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.
Highlights
Few-body systems and problems [1,2] are ubiquitous in almost all fields of physics; they arise, for example, in particle physics, nuclear physics, atomic physics, condensedmatter physics, and so on
This was first addressed by Hylleraas [3] by variational methods and pushed recently to extraordinary precision [4]. Another quite similar case is the H− ion [5], which is remarkable for its very weakly bound ground state and is of astrophysical interest [6]. Another example is found in semiconductor physics, where the trion, i.e., a bound state of an exciton and an electron [7], is observed through its absorption or emission spectrum [8,9]
Three-body systems arise because they may have their own intrinsic interest, such as the well-known Efimov trimers [2,10], with their remarkable scaling properties, which have been the subject of much recent activity in nuclear physics and in cold-atom physics
Summary
Few-body systems and problems [1,2] are ubiquitous in almost all fields of physics; they arise, for example, in particle physics, nuclear physics, atomic physics, condensedmatter physics, and so on. Problem: it makes much more sense to use the already known solution of the two-body problem rather than start again from the beginning, as if the two-body problem had not been solved This feature is so attractive that it is worthwhile to explore whether it can be extended with the same advantages to the case of a general interaction potential, getting rid of the simplified contact interaction suited to cold gases. When we come to explicit use, we consider the specific case of the 3D Coulomb potential, which is appropriate to the case of the helium ground state we consider explicitly, and to the case of the trion, which we have mainly in mind This Coulomb potential case turns out to be convenient since there is a simple analytic expression found by Schwinger [22] for the T matrix, which sums up the solution of the two-body Coulomb problem. The translation to any other physical situations of interest is obvious
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