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 $|{\mathrm{r}}_1-{\mathrm{r}}_2|=a$, $|{\mathrm{r}}_2-{\mathrm{r}}_3|=a$, $|{\mathrm{r}}_1-{\mathrm{r}}_3|=a$. 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 eigen-functions 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.