The Second Virial Coefficient for Quantum-Mechanical Sticky Spheres

Abstract

This paper gives a non-rigorous demonstration of a result announced in poster form at the ‘On three levels’ conference in July 1993; it may be useful since the rigorous proof given in [1] uses probabilistic techniques which are not familiar to all physicists. The result concerns the second virial coefficient B _D,a for a quantum-mechanical system of hard spheres (a true mathematician would perhaps call them ‘hard balls’) with an attractive two-body interaction, typically a square well of width a. More precisely, the interaction between any pair of molecules with relative displacement r is taken to be {x_n} = {f_n}({x_n} - 1) where the colon indicates a definition, |r| denotes the magnitude of the vector r, ε denotes the depth of the well, a its width, D the diameter of the hard spheres, and u is the ‘unit square well’ potential defined by

$$u(x): = - {1_{(0.1]}}(x) = \left\{ {\matrix{ \hfill { - 1} & \hfill {{\rm{if}}} \cr \hfill 0 & \hfill {{\rm{if}}} \cr } } \right.{\rm{ }}\matrix{ {0 < x \le 1,} \hfill \cr {1 < x.} \hfill \cr } $$

((2))

The potential (1) has been much used as a model of real fluids [2,3], colloids [4] and some biological systems [5].