1D toy model for magnetic trapping

We study, both classically and quantum mechanically, the problem of a neutr al particle with a spin angular momentum S, mass m, and magnetic moment mu, moving in one dimension in an inhomogeneous magnetic field given by B=B-0( z) over cap+ B-perpendicular to'x (y) over cap. This problem serves for us as a toy model to study the trapping of neutral particles. We identify K = root[S-2(B-perpendicular to')(2)/mu mB(0)(3]), which is the ratio between t he precessional frequency of the particle and its vibrational Frequency, as the relevant parameter of the problem. Classically, we find that when mu i s antiparallel to B, the particle is trapped provided that K<0.5. We also E nd that viscous friction, be it translational or precessional, destabilizes the system. Quantum mechanically, we study the problem of a spin S=h/2 par ticle in the same field. Treating K as a small parameter for the perturbati on from the adiabactic Hamiltonian, we find that the lifetime T-esc of the particle in its trapped ground state is T-esc =(T-vib/2 pi) x(1/root 8 pi K )exp(2/K), where T-vib = 2 pi root mB(0)/mu(B-perpendicular to')(2) is the classical period of the particle when placed in the adiabatic potential V = mu/B/. (C) 2000 American Association of Physics Teachers.