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+---
+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.