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Bulletin de la Classe des sciences

L'influence de conditions aux limites à distance finie sur la quantification de l'atome d'hydrogène

Paul Janssens, R. Van Dooren, Monique Delchambre

Published Jan 1, 1971• 0 citations
Publisher

Abstract

The quantum states of a bounded particle in a spherical symmetric potential are studied under the influence of different non classical boundary conditions, such as annuling the wave function upon a given sphere. The cases of Schrödinger, Klein-Gordon and Dirac equations are examined.
DOI
10.3406/barb.1971.61922

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The Schrödinger, Klein–Gordon, and Dirac equations for the hydrogen atom are solved in D dimensions, where D may be noninteger. For D=1 the nondegenerate ground state with infinite binding energy is obtained as a limit as D tends to 1 for the Schrödinger equation. The corresponding solution of the Klein–Gordon equation is shown to be unacceptable, contradicting previous work. Unexpectedly, no bound state solutions of the Dirac equation with D=1 are found at all. A few comments are made on the D=3 and 2 cases.
In this article, the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential domain is investigated within the framework of ordinary quantum mechanics. The Klein-FockGordon equation, which describes the dynamics of relativistic scalar particles, is of particular interest in the case of interactions with short-range potentials of the central type. The Hülten potential, on the other hand, has an exponentially decreasing form and is widely used in modeling atomic, molecular, and nuclear systems due to its approximation to the short-range Coulomb potential and its rapid decay at large distances. In this work, the radial Klein-Fock-Gordon equation is separated in spherical coordinates and the differential equation for the Hülten potential is reduced to a hypergeometric type equation. By applying the appropriate boundary conditions, the quantization condition for the energy spectrum is obtained and the analytical expression of the energy levels for the bound states is derived. The normalized form of the wave functions is expressed by means of special functions. Analysis of the obtained results shows that the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential leads to a clear manifestation of relativistic effects in the energy spectrum of the system and the structure of the wave functions. The energy spectrum depends not only on the parameters and quantum numbers of the potential, but also on relativistic factors introduced through the speed of light c. A comparison of the wave functions shows that the radial solutions obtained by the Klein-Fock-Gordon equation differ somewhat from those obtained by the Schrödinger equation. The shape and maximum of the radial wave function, which characterizes the probability of a particle being at a certain distance from the center, may change slightly compared to the non-relativistic case. These changes are especially noticeable in regions where the potential is deep and the interaction is stronger. Thus, the analytical solution of the Klein-Fock-Gordon equation in the Hülten potential field both clarifies the nature of relativistic effects and demonstrates agreement with the results of nonrelativistic quantum mechanics in terms of theoretical consistency and accuracy of the limit transition. This indicates that the model is a suitable theoretical framework for a more fundamental and accurate description of short-range interactions in molecular systems. Keywords: Klein-Fock-Gordon equation, Hülten potential, analytical solutions, energy spectrum, relativistic quantum mechanics.