  In this paper, we define four different notions of controllability of
physical interest for multilevel quantum mechanical systems. These notions
involve the possibility of driving the evolution operator as well as the state
of the system. We establish the connections among these different notions as
well as methods to verify controllability.
  The paper also contains results on the relation between the controllability
in arbitrary small time of a system varying on a compact transformation Lie
group and the corresponding system on the associated homogeneous space. As an
application, we prove that, for the system of two interacting spin 1/2
particles, not every state transfer can be obtained in arbitrary small time.
