Quantum-Mechanical Three-Body Problem

Abstract

We treat the quantum-mechanical problem of three spinless particles, with the boundary condition that the logarithmic derivative of the wave function be a prescribed constant at each of the three boundaries |ri-tz\ =a, |r2-rs| -a, |r x -r 3 | -a. This boundary condition is discussed; it is roughly equivalent to an interparticle potential which consists of a hard core plus a strong short-range attractive part. The eigenfunctions and eigenvalues of the system are given by the solutions of an infinite set of coupled homogeneous integral equations. The equations involve partial wave expansions in the interparticle distances and can often be truncated with good approximation by taking only a finite number of partial waves. We discuss the solution of these equations for the ground state of the system, taking relative S-waves only, for which case the infinite set of equations reduces to a single integral equation in one variable.

Few-Body Systems in Condensed Matter Physics

“…It was shown that the eigenfunctions of the Hamiltonian of a threeparticle system with pair interaction can be represented in a natural fashion as the sum of three terms, for each of which there exists a coupled set of equations. It should be noted that the natural division of the wave function in the three-particle problem into three terms had also been considered earlier by Eyges [3] and Gribov [4]. However, by using this division of the wave function in the three particle problem into three terms Faddeev obtained a set of integral equations with an unambiguous solution.…”

“…For indirect excitons, (11) can be solved only numerically for both types of potentials. One can consider a spatially separated electron-hole pair in two parallel TMDC layers at large distances D ≫ a B , where a B is the 2D Bohr radius of a dipolar exciton and use the oscillatory approximation (3). For TMDC materials the Bohr radius of the dipolar exciton is found to be in the range from 1.5Å for MoTe 2 [69] up to 3.9Å for MoS 2 [70].…”

“…iii) depending on the values of ∆ ′ the Hamiltonian (8) describes the interacting the electron-hole system via the RK potential [46] in a monolayer TMDC or Xenes. In the case of indirect excitons, when the electron and hole are located in two different monolayers with the interlayer separation D, one can consider the electron-hole interaction via the RK or Coulomb potentials (2) or use the OA (3).…”