Quantum Mechanical Three-Body Problem. II

Abstract

We present a method for treating the following quantum mechanical three-body problem: to find the ground-state eigenvalue and eigenfunction for a system of three identical particles between any pair of which there is an attractive central force. An essential point of the method is to assume the wave function $\Psi$ has a special analytic form, $\Psi =\psi ({\mathrm{r}}_{12}, {\rho }_3)+\psi ({\mathrm{r}}_{13}, {\rho }_2)+\psi ({\mathrm{r}}_{23}, {\rho }_1)$, where ${\mathbf{r}}_{12}$=${\mathbf{r}}_1$-${\mathbf{r}}_2$, ${\rho }_3={\mathrm{r}}_3-\frac{1}{2}({\mathrm{r}}_1+{\mathrm{r}}_2)$ and ${\mathbf{r}}_{13}$, ${\rho }_2$ and ${\mathbf{r}}_{23}$, ${\rho }_1$ are defined analogously. The Schrödinger equation for the system can then be written as an integral equation for $\phi (\mathrm{k}, \kappa )$, the Fourier transform of $\psi$. We expand this in Legendre polynomials, $\phi (\mathrm{k}, \kappa )=\Sigma \stackrel{\infty }{l=0\mathrm{}}{\phi }_l(k, \kappa )P_l\left(cos\gamma \right),$ and this yields a set of coupled integral equations for the ${\phi }_l(k, \kappa )$. These can be truncated and to a good approximation one can neglect all ${\phi }_l$ except ${\phi }_0$, thereby reducing the problem to a single integral equation for a function of two variables.