  The existence of a maximal acceleration for massive objects was conjectured
by Caianiello 30 years ago based on the Heisenberg uncertainty relations. Many
consequences of this hypothesis have been studied, but until now, there has
been no evidence that boundedness of the acceleration may lead to quantum
behavior. In previous research, we predicted the existence of a universal
maximal acceleration and developed a new dynamics for which all admissible
solutions have an acceleration bounded by the maximal one. Based on W.
K\"{u}ndig's experiment, as reanalyzed by Kholmetskii et al, we estimated its
value to be of the order $10^{19}m/s^2$.
  We present here a solution of our dynamical equation for a classical
hydrogen-like atom and show that this dynamics leads to some aspects of quantum
behavior. We show that the position of an electron in a hydrogen-like atom can
be described only probabilistically. We also show that in this model, the
notion of "center of mass" must be modified. This modification supports the
non-existence of a magnetic moment in the atom and explains the relevance of
the conformal group in the quantum region.
