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Journal of Physics B Atomic and Molecular Physics

Transition moment integrals and the hypervirial theorem. I

Charles Alfred Coulson, John C. Nash

Published May 1, 1972• 13 citations
Publisher

Abstract

Hypervirial theorems are used to derive a differential equation for the transition-moment generating function (A mod exp(itx) mod B) associated with eigenstates A and B in a one dimensional wave- mechanical system. Explicit solution of the equation is possible for the harmonic oscillator, and leads to a simple expression for the overlap integral between two similar oscillators on different centres.

DOI
10.1088/0022-3700/5/5/012

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Hypervirial theorems are used to derive a differential equation for the transition-moment generating function (A mod exp(itx) mod B) associated with eigenstates A and B in a one dimensional wave- mechanical system. Explicit solution of the equation is possible for the harmonic oscillator, and leads to a simple expression for the overlap integral between two similar oscillators on different centres.
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 manner in which probability amplitudes of paths sum up to form wave functions of a harmonic oscillator, as well as other, simple 1-dimensional problems, is described. Using known, closed-form, path-based propagators for each problem, an integral expression is written that describes the wave function. This expression conventionally takes the form of an integral over initial locations of a particle, but it is re-expressed here in terms of a characteristic momentum associated with motion between the endpoints of a path. In this manner, the resulting expression can be analyzed using a generalization of stationary-phase analysis, leading to distributions of paths that exactly describe each eigenstate. These distributions are valid for all travel times, but when evaluated for long times they turn out to be real, non-negative functions of the characteristic momentum. For the harmonic oscillator in particular, a somewhat broad distribution is found, peaked at value of momentum that corresponds to a classical energy which in turn equals the energy eigenvalue for the state being described.
Abstract We discuss the application of operator methods to one-dimensional systems other than the harmonic oscillator. In Section 3.1 the method of ‘oneway factorization’ is used to obtain generalized quantum harmonic potentials. In Section 3.2 factorization and shift operators for an arbitrary one-dimensional potential are considered, and a classification is presented of the problems that can be treated by this method [1]. The use of the hypervirial theorem in the calculation of expectation values for one-dimensional systems is discussed in Section 3.3. In Sections 3.4–3.7 four examples are used to illustrate the applications of the preceding material. In these examples we consider in some detail how shift operators for energy can be derived and we also use these shift operators to calculate wavefunctions. Supersymmetric quantum mechanics and shape invariance are discussed in Sections 3.8 and 3.9.
Multipole moments for the harmonic oscillator as well as for other familiar quantum systems can be obtained without knowledge of the wavefunction by means of simple recursion relations. The third-order differential equation satisfied by the probability density is first obtained and from it, the recursion relation for the moments is directly deduced. A general and presumably novel formula giving all the multipole moments is obtained by solving the finite difference equation generated by the recursion relation.