda \psi_n}$:

$$\begin{aligned}
    0
    = \nabla_\lambda \delta_{mn}
    = \nabla_\lambda \Inprod{\psi_m}{\psi_n}
    = \Inprod{\nabla_\lambda \psi_m}{\psi_n} + \Inprod{\psi_m}{\nabla_\lambda \psi_n}
\end{aligned}$$

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
to minimize 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](/know/concept/berry-phase/):

$$\begin{aligned}
    \boxed{
        (E_n - E_m) \Inprod{\psi_m}{\nabla_\lambda \psi_n}
        = \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n}
    }
\end{aligned}$$



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