Elementary derivation for passage times

When a quantum system undergoes unitary evolution in accordance with a prescribed Hamiltonian, there is a class of states \psi> such that, after the passage of a certain time, \psi> is transformed into a state orthogonal to itself. The shortest time for which this can occur, for a given system, is called the passage time. We provide an elementary derivation of the passage time, and demonstrate that the known lower bound, due to Fleming, is typically attained., except for special cases in which the energy spectra have particularly simple structures. It is also shown, using a geodesic argument, that the passage times for these exceptional cases are necessarily larger than the Fleming bound. The analysis is extended to passage times for initially mixed states.