  We consider some basic problems associated with quantum mechanics of systems
having a time-dependent Hilbert space. We provide a consistent treatment of
these systems and address the possibility of describing them in terms of a
time-independent Hilbert space. We show that in general the Hamiltonian
operator does not represent an observable of the system even if it is a
self-adjoint operator. This is related to a hidden geometric aspect of quantum
mechanics arising from the presence of an operator-valued gauge potential. We
also offer a careful treatment of quantum systems whose Hilbert space is
obtained by endowing a time-independent vector space with a time-dependent
inner product.
