1D Toy Model For Trapping Neutral Particles

We study, both classically and quantum-mechanically, the problem of a neutral particle with spin, moving in one-dimension in an inhomogeneous magnetic field. This problem serves for us as a toy model to study the trapping of neutral particles. We identify K, the ratio between the precessional frequency of the particle and its vibration frequency, as the relevant parameter of the problem. Classicaly, we find that when the magnetic moment is antiparallel to the direction of the magnetic field, the particle is trapped provided that K<0.5. We also find that viscous friction, be it translational or precessional, destabilizes the system. Quantum-mechanically, we study the problem of spin half particle in the same field. Treating K as a small parameter for the perturbation from the adiabatic Hamiltonian, we find the lifetime of the particle in its trapped ground-state.