Boundary conditions on wavefunctions for three bodies at singular configurations

Abstract

Singular configurations have long been out of consideration in the study of many-particle systems. In this paper we show that a group theoretical method can provide boundary conditions on wavefunctions at singular configurations of many particles. The consequences of this method are qualitatively stated as follows. (i) If the projection of angular momentum on an axis does not vanish, the particles will not be aligned on the axis, i.e. the wavefunction vanishes at collinear configurations along the axis. (ii) If the total angular momentum does not vanish, the simultaneous collision at a point will not take place, i.e. the wavefunction vanishes at the configuration of simultaneous collision. Furthermore, on the assumption of analyticity of the wavefunction at singular configurations, the behaviour of the wavefunction around the singular configurations shows that (a) the larger the projection of angular momentum on an axis becomes, the less the particles are likely to be aligned on the line, and that (b) the more the total angular momentum grows, the less likely the particles are to collide at a point. The proof is carried out for three-body systems by transforming the power series expansion of a wavefunction into a Fourier series expansion in terms of the angular momentum eigenfunctions, in both cases of collinear configurations and of the triple collision, without reference to the Hamiltonian operator. These results are described quantitatively in terms of the angular momentum quantum numbers in propositions 1 and 3 in the text.