The problem of a particle in the three-dimensional ring-shaped potential 17u2(2a,/ rqa:/rz sin' @ ) E~ introduced by Hartmann is transformed into the problem of a coupled pair of two-dimensional harmonic oscillators wjth inverse quadratic potentials by using a nonbijective canonical transformation, viz., the Kustaanheimo-Stiefel transformation. The energy E of the levels for the ring-shaped potential is obtained in a straightforward way from the one for the two-dimensional potential -( 4~p *
Recent works on the hydrogen-oscillator connection are extended to cover in a systematic (and easily computarizable) way the problem of the expansion of an R3 hydrogen wave function in terms of R4 oscillator wave functions. Passage formulas from oscillator to hydrogen wave functions are obtained in six cases resulting from the combination of the following coordinate systems: spherical and parabolic coordinate systems for the hydrogen atom in three dimensions, and Cartesian, double polar, and hyperspherical coordinate systems for the isotropic harmonic oscillator in four dimensions. These coordinate systems are particularly useful in physical applications (e.g., Zeeman and Stark effects for hydrogenlike ions and coherent state approaches to the Coulomb problem).
We show that our recently published Arithmetic Model of the genetic code based on Gödel Encoding is robust against symmetry transformations; specially Rumer's one U«G, A«C and constitutes a link between the degeneracy structure and the chemical composition of the 20 canonical amino acids. As a result, several remarkable atomic patterns involving hydrogen, carbon, nucleon and atom numbers are derived. This study has no obvious practical application(s) but could, we hope, add some new knowledge concerning the physico-mathematical structure of the genetic code.
Abstract. A building principle working for both atoms and monoatomic ions is proposed in this Letter. This principle relies on the q-deformed "chain" SO(4) > SO(3) q .
The connection between the three-dimensional hydrogen atom and a four-dimensional harmonic oscillator obtained in previous works, from a hybridization of the infinitesimal Pauli approach to the hydrogen system with the Schwinger approach to spherical and hyperbolical angular momenta, is worked out in the case of the zero-energy point of the hydrogen atom. This leads to the equivalence of the threedimensional hydrogen problem with a four-dimensional free-particle problem involving a constraint condition. For completeness, the latter result is also derived by using the Kustaanheimo-Stiefel transformation introduced in celestial mechanics. Finally, it is shown how the Lie algebra of SO(4,2) quite naturally arises for the whole spectrum (discrete plus continuum plus zero-energy point) of the three-dimensional hydrogen atom from the introduction of the constraint condition into the Lie algebra of Sp(8, IR) associated with the four-dimensional harmonic oscillator.
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