Extensions of the Feynman–Hellman theorem and applications

Abstract

Epstein’s [Am. J. Phys. 22, 613 (1954)] off-diagonal and higher-order extensions of the Feynman–Hellmann theorem, obtained by using the basic technique of parameter differentiation under the integral sign, are further pursued. Epstein’s rederivation of the Rayleigh–Schrödinger perturbation expansion is also extended to include the degenerate case. The same approach is also used to obtain the Lennard-Jones–Brillouin–Wigner perturbation theory. The quantum virial theorem and its off-diagonal generalization is deduced and its application is illustrated by taking the example of the linear harmonic oscillator. The semiclassical expression for the kinetic energy is obtained directly from the quantization condition.