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Journal of Applied Mathematics and Physics

Angular Decomposition and Sign Solved Propagator of the 2D and 3D Helium Atom Path Integrals

Emmanouil George Thrapsaniotis

Published Jan 1, 2022• 2 citations
Publisher

Abstract

In the present paper on the one hand we apply the central limit theorem to the solution of the sign problem of a path integral of two-interacting particles in potential and give an expression for the sign solved propagator (SSP) derived from that solution and on the other hand we perform the angular decomposition of the path integrals of the 2D and 3D Helium atoms.Finally, we combine those two results and derive the SSPs of the 2D and 3D Helium atoms.

DOI
10.4236/jamp.2022.102040

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In the present paper on the one hand we apply the central limit theorem to the solution of the sign problem of a path integral of two-interacting particles in potential and give an expression for the sign solved propagator (SSP) derived from that solution and on the other hand we perform the angular decomposition of the path integrals of the 2D and 3D Helium atoms.Finally, we combine those two results and derive the SSPs of the 2D and 3D Helium atoms.
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We present a two-dimensional analysis of the two-electron problem which comes from the classical conservation theorems and from which we obtain a version of the Dirac equation for the helium atom. Approximate solutions for this equation are discussed in two different methods, although in principle it can be solved analytically. One method is variational, of the Hylleraas type, the execution of which is left for a later communication. In contrast, the other method will have a more complete treatment, in which the set of equations will be separated into its angular and radial components. Furthermore, an exact solution for the angular component will be displayed as well as an approximate solution for the radial component, valid only for the fundamental state of the atom.