  We find a relationship between unitary transformations of the dynamics of
quantum systems with time-dependent Hamiltonians and gauge theories. In
particular, we show that the nonrelativistic dynamics of spin-$\frac12$
particles in a magnetic field $B^i (t)$ can be formulated in a natural way as
an SU(2) gauge theory, with the magnetic field $B^i(t)$ playing the role of the
gauge potential A^i. The present approach can also be applied to systems of n
levels with time-dependent potentials, U(n) being the gauge group. This
geometric interpretation provides a powerful method to find exact solutions of
the Schr\"odinger equation. The root of the present approach rests in the
Hermiticity property of the Hamiltonian operators involved. In addition, the
relationship with true gauge symmetries of n-level quantum systems is
discussed.
