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Journal of Physics A Mathematical and General

A hypervirial JWKB calculation

John P. Killingbeck

Published Feb 21, 1987• 28 citations
Publisher

Abstract

By a combination of mathematical analysis and microcomputer experiment it is shown that the hypervirial perturbation method can yield, not only the accurate quantum mechanical values, but also the zeroth-order JWKB values of E and (x 2 ) for the case of a harmonic oscillator with a lambda x 4 perturbation.
DOI
10.1088/0305-4470/20/3/022

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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.
The detailed transition from classical mechanics to quantum mechanics, particularly in regard to the question of quantum corrections to the semiclassical quantization, can be completely understood for the x4{x}^{4}x4 perturbed harmonic oscillator. This is a consequence of five facts: (i) The quantum perturbation series for the energy can be solved order by order in closed form. (ii) The quantum series rearranges directly into the classical canonical perturbation series plus quantum corrections proportional to successively higher powers of ${\ensuremath{\Elzxh}}^{2}$. (iii) The classical series and the subseries for the quantum corrections converge (subseries by subseries) to the terms of the Jeffreys-Wentzel-Kramers-Brillouin (JWKB) expansion for the energy (to all orders of \ensuremath{\Elzxh}). (iv) The (all-order) JWKB expansion, conversely, yields the quantum perturbation series and consequently uniquely determines the exact eigenvalue. (v) The ratio of the quantum energy to the classical energy at action (n+1/2)\ensuremath{\Elzxh} differs from 1 asymptotically by 1/[9\ensuremath{\pi}(n+1/2)2{+1/2)}^{2}+1/2)2], independent of both \ensuremath{\Elzxh} and the perturbation coupling constant, as the harmonic-oscillator quantum number n approaches \ensuremath{\infty}. An interesting additional feature is the mechanism by which the convergence of the classical perturbation series and of the series for the quantum corrections gets turned into the divergence of the quantum perturbation series.
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.
The accuracy of the renormalized perturbation series for anharmonic oscillators and other simple quantum-mechanical models increases with the quantum number when the adjustable parameter is determined by a simple variational criterion. In the semiclassical limit of quantum mechanics such an expansion converges towards the result of the Jeffreys–Wentzel–Kramers–Brillouin method to all orders. The hypervirial and Hellmann–Feynman theorems facilitate the derivation of the semiclassical series from Rayleigh–Schrödinger perturbation theory.
This chapter gives a brief sketch of some parts of Rayleigh-Schrödinger perturbation theory, the author have written in detail about perturbation theory elsewhere and concentrate here on a few ideas which are directly useful in numerical work on a microcomputer. The intellectual puzzle of how to reduce everything to an energy calculation gets really interesting if we want to calculate a local quantity, e.g. the value of the (normalised) wavefunction at the origin. The renormalised series trick seems to give the results to greater accuracy and more easily. The program makes an essential use of arrays, but since the array subscripts on the Pet cannot be negative the subscripts have to be changed from their values in the theoretical equations. In many-body theory, then, the sum-overstates formalism can be useful for giving a semi-quantitative account of various important effects, and the so-called diagrammatic perturbation theory used in atomic theory leans heavily on that formalism.