Series Solutions for Atom Guides

In an atomic waveguide based on magnetic interactions, the forces on different spin components have opposite signs. Thus, if one component of a spinor is bound near the guiding center, another component of the spinor is repelled from the center. This is simply an example of the Stern-Gerlach effect commonly used to separate particles in different spin states. One difference is that, in the atomic guide, the spatial degrees of freedom must be quantized as well as the spin degrees of freedom. In order to accurately treat the complete quantum problem both inside and outside the guiding region, a power series solution valid for the whole real axis is developed. Thus, in principle, both the bound and unbound components of the full spinor solution can be treated within the same framework. The power series converges and can be readily summed over a large range of the radial coordinate by making use of arbitrary precision arithmetic. This technique produces high precision eigenvalues as well as detailed information about the behavior of the spinor wavefunction at both large and small distances from the origin, making it possible to perform detailed studies of atom propagation without resorting to the adiabatic approximation.