= Hellmann–Feynman theorem
{c}
{title2=$E'(\lambda)=\langle\psi,H'(\lambda)\psi\rangle$}
{wiki}
For a differentiable normalized <eigenstate> of a differentiable <self-adjoint> <Hamiltonian>, differentiate $H\psi=E\psi$ and pair with $\psi$. The terms involving $\psi'$ cancel because $H$ is <self-adjoint>, leaving the stated formula. It turns energy <derivatives> into observable expectations. A smoothly chosen eigenbranch and suitable operator-domain regularity are required, particularly at degeneracies or crossings.
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