Hypervirial and Hellmann-Feynman Theorems Applied to Anharmonic Oscillators

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Abstract
A set of hypervirial theorems plus the Hellman-Feynman theorem are applied to a general anharmonic oscillator. The exact energy and expection values of powers of the position coordinate are expanded in a power series of the anharmonic coupling constant and it is shown that use of the above theorems enables one to express each term in these expansions solely in terms of the unperturbed energy and known constants. This procedure thus eliminates the usual tedious calculations of sums over intermediate states of products of matrix elements which arise in nth order Rayleigh-Schrödinger perturbation theory.

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