• Open Access

Three-Body Coulomb Problem

R. Combescot
Phys. Rev. X 7, 041035 – Published 15 November 2017

Abstract

We present a general approach for the solution of the three-body problem for a general interaction and apply it to the case of the Coulomb interaction. This approach is exact, simple, and fast. It makes use of integral equations derived from the consideration of the scattering properties of the system. In particular, this makes full use of the solution of the two-body problem, the interaction appearing only through the corresponding known T matrix. In the case of the Coulomb potential, we make use of a very convenient expression for the T matrix obtained by Schwinger. As a check, we apply this approach to the well-known problem of the helium atom ground state and obtain a perfect numerical agreement with the known result for the ground-state energy. The wave function is directly obtained from the corresponding solution. We expect our method to be, in particular, quite useful for the trion problem in semiconductors.

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  • Received 23 May 2017

DOI:https://doi.org/10.1103/PhysRevX.7.041035

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalNuclear PhysicsGeneral Physics

Authors & Affiliations

R. Combescot

  • Laboratoire de Physique Statistique, Ecole Normale Supérieure, PSL Research University, UPMC Paris 06, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France

Popular Summary

Among the few-body problems, which describe the dynamics of several bodies under mutual forces, the simplest ones are those involving just two objects, such as the motion of Earth around the Sun under Newtonian gravitational forces. These are easy to solve because they can be reduced to a one-body problem by taking the center of mass of the two bodies as fixed. In general, writing the equations for the problem is easy, but solving them is quite difficult—all the more since the number of variables increases rapidly with the number of objects considered. And this is much more difficult when quantum mechanics, rather than classical mechanics, must be used. Nevertheless, the one-body problem can be systematically solved because, when appropriate mathematical answers cannot be found, computers can be used to produce satisfactory numerical results. When dealing with the three-body problem, one usually starts from scratch, as if nothing is known about the solution to the two-body problem. We show how to make use of the solution to the two-body problem to more easily solve the three-body problem.

This approach leads to a set of integral equations obtained from the consideration of the scattering properties of the three-body system. We use this method specifically to find the ground-state energy of the helium atom, and we obtain perfect numerical agreement with the known result. The electronic wave function is also found from the corresponding solution.

In the near future, we expect to apply our approach to other three-body problems of interest, such as two electrons and one hole (forming a “trion”) in semiconductors. We will also look for a possible generalization to the four-body problem.

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Vol. 7, Iss. 4 — October - December 2017

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