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## Brazilian Journal of Physics

*On-line version* ISSN 1678-4448

#### Abstract

DIAZ BULNES, J. and OLIVEIRA, I.S.. **Construction of exact solutions for the Stern-Gerlach effect**.* Braz. J. Phys.* [online]. 2001, vol.31, n.3, pp.488-495.
ISSN 1678-4448. http://dx.doi.org/10.1590/S0103-97332001000300023.

We obtain exact solutions for the Schrödinger-Pauli matrix equation for a neutral particle of spin 1/2 in a magnetic eld 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, |Y_{+}|^{2} and |Y_{-}|^{2 }, and the eigenenergies are calculated. These include a small contribution from the field gradient, a, proportional to (a)^{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 D_{z} » 4 mm, in accordance to a real Stern-Gerlach experiment.