  A d-dimensional finite quantum model system confined to a general
hypercubical geometry with linear spatial size L and ``temporal size'' 1/T (T -
temperature of the system) is considered in the spherical approximation under
periodic boundary conditions. Because of its close relation with the system of
quantum rotors it represents an effective model for studying the
low-temperature behaviour of quantum Heisenberg antiferromagnets. Close to the
zero-temperature quantum critical point the ideas of finite-size scaling are
used for studying the critical behaviour of the model. For a film geometry in
different space dimensions $\half\sigma <d<\threehalf\sigma$, where
$0<\sigma\leq2$ controls the long-ranginess of the interactions, an analysis of
the free energy and the Casimir forces is given.
