Quantum Mechanical Three-Body Problem. II

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 Ψ has a special analytic form, ${Ψ}={ψ}({r}₁₂, {{ρ}}₃)+{ψ}({r}₁₃, {{ρ}}₂)+{ψ}({r}₂₃, {{ρ}}₁)$, where ${r}₁₂$=${r}₁$-${r}₂$, ${{ρ}}₃={r}₃{-}1/2({r}₁+{r}₂)$ and ${r}₁₃$, ${{ρ}}₂$ and ${r}₂₃$, ${{ρ}}₁$ are defined analogously. The Schr\"odinger equation for the system can then be written as an integral equation for ${φ}(k, {κ})$, the Fourier transform of ${ψ}$. We expand this in Legendre polynomials, ${φ}(k, {κ})={Σ}{{∞}}{l=0}{{φ}}ₗ(k, {κ}){P}ₗ(cos{γ}),$ and this yields a set of coupled integral equations for the ${{φ}}ₗ(k, {κ})$. These can be truncated and to a good approximation one can neglect all ${{φ}}ₗ$ except ${{φ}}₀$, thereby reducing the problem to a single integral equation for a function of two variables.We propose an iterative scheme for solving this equation for the ground-state eigenfunction, and suggest a simple but accurate nonvariational method for deriving the energy eigenvalue therefrom. We test this proposed solution by working it out in detail for the case of exponential interparticle potentials. The results for the eigenvalue compare favorably with variational calculations by other authors. Finally, we discuss the accuracy of the approximations and the possible sources of error in the wave function.