Hellmann-Feynman theorem

The Hellmann-Feynman theorem relates the derivative of the eigenvalues of a Hamiltonian which depends smoothly on a parameter $\lambda$ to the expectation of the derivative of the Hamiltonian $H_\lambda$ .

\begin{align*} \frac{d E_\lambda}{d \lambda} &= \left\langle \psi_\lambda \left| \frac{dH_\lambda}{d\lambda} \right| \psi_\lambda \right\rangle. \end{align*}

The theorem is easy to prove but annoying to type. Prove it yourself.