Construction of exact solutions for the Stern-Gerlach effect

Abstract

We obtain exact solutions for the Schrödinger-Pauli matrix equation for a neutral particle of spin 1/2 in a magnetic field with a field gradient. The analytical wavefunctions are written on the symmetry plane Y = 0, which contains the incident and splitted beams, in terms of the Airy functions. The time-evolution of the probability densities, |ψ+|2 and |ψ-|2, and the eigenenergies are calculated. These include a small contribution from the field gradient, α, proportional to (αℏ)2/3, which amounts to equal energy displacements on both magnetic levels. The results are generalized for spin S = 3/2, and in this case we found that the m = ±1/2 and m = ±3/2 magnetic sublevels are unequaly splitted by the field gradient, being the difference in energy of the order 0.4 MHz. Replacing real experimental parameters we obtained a spatial splitting of the spin up and spin down states of the order Δz ≈ 4 mm, in accordance to a real Stern-Gerlach experiment.