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Journal of Physics A Mathematical Nuclear and General

Hypervirial theorems applied to the linear oscillator with velocity-dependent anharmonicity

Awele Maduemezia

Published Jun 1, 1973• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/6/6/006

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•Jan 1, 1972•0 citations
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 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 is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1/2 (omega 2 q 2 + lambda q 4 ). As a lowest-order approximation an extremely simple formula for energy levels, E i (0) = (i + 1/2)1/4(3/alpha i + alpha i ), is derived (i being the quantum number of the energy level), which covers any (lambda, i). alpha i is the real positive root of a cubic equation: y i alpha i 3 + alpha i 2 - 1 = 0, with y i = 6lambda(2i 2 + 2i + 1)/(2i+1). This formula reproduces the exact energy levels within an error of about 1% for any ( lambda ,i) (the worst case is 2% for i = 0, lambda to infinity). Systematically higher orders of this perturbation theory are developed, which contains the 'usual' perturbation theory for the limiting case of small lambda , but this perturbation theory is valid for any (i, lambda). The second-order perturbation theory reduces the errors of our lowest-order results by a factor of about 1/5 in general. Various ranges (large, intermediate, small) of (i, lambda ) are investigated and compared with the exact values obtained by the Montroll group (1975, 1978). For i = 0, 1, even the fourth-order perturbation calculation can be elaborated explicitly, which reduces the error to about 0.01% for any lambda . For small lambda it gives correct numerical coefficients up to lambda 4 terms, as it should.
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.
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.