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Physical Review Letters

Quantum-Mechanical Analysis of a Longitudinal Stern-Gerlach Effect

G. A. Gallup, Herman Batelaan, T. J. Gay

Published May 14, 2001• 41 citations
Publisher

Abstract

We present the results of a rigorous quantum-mechanical calculation of the propagation of electrons through an inhomogeneous magnetic field with axial symmetry. A complete spin polarization of the beam is demonstrated assuming that a Landau eigenstate can be inserted into the field. This is in contrast with the semiclassical situation, where the spin splitting is blurred.

DOI
10.1103/physrevlett.86.4508

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We present the results of a rigorous quantum-mechanical calculation of the propagation of electrons through an inhomogeneous magnetic field with axial symmetry. A complete spin polarization of the beam is demonstrated assuming that a Landau eigenstate can be inserted into the field. This is in contrast with the semiclassical situation, where the spin splitting is blurred.
The splitting of a beam of potassium atoms in the classical Stern-Gerlach scheme under varying inhomogeneity of the magnetic field is experimentally investigated in this work. First, the basic shape of the beam in the absence of an effective field is recorded, which makes it possible to introduce and calibrate the geometrical parameters of the channel. Then, when the current in the magnet windings increases and the field gradient grows, a systematic shift of the beam density maxima is observed, described by the model function F(u), which includes the parameter q, which characterizes the strength of interaction of atoms with the field. Theoretical calculations based on this function showed good agreement with the experimental results, including the asymmetry of the distribution due to the nonideal symmetry of the magnetic system. The obtained dependences of the position of the intensity maxima on ∇B confirmed the validity of both linear and asymptotic approximation for different modes of magnet operation. These conclusions have both fundamental importance for understanding the quantum mechanical aspects of beam splitting and applied significance in the development of methods for precise control of spin-polarized atomic beams in spectroscopy and spintronics.
The motion of neutral particles with magnetic moments in an inhomogeneous magnetic field is described in a quantum mechanical framework. The validity of the semiclassical approximations which are generally used to describe these phenomena is discussed. Approximate expressions for the evolution operator are derived and compared to the exact calculations. Focusing and spin-flip phenomena are predicted. The reliability of Stern-Gerlach experiments to measure spin projections is assessed in this framework.
The conflict between Bohr's assertion that the magnetic moment of the electron cannot be measured with experiments based on the concept of classical trajectories, and the measurement of the magnetic moment of electrons in a modified Penning trap by Dehmelt et al. has led us to reevaluate other implications of Bohr's assertion. We show that, contrary to the analysis of Bohr and Pauli, the assumption of classical trajectories in a Stern-Gerlach--like device can result in a high degree of spin separation for an electron beam. This effect may persist within a fully quantum-mechanical analysis. The magnetic fields considered are such that a tabletop Stern-Gerlach electron spin filter is feasible.
In the optical Stern-Gerlach effect the interaction of a two-level atom with an optical-field gradient causes a state-selective deflection that leads to a splitting of the atomic trajectory. The semiclassical approach to this problem actually predicts a splitting of the atomic beam into two paths, similar to the well-known magnetic Stern-Gerlach effect for spin 1/2. Here we study a model in which the quantum nature of the field is taken into account and we show, in particular, that for an initial coherent state of the cavity light the optical Stern-Gerlach effect manifests itself as a splitting of the atomic beam into two groups containing, however, many spots. \textcopyright{} 1996 The American Physical Society.
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, alpha, proportional to (alpha )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 deltaz ~ 4 mm, in accordance to a real Stern-Gerlach experiment.