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Physical Review A

Classical mnemonic approach for obtaining hydrogenic expectation values of rP

Lorenzo J. Curtis

Published Jan 1, 1991• 11 citations
Publisher

Abstract

A recent paper by Drake and Swainson [Phys. Rev. A 42, 1123 (1990)] presents a recursion relation that generates a set of coefficients needed to obtain general closed-form expressions for hydrogenic expectation values for ${\mathit{r}}^{\mathit{P}}$. By combining this recursion relation with an earlier semiclassical formulation of this calculation [J. Phys. B 14, 1373 (1981)], it is shown that the correct expectation values can also be obtained by a simple substitution of prescribed angular momentum operators into the classical Kepler formula, which provides a transparent connection to the correspondence limit.

DOI
10.1103/physreva.43.568

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A recent paper by Drake and Swainson [Phys. Rev. A 42, 1123 (1990)] presents a recursion relation that generates a set of coefficients needed to obtain general closed-form expressions for hydrogenic expectation values for rP{\mathit{r}}^{\mathit{P}}rP. By combining this recursion relation with an earlier semiclassical formulation of this calculation [J. Phys. B 14, 1373 (1981)], it is shown that the correct expectation values can also be obtained by a simple substitution of prescribed angular momentum operators into the classical Kepler formula, which provides a transparent connection to the correspondence limit.
The generalized and canonical momentum operators in spherical polar coordinates are derived and discussed, and the radial operators applied to the evaluation of the Heisenberg uncertainty relation for bound states of hydrogenic atoms. The two Pasternack relations, which connect the expectation values 〈rk〉 for values of k differing by an integer, are examined, as well as a generalization of the first relation, and the advantages which they hold for computational purposes discussed. An earlier calculation by the present author, of the electron impact ionization cross section for hydrogen atoms in the 2S metastable state, is compared with other experimental data from the literature. The calculation of the wavelengths of transitions between hydrogen-like levels in highly excited ions possessing a few additional, tightly bound, electrons is described and compared with experimental data.
The radial functions for the discrete levels of Dirac's relativistic hydrogenic atom are expressed in terms of the generalized Laguerre polynomials which have known properties. This leads to an easy method of evaluating the normalization constants and the average values of rq{r}^{q}rq.
Heisenberg's form of the correspondence principle is applied to the matrix elements of tensor operators. A correspondence limit of the Wigner-Eckart theorem is obtained. It is shown that the Clebsch-Gordan coefficient tends in this limit to the product of a reduced rotation matrix element with a simple multiplicative factor. A similar, but not identical, expression has long been known as a formal limit obtaining when two of the three angular momenta involved become large. It is characteristic of correspondence principle expressions that the initial and final quantal states must be represented by some 'mean' classical orbit, the choice of which remains ambiguous. Here recursion relations are used to arrive at the optimal choice. In this way, a correspondence principle approximation to the Clebsch-Gordan coefficient is obtained which may be used with success far from the limit to which it nominally refers. This approximation is tested numerically against both exact and semiclassical computations.
Formulae for the calculation of expectation values of arbitrary powers of r for hydrogen-like wavefunctions are derived both in quantum mechanics and within the framework of the WKB approximation. The quantum to classical correspondence is then investigated in the limit n to infinity for finite n r , as well as for finite l. For negative powers and finite l, an asymptotically remaining deviation of the semiclassical formulae from the quantum expressions is found. A modification in the eccentricity of the classical orbit is introduced which corrects the asymptotic behaviour of the WKB approximation.