  We study the influence of diffusion on NMR experiments when the molecules
undergo random motion under the influence of a force field, and place special
emphasis on parabolic (Hookean) potentials. To this end, the problem is studied
using path integral methods. Explicit relationships are derived for commonly
employed gradient waveforms involving pulsed and oscillating gradients. The
Bloch-Torrey equation, describing the temporal evolution of magnetization, is
modified by incorporating potentials. A general solution to this equation is
obtained for the case of parabolic potential by adopting the multiple
correlation function (MCF) formalism, which has been used in the past to
quantify the effects of restricted diffusion. Both analytical and MCF results
were found to be in agreement with random walk simulations. A multi-dimensional
formulation of the problem is introduced that leads to a new characterization
of diffusion anisotropy. Unlike for the case of traditional methods that employ
a diffusion tensor, anisotropy originates from the tensorial force constant,
and bulk diffusivity is retained in the formulation. Our findings suggest that
some features of the NMR signal that have traditionally been attributed to
restricted diffusion are accommodated by the Hookean model. Under certain
conditions, the formalism can be envisioned to provide a viable approximation
to the mathematically more challenging restricted diffusion problems.
