Hellmann-Feynman theorem

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The Hellmann-Feynman theorem is a theorem in quantum mechanics, which relates the energy eigenvalues of a time-independent Hamiltonian operator to the parameters composing it. In general, the theorem states that,

\frac{\partial {E_n}}{\partial {\lambda}}=\int{\psi_n^*\frac{\partial{\hat{H}}}{\partial{\lambda}}\psi_nd\tau}

where \hat{H} is the parameterized Hamiltonian operator,

En is the nth Hamiltonian eigenvalue,

ψn is the nth Hamiltonian eigenvector,

λ is the parameter of interest,

and dτ implies an integration over the complete domain of the eigenvectors.

[edit] The proof

The proof is actually very easy. Using the Dirac's bra-ket notation, we can write

\frac{\partial E}{\partial\lambda} = \frac{\partial}{\partial\lambda}\langle\psi|\hat{H}|\psi\rangle
= \langle\frac{\partial\psi}{\partial\lambda}|\hat{H}|\psi\rangle + \langle\psi|\frac{\partial\hat{H}}{\partial\lambda}|\psi\rangle + \langle\psi|\hat{H}|\frac{\partial\psi}{\partial\lambda}\rangle
= E \langle\frac{\partial\psi}{\partial\lambda}|\psi\rangle + \langle\psi|\frac{\partial\hat{H}}{\partial\lambda}|\psi\rangle + E \langle\psi|\frac{\partial\psi}{\partial\lambda}\rangle
= E \frac{\partial}{\partial\lambda}\langle\psi|\psi\rangle + \langle\psi|\frac{\partial\hat{H}}{\partial\lambda}|\psi\rangle
= \langle\psi|\frac{\partial\hat{H}}{\partial\lambda}|\psi\rangle.

[edit] See also

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