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Journal of Optics B Quantum and Semiclassical Optics

Radial coherent states for Dirac hydrogen-like atom

G. E. Drăgănescu, ANTONINO MESSINA, Anna Napoli

Published Jun 10, 2002• 7 citations
Publisher

Abstract

In this paper we use an su (2) representation of the radial eigenfunction of the Dirac hydrogen-like atom and we build the Glauber coherent states and the displacement operator coherent states. We also calculate the average values of some observables corresponding to these states.
DOI
10.1088/1464-4266/4/5/302

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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.
Previous attempts to represent graphically the Schrodinger electron cloud for hydrogen-like atoms have been so successful that similar graphical representations have been looked for on the Dirac theory. With a single electron specified on Dirac's theory by a set of four wave functions instead of one, the probability density $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}={\ensuremath{\psi}}_{I}{{\ensuremath{\psi}}_{I}}^{*}+{\ensuremath{\psi}}_{2}{{\ensuremath{\psi}}_{2}}^{*}+{\ensuremath{\psi}}_{3}{{\ensuremath{\psi}}_{3}}^{*}+{\ensuremath{\psi}}_{4}{{\ensuremath{\psi}}_{4}}^{*}$ is treated by considering, as on the Schrodinger theory, the independent angular and radial factors separately. Tables and graphs are given for $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a function of $\ensuremath{\theta}$ alone. The graphs are compared with and shown to correspond closely to the space quantization of the classical electron orbits. Graphs representative of $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$ as a function of rrr alone are given and compared with the electronnucleus distances of the corresponding electron orbits. The radial and angular factors are brought together by means of a mechanical device described previously, which when photographed represents closely the electron-cloud $\ensuremath{\Psi}{\ensuremath{\Psi}}^{*}$. Photographs are given for the states 12S121^{2}S_{\frac{1}{2}}12S21​​, 22P12,322^{2}P_{\frac{1}{2}, \frac{3}{2}}22P21​,23​​, 32D32,523^{2}D_{\frac{3}{2}, \frac{5}{2}}32D23​,25​​ 42F52,724^{2}F_{\frac{5}{2}, \frac{7}{2}}42F25​,27​​, and 52G725^{2}G_{\frac{7}{2}}52G27​​, with $m=\ifmmode\pm\else\textpm\fi{}\frac{1}{2},\ifmmode\pm\else\textpm\fi{}\frac{3}{2},\ensuremath{\cdots}\ifmmode\pm\else\textpm\fi{}j$
We add a phase variable and its corresponding operator to the description of the hydrogen atom. With the help of these additions, we device operators that act as ladder operators for the radial system. The algebra defined by the commutation relations between those operators has a Casimir operator coincident with the radial Hamiltonian of the problem. The algebra happens to be the well-known su(1,1) Lie algebra, hence the phase-dependent eigenfunctions calculated within our scheme belong in a representation of that algebra, a fact that may be useful in certain applications. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem 107: 1608 -1613, 2007
We study the radial part of the Dunkl–Coulomb problem in two dimensions and show that this problem possesses the SU(1,[Formula: see text]1) symmetry. We introduce two different realizations of the su(1,[Formula: see text]1) Lie algebra and use the theory of irreducible representations to obtain the energy spectrum and eigenfunctions. For the first algebra realization, we apply the Schrödinger factorization to the radial part of the Dunkl–Coulomb problem to construct the algebra generators. In the second realization, we introduce three operators, one of them proportional to the Hamiltonian of the radial Schrödinger equation. Finally, we use the SU(1,[Formula: see text]1) Sturmian basis to construct the radial coherent states in a closed form.