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American Journal of Physics

Some Applications of Hypervirial Theorems to the Calculation of Average Values

Jean H. Epstein, Saul T. Epstein

Published Apr 1, 1962• 87 citations
Publisher

Abstract

Hypervirial theorems are used to compute various average values for the nonrelativistic hydrogenic atom, the isotropic harmonic oscillator, and the relativistic hydrogenic atom. In the latter case many new results are found.
DOI
10.1119/1.1941987

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Two applications of the quantum mechanical hypervirial theorem are discussed. This states that for any system having eigenstates A and B of Hamiltonian H, an operator W has the matrix element relation (A,(HW andminus; WH) B) = (EA andminus; EB) (A, W B) providing W is such that H is Hermitean between A and W B. The first application uses the operator W = eitx to derive a further relationship between certain matrix elements. For one dimensional systems in which the potential V(x) involves only positive powers of x, this relationship becomes a differential equation in f(t) = (A, eitx B) This differential equation has been solved analytically for the harmonic oscillator and in series for the quartic oscillator which were taken as example systems. The second application is the off-diagonal hypervirial method, which seeks to require a wave function A to satisfy a number of hypervirial relations. In particular the wave function A is forced to obey as many conditions as it has variable parameters. To accomplish this a computational algorithm was developed which allowed one of these parameters to be non-linear. The results for oscillator test systems are not encouraging. The systems tested were the quartic and sixth power oscillators with a trial basis of harmonic oscillator eigenfunctions. In general the results were not as good as those obtainable by the variational method.
An alternative derivation is given of a perturbation theoretic scheme for the anharmonic oscillator recently proposed by Swenson and Danforth (see Abstr. A64093 of 1972). The scheme uses the hypervirial and Hellmann-Feynman theorems. The calculation is extended to velocity-dependent perturbations. As a special case, the linear relativistic quantum oscillator is studied to second order, and the corresponding energies worked out. Various powers-of-momentum expectation values are also worked out in this order.