Hellmann–Feynman theorem ($E^{\prime }(\lambda )=⟨\psi ,H^{\prime }(\lambda )\psi ⟩$)

For a differentiable normalized eigenstate of a differentiable self-adjoint Hamiltonian, differentiate $H\psi =E\psi$ and pair with $\psi$. The terms involving ${\psi }^{\prime }$ 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.