Categories: Physics, Quantum mechanics.

Hellmann-Feynman theorem

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:

\[\begin{aligned} \matrixel{\psi_m}{\hat{H}}{\psi_n} = E_n \braket{\psi_m}{\psi_n} = \delta_{mn} E_n \end{aligned}\]

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

\[\begin{aligned} \delta_{mn} \nabla_\lambda E_n &= \nabla_\lambda \matrixel{\psi_m}{\hat{H}}{\psi_n} \\ &= \matrixel{\nabla_\lambda \psi_m}{\hat{H}}{\psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n} + \matrixel{\psi_m}{\hat{H}}{\nabla_\lambda \psi_n} \\ &= E_m \braket{\psi_m}{\nabla_\lambda \psi_n} + E_n \braket{\nabla_\lambda \psi_m}{\psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n} \end{aligned}\]

In order to simplify this, we differentiate the orthogonality relation \(\braket{\psi_m}{\psi_n} = \delta_{mn}\), which ends up telling us that \(\braket{\nabla_\lambda \psi_m}{\psi_n} = - \braket{\psi_m}{\nabla_\lambda \psi_n}\):

\[\begin{aligned} 0 = \nabla_\lambda \delta_{mn} = \nabla_\lambda \braket{\psi_m}{\psi_n} = \braket{\nabla_\lambda \psi_m}{\psi_n} + \braket{\psi_m}{\nabla_\lambda \psi_n} \end{aligned}\]

Using this result to replace \(\braket{\nabla_\lambda \psi_m}{\psi_n}\) in the previous equation leads to:

\[\begin{aligned} \delta_{mn} \nabla_\lambda E_n &= (E_m - E_n) \braket{\psi_m}{\nabla_\lambda \psi_n} + \matrixel{\psi_m}{\nabla_\lambda \hat{H}}{\psi_n} \end{aligned}\]

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:

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

References

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

© Marcus R.A. Newman, a.k.a. "Prefetch". Available under CC BY-SA 4.0.
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