  We calculate the rate of decrease of the expectation value of the transverse
component of spin for spin-1/2 particles in a magnetic field with a spatial
gradient, to determine the conditions under which a previous classical
description is valid. A density matrix treatment is required for two reasons.
The first arises because the particles initially are not in a pure state due to
thermal motion. The second reason is that each particle interacts with the
magnetic field and the other particles, with the latter taken to be via a
2-body central force. The equations for the 1-body Wigner distribution
functions are written in a general manner, and the places where quantum
mechanical effects can play a role are identified. One that may not have been
considered previously concerns the momentum associated with the magnetic field
gradient, which is proportional to the time integral of the gradient. Its
relative magnitude compared with the important momenta in the problem is a
significant parameter, and if their ratio is not small some non-classical
effects contribute to the solution.
  Assuming the field gradient is sufficiently small, and a number of other
inequalities are satisfied involving the mean wavelength, range of the force,
and the mean separation between particles, we solve the integro- partial
differential equations for the Wigner functions to second order in the strength
of the gradient. When the same reasoning is applied to a different problem with
no field gradient, but having instead a gradient to the z-component of
polarization, the connection with the diffusion coefficient is established, and
we find agreement with the classical result for the rate of decrease of the
transverse component of magnetization.
