  The Hamiltonian for a particle constrained to motion near a toroidal helix
with loops of arbitrary eccentricity is developed. The resulting three
dimensional Schr\"odinger equation is reduced to a one dimensional effective
equation inclusive of curvature effects. A basis set is employed to find
low-lying eigenfunctions of the helix. Toroidal moments corresponding to the
individual eigenfunctions are calculated. The dependence of the toroidal
moments on the eccentricity of the loops is reported. Unlike the classical
case, the moments strongly depend on the details of loop eccentricity.
