  The important role of geometric phases in searches for a permanent electric
dipole moment of the neutron, using Ramsey separated oscillatory field nuclear
magnetic resonance, was first noted by Commins and investigated in detail by
Pendlebury et al. Their analysis was based on the Bloch equations. In
subsequent work using the spin density matrix Lamoreaux and Golub showed the
relation between the frequency shifts and the correlation functions of the
fields seen by trapped particles in general fields (Redfield theory). More
recently we presented a solution of the Schr\"odinger equation for spin-$1/2$
particles in circular cylindrical traps with smooth walls and exposed to
arbitrary fields [Steyerl et al.] Here we extend this work to show how the
Redfield theory follows directly from the Schr\"odinger equation solution. This
serves to highlight the conditions of validity of the Redfield theory, a
subject of considerable discussion in the literature [e.g., Nicholas et al.]
Our results can be applied where the Redfield result no longer holds, such as
observation times on the order of or shorter than the correlation time and
non-stochastic systems and thus we can illustrate the transient spin dynamics,
i.e. the gradual development of the shift with increasing time subsequent to
the start of the free precession. We consider systems with rough, diffuse
reflecting walls, cylindrical trap geometry with arbitrary cross section, and
field perturbations that do not, in the frame of the moving particles, average
to zero in time. We show by direct, detailed, calculation the agreement of the
results from the Schr\"odinger equation with the Redfield theory for the cases
of a rectangular cell with specular walls and of a circular cell with diffuse
reflecting walls.
