×

Semiclassical modeling of the states and spectra of para helium singlets with an algebraic formula for excited states. (English) Zbl 0637.65128

This paper presents a semiclassical model for the spectra of Para Helium. For the ground state two electrons are considered as paired and the position at which their orbital motions are equidistant from the nucleus at some time T, r, is such that the centripetal and electrostatic forces balance. In addition at this time T it is assumed that the speed of each electron satisfies the Bohr relationship \(mvr=nh/2\pi\) \(n=1,2,3... \). The pairing of the electrons is assumed to give rise to a generalized Coulombic force \(F_{12}=e^ 2/(2r)^{2+\alpha}.\)
These assumptions lead to two nonlinear simultaneous equations for \(\alpha\) and r which are solved using Newton’s method. This model provides an estimate for energy. This is improved by considering r to be the average distance of an electron from the nucleus and for a distance of 2r assuming that the velocity of any electron is zero. This is the bounding constraint. A calculation of the energy for \(v=0\), a radius of 2r for one electron and a new value of \(\alpha\) from a least squares relation between r and \(\alpha\), is given. This turns out to be a very accurate estimate. Additional reasonable semiclassical assumptions give excited state values with surprising accuracy and algebraic formulae for excited state estimates.
Reviewer: B.Burrows

MSC:

65Z05 Applications to the sciences
81V45 Atomic physics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q99 Partial differential equations of mathematical physics and other areas of application
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pearson, R. G., Semiclassical model for atoms, (Proc. Nat. Acad. Sci. U.S.A., 78 (1981)), 4002
[2] Leopold, J. G.; Percival, I. C., The semiclassical two-electron atom and the old quantum theory, J. Phys. B, 13, 1037 (1980)
[3] van der Merwe, P.du T., Semiclassical theory of the helium atomic spectrum, J. Chem. Phys., 81, 5976 (1984)
[4] Greenspan, D., Deterministic modeling and computations for the states of two-electron systems with an explicit formula for the excited states of helium, J. Chem. Phys., 84, 300 (1986)
[5] Green, M. B., Unification of forces and particles in superstring theories, Nature, 314, 409 (1985)
[6] Greenspan, D., A mathematical curiosity in estimating the radius of first ring electrons of an arbitrary atom, J. Comput. Appl. Math., 15, 103 (1986) · Zbl 0589.65040
[7] Greenspan, D., An elementary algebraic method for approximating average radii of first and second ring electrons, J. Comput. Appl. Math., 16, 117 (1986) · Zbl 0611.65036
[8] French, A. P.; Taylor, E. F., An Introduction to Quantum Physics (1978), Norton: Norton New York
[9] Landau, L. D.; Lifschitz, E. M., Mechanics (1960), Addison-Wesley: Addison-Wesley Reading, Mass · Zbl 0997.70500
[10] Greenspan, D., Discrete Numerical Methods in Physics and Engineering (1974), Academic: Academic New York · Zbl 0288.65001
[11] Woodgate, G. K., Elementary Atomic Structure (1980), Clarendon: Clarendon Oxford
[12] Atkins, P. W., Physical Chemistry (1982), Freeman: Freeman San Francisco
[13] Litzen, U., Improved experimental values for some hydrogen-like levels of He I, Phys. Scripta, 2, 103 (1970)
[14] Martin, W. C., Energy levels and spectrum of neutral helium, J. Res. Nat. Bur. Standards A, 64, 19 (1960)
[15] (Weast, W. C., CRC Handbook of Chemistry and Physics (1985-1986), CRC Press: CRC Press Boca Raton, Fla)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.