We consider a simple quantum model of atom-molecule conversion where bosonic
atoms can combine into diatomic molecules and vice versa. The many-particle
system can be expressed in terms of the generators a deformed $SU(2)$ algebra,
and the mean-field dynamics takes place on a deformed version of the Bloch
sphere, a teardrop shaped surface with a cusp singularity. We analyse the
mean-field and many-particle correspondence, which shows typical features of
quantum-classical correspondence. We demonstrate that semiclassical methods can
be employed to recover full many-particle features from the mean-field
description in cold atom systems with atom-molecule conversion, and derive an
analytic expression for the many-particle density of states in the limit of
large particle numbers.