Improved Method for Quantum Mechanical Three-Body Problems. I. Attractive Potentials

Abstract

The quantum-mechanical ground-state problem for three identical particles bound by attractive interparticle potentials is discussed. For this problem it has previously been shown that it is advantageous to write the wavefunction in a special functional form, for which an integral equation which is equivalent to the Schrödinger equation was derived. In this paper a new method for solving this equation is presented. The method involves an expansion in a set of ``effective'' two-body functions; these are the eigenfunctions of a two-body problem with a potential of the same shape as the interparticle potential in the three-body problem, but of enhanced strength. The integral equation then becomes a set of coupled equations for certain functions fl(k), the ``coefficients'' in this expansion. By choosing the effective two-body strength properly, one can optimize the convergence of this expansion so that a good approximation is got by retaining only the lowest (l̂ = 0) term. The resultant single integral equation for f0(k) can then be solved by an approximate method. As a test and check, the one-dimensional three-body problem with δ-function interactions is treated and the results are compared with the known exact one. It is found that the first approximation in the method, which is almost trivial to apply, yields an eigenvalue and eigenfunction accurate to a few percent. The method has also been applied to the three-dimensional problem with exponential interparticle potentials, and a comparison of the results with the best numerical calculations is given.