  We ask the question how the (weak) equivalence principle established in
classical gravitational physics should be reformulated and interpreted for
massive quantum objects that may also have internal degrees of freedom (dof).
This inquiry is necessary because even elementary concepts like a classical
trajectory are not well defined in quantum physics -- trajectories originating
from quantum histories become viable entities only under stringent decoherence
conditions. From this investigation we posit two logically and operationally
distinct statements of the equivalence principle for quantum systems: Version
A: The probability distribution of position for a free-falling particle is the
same as the probability distribution of a free particle, modulo a
mass-independent shift of its mean. Version B: Any two particles with the same
velocity wave-function behave identically in free fall, irrespective of their
masses. Both statements apply to all quantum states, including non-classical
ones, and also for composite particles with quantum internal dof. We also
investigate the consequences of the interaction between internal and external
dof induced by free fall. For a class of initial states, we find a dephasing
for the translational dof, namely, the suppression of the off-diagonal terms of
the density matrix, in the position basis. We also find a gravitational phase
shift in the reduced density matrix of the internal dof that does not depend on
the particle's mass. For classical states, the phase shift has a natural
classical interpretation in terms of gravitational red-shift and special
relativistic time-dilation.
