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arXiv:1005.2792

Conformal Mapping of Relativistic Quantum Bound Systems to Eliminate Potential Fields

Ducharme, Robert J.

Published May 17, 2010• 0 citations
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Abstract

In two recent papers, an isometric conformal transformation has been introduced that eliminates potential interaction terms from the Schrödinger equation for central potential problems. The method has been demonstrated for both the hydrogen atom and three-dimensional harmonic oscillator. Here, it is shown that the same transformation technique can also be applied to central potential problems formulated using the Klein-Gordon equation.

DOI
10.48550/arxiv.1005.2792

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In two recent papers, an isometric conformal transformation has been introduced that eliminates potential interaction terms from the Schrödinger equation for central potential problems. The method has been demonstrated for both the hydrogen atom and three-dimensional harmonic oscillator. Here, it is shown that the same transformation technique can also be applied to central potential problems formulated using the Klein-Gordon equation.
In a recent paper, it has been shown the Schrödinger equation for the three-dimensional harmonic oscillator can be simplified through the use of an isometric conformal transformation. Here, it is demonstrated that the same transformation technique is also applicable to the Schrödinger equation for the hydrogen atom. This approach has two interesting features. Firstly, it eliminates potential fields from the Schrödinger equation. The Coulomb and harmonic binding terms are instead represented as imaginary parts of complex time. Secondly, the method leads to a general relationship between potential energy and ground state energy that encompasses both the hydrogen atom and the harmonic oscillator as special cases.
Quantum relativistic solutions for a particle in a one-dimensional Morse potential are presented using methods previously proposed and applied to a particle in a harmonic oscillator. The methods lead to both numerical and analytical solutions, with the latter allowing smooth variation of the system parameters from non-relativistic to ultrarelativistic limits. Analytical expressions for the energy levels and wavefunctions are obtained, as solutions to a Schrödinger-type equation, including relativistic effects through a state dependent rescaled mass. The eigenstates of the Morse potential exhibit suitable and smooth behavior, and approach the corresponding harmonic oscillator solutions as the depth of the Morse potential well increases, as expected. A comparison is also presented between the relativistic harmonic oscillator obtained with this method, and the so-called Klein-Gordon oscillator.
Quantization of the conformal Kepler problem is defined and studied in order that the quantized system, which will be referred to as a conformal hydrogen atom, may associate the harmonic oscillator with the hydrogen atom. The conformal hydrogen atom shares with the harmonic oscillator the eigenspaces of negative energies. The four-dimensional conformal hydrogen atom reduces to the three-dimensional ordinary hydrogen atom. The symmetry group SO(4) of the hydrogen atom is brought out from the symmetry subgroup of the harmonic oscillator. The conformal hydrogen atom gives an example of those quantum systems of which the configuration spaces are curved Riemannian spaces with nonconstant scalar curvatures and of which the Hamiltonian operators depend on the scalar curvatures.
The use of complex variables and conformal mapping have long been useful tools in many branches of physics. However, these methods have not been widely applied in mechanics and elementary quantum mechanics. In this paper, it is shown that there are many useful applications of complex variables and conformal mapping in these subjects. It will be proven that central force problems in the plane have conformal duals. The inverse fifth power law force is self-dual in all dimensions. These results are extended to noncentral forces. Many of these results also apply to the quantum mechanics of a particle. For central force problems and certain noncentral force problems in the plane the Schrödinger equation preserves its form under conformal mapping and the inverse fifth power law force is again self-dual in all dimensions. The methods may also be applied in the presence of magnetic fields.