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.
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 1932 Pauli and Bohr declared that electromagnetic fields cannot be used to separate electrons according to their spin. A Stern–Gerlach magnet can split atomic beams into beams with different polarizations, but the Lorentz force blurs the splitting for charged particles such as electrons. Brillouin suggested that electrons could be separated by applying magnetic gradients along the electron beam, but Pauli disagreed. He wrote in 1958: "It is impossible to observe the spin of the electron, separated fully from its orbital momentum, by means of experiments based on the concept of classical particle trajectories."
In the late 1920s Niels Bohr propagated the idea that the magnetic moment of a free electron could not be observed. This derived from the idea that the spin degree of freedom characterized the electron only when it is bound in an atom. This view initiated a lively discussion, which involved many of the most prominent theoreticians of the time. The independent existence of the electron spin became an issue of principle. In particular it was deemed that quantum effects would destroy the separated classical trajectories in a Stern-- Gerlach type of experiment. We review these discussions and some later developments. Quantum effects do prove to be essential, but they do not overwhelm the magnetic effects of spin. In addition to these arguments, it has been possible experimentally to determine the electron g -factor with high accuracy in electromagnetic traps. In fact no principle seems to prevent the observation of the magnetic moment of the free electron.
The Stern-Gerlach experiment is the fundamental experiment in order to exhibit the quantization of spin and understand the measurement problem in quantum mechanics. However, although the Stern-Gerlach experiment plays an essential role in the teaching of quantum mechanics, no complete analysis of this experiment using Pauli spinors is presented in the pedagogical literature. This paper presents such an analysis and develops implications for the theory of quantum measurement. We first propose an analytic expression of both the wave function and the probability density in the Stern-Gerlach experiment. Our explicit solution is obtained via a complete integration of the Pauli equation over time and space. The probability density evolution describes a slipping of the wave packet into two separate packets due to the measurement device, but it cannot account for impacts. We therefore calculate the de Broglie-Bohm trajectories, which not only explain impacts naturally, but also accounts for the spin quantization following the magnetic field gradient. It is then possible to propose a clear explanation of measurement effects in the Stern-Gerlach experiment.
A nondestructive type of Stern-Gerlach ef- fect for an individual electron is described that has been thoroughly demonstrated in experiments at the University of Washington. This Stern-Gerlach makes use of an inhomogeneous magnetic field provided by a weak auxiliary magnetic bottle and is nondestructive in contrast to all previous versions of the effect. As in the classic Stern-Gerlach effect, changes in the spin state are detected via changes in classical particle trajectories; however, what is observed now is not a deflection of the orbit but rather a change of its frequency in the storage well. A simplified model of the continuous Stern-Gerlach effect at zero temperature is introduced to develop the relation between minimum measurement time required for determination of spin state, driven oscillation amplitude, and zero-point fluctuations in the storage well. The problem of the reduction of the wavefunction by the interaction of the electron with the apparatus is addressed following W. Pauli.
In the Stern–Gerlach experiment, silver atoms were separated according to their spin state (Gerlach and Stern 1922 Z. Phys. 9 353–355). This experiment demonstrates the quantization of spin and relies on the classical description of motion. However, so far, no design has led to a functional Stern–Gerlach magnet for free electrons. Bohr and Pauli showed in the 1930 Solvay conference that Stern–Gerlach magnets for electrons cannot work, at least if the design is based on classical trajectories (Pauli W 1932 Proc. of the 6th Solvay Conf. 2 (1930) (Brussels: Gauthier-Villars) pp 183–86, 217–20, 275–80; Pauli W 1964 Collected Scientific Papers ed R Kronig and V F Weiskopf, vol 2 (New York: Wiley)). Here, we present ideas for the realization of a Stern–Gerlach magnet for electrons in which spin and motion are treated fully quantum mechanically . We show that a magnetic phase grating composed of a regular array of microscopic current loops can separate electron diffraction peaks according to their spin states. The experimental feasibility of a diffractive approach is compared to that of an interferometric approach. We show that an interferometric arrangement with magnetic phase control is the functional equivalent of an electron Stern–Gerlach magnet.
Abstract In 1921, Otto Stern conceived the idea for an experiment that would decide between a classical and a quantum description of atomic behavior, as epitomized by the Bohr–Sommerfeld–Debye model of the atom. This model entailed not only the quantization of the magnitude of the orbital electronic angular momentum but also of the projection of the angular momentum on an external magnetic field – the so‐called space quantization. Stern recognized that space quantization would have observable consequences: namely, that the magnetic dipole moment due to the orbital angular momentum would be space quantized as well, taking two opposite values for atoms whose only unpaired electron has just one quantum of orbital angular momentum. When acted upon by a suitable inhomogeneous magnetic field, a beam of such atoms would be split into two beams consisting of deflected atoms with opposite projections of the orbital angular momentum on the magnetic field. In contradistinction, if atoms behaved classically, the atomic beam would only broaden along the field gradient and have maximum intensity at zero deflection, i. e., where there would be a minimum or no intensity for a beam split due to space quantization. Stern anticipated that, although simple in principle, the experiment would be difficult to carry out – and invited Walther Gerlach to team up with him. Gerlach's realism and experimental skills together with his sometimes stubborn determination to make things work proved invaluable for the success of the Stern–Gerlach experiment (SGE). After a long struggle, Gerlach finally saw, on 8 February 1922, the splitting of a beam of silver atoms in a magnetic field. The absence of the concept of electron spin confused and confounded the interpretation of the SGE, as the silver atoms were, in fact, in a 2S state, with zero orbital and spin angular momentum. However, a key quantum feature whose existence the SGE was designed to test – namely space quantization of electronic angular momentum – was robust enough to transpire independent of whether the electronic angular momentum was orbital or due to spin. The SGE entails other key aspects of quantum mechanics such as quantum measurement, state preparation, coherence, and entanglement. Confronted with the outcome of the SGE, Stern noted: “I still have objections to the idea of beauty of quantum mechanics. But she is correct.”