Quantized motion of atoms in a quadrupole magnetostatic trap

We consider quantized motion of neutral atoms cooled below the recoil limit in a quadrupole magnetostatic trap. Because of Majorana transitions to untrapped levels near the point of zero field at the trap center, all quantum levels have a nonzero decay rate. The Schroedinger equation associated with the potential {ital g}{mu}{bold B}{center dot}{bold S} ({bold S} is the total atomic spin) takes the form of coupled equations in {ital r} when the spinor components are expanded in spherical harmonics. We integrate the multichannel problem numerically to obtain asymptotic phase shifts, resonance energies, and widths. For {ital S}=1/2, the lowest levels have widths somewhat less than their spacing. Thus the trap quantum-level structure might possibly be observable if the atoms are sufficiently cold, namely, in the 0.1-{mu}K regime for most atoms and attainable trap field gradients. The width decreases rapidly with increasing {ital M}{sub {ital J}}, the angular momentum about the symmetry axis. Spectroscopic linewidths of a few hertz are possible if there is enough population in the lowest levels with a few {ital M}{sub {ital J}} quanta. The decay rate of the lowest levels, however, is probably too rapid for studying Bose--Einstein condensation in such a trap.