Quantum-Mechanical Sum-Rule for Infinite Sums Involving the Operator∂H∂λ

The sum-rule $\Sigma \stackrel{\prime }{n}\frac{〈m\left|A\right|n〉〈n\left|\frac{\partial \Omega }{\partial \lambda }\right|m〉+〈m\left|\frac{\partial \Omega }{\partial \lambda }\right|n〉〈n\left|A\right|m〉}{{\epsilon }_n-{\epsilon }_m}=〈〉\left(m\left|\frac{\partial A}{\partial \lambda }\right|m\right)-\frac{\partial }{\partial \lambda }\left(〈m\right|A\left|m〉\right)$ is derived. In this relation $|m〉, \cdots , |n〉, \cdots$ form a complete set of orthonormal vectors, which are the eigenvectors of the Hermitian linear operator $\Omega$, with eigenvalues ${\epsilon }_m, \cdots {\epsilon }_n, \cdots$; $\lambda$ is a parameter which occurs in $\Omega$, and $A$ is an arbitrary linear operator. In many sums of this type, $\Omega$ is the Hamiltonian operator $H$. Particular examples are considered, and a differential equation, relating the mass dependence and coordinate dependence of the wave function $\psi$, is derived.