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arXiv:2311.13406 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

Zig-zag dynamics in a Stern–Gerlach spin measurement

Simon Krekels, Christian Maes, Kasper Meerts, Ward Struyve

Published Mar 1, 2024• 3 citations
arXivHTMLPDF

Abstract

The century-old Stern–Gerlach setup is paradigmatic for a quantum measurement. We visualize the electron trajectories following the Bohmian zig-zag dynamics. This dynamics was developed in order to deal with the fundamentally massless nature of particles (with mass emerging from the Brout–Englert–Higgs mechanism). The corresponding trajectories exhibit a stochastic zig-zagging, as the result of the coupling between left- and right-handed chiral Weyl states. This zig-zagging persists in the non-relativistic limit, which will be considered here, and which is described by the Pauli equation for a non-uniform external magnetic field. Our results clarify the different meanings of ‘spin’ as a property of the wave function and as a random variable in the Stern–Gerlach setup, and they illustrate the notion of effective collapse. We also examine the case of an EPR-pair. By letting one of the entangled particles pass through a Stern–Gerlach device, the non-local influence (action-at-a-distance) on the other particle is manifest in its trajectory, e.g. by initiating its zig-zagging.

DOI
10.1098/rspa.2023.0861

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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.

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The century-old Stern–Gerlach setup is paradigmatic for a quantum measurement. We visualize the electron trajectories following the Bohmian zig-zag dynamics. This dynamics was developed in order to deal with the fundamentally massless nature of particles (with mass emerging from the Brout–Englert–Higgs mechanism). The corresponding trajectories exhibit a stochastic zig-zagging, as the result of the coupling between left- and right-handed chiral Weyl states. This zig-zagging persists in the non-relativistic limit, which will be considered here, and which is described by the Pauli equation for a non-uniform external magnetic field. Our results clarify the different meanings of ‘spin’ as a property of the wave function and as a random variable in the Stern–Gerlach setup, and they illustrate the notion of effective collapse. We also examine the case of an EPR-pair. By letting one of the entangled particles pass through a Stern–Gerlach device, the non-local influence (action-at-a-distance) on the other particle is manifest in its trajectory, e.g. by initiating its zig-zagging.
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.
The classical Stern-Gerlach experiment is analyzed with an emphasis on the spin dynamics. The central question asked is whether there occurs a relaxation of the spin angular momentum during the time the particle passes through the Stern-Gerlach magnet. We examine in particular the transverse relaxation, involving angular momentum exchange between the spin of the particles and the spins of the magnet. A method is presented describing relaxation effects at an individual particle level. This leads to a stochastic equation of motion for the spins. This is coupled to a classical equation of motion for the particle translation. The experimental situation is then modeled through simulations of individual trajectories using two sets of parameter choices and three different sets of initial conditions. The two main conclusions are: (A) if the coupling between the magnet and the spin is solely described by the Zeeman interaction with the average magnetic field the simulations show a clear disagreement with the experimental observation of Stern and Gerlach. (B) If one, on the other hand, also allows for a T(2) relaxation time shorter than the passage time one can obtain a practically quantitative agreement with the experimental observations. These conclusions are at variance with the standard textbook explanation of the Stern-Gerlach experiment.
The Stern-Gerlach experiment (SGE) has been known for decades, but a complete theory that explains all facets of it remains enigmatic. In its classic realization, each spin state of an atom follows a unique path or trajectory in an SGE. The enigmatic part is the final output of SGE. Feynman envisaged the outcome to be an entangled state, a superposition of $|\text{spin}\phantom{\rule{4.pt}{0ex}}\text{state}\ensuremath{\rangle}\ensuremath{\bigotimes}|\text{corresponding}\phantom{\rule{4.pt}{0ex}}\text{path}\ensuremath{\rangle}$. Later works described the final state of SGE as a mixed state, a weighted average of states belonging to separate beams or trajectories. Notably, Schwinger, Englert, and Scully compared the difficulty of obtaining a coherent output (a pure state) to ``putting together Humpty Dumpty'' again. Yet recent experiments using Bose-Einstein condensates indicate the emerging beams are partly coherent, favoring Feynman's description. In this work, we revisit SGE using a recent version of the quantum master equation. We show that both outcomes are probable depending on the time of flight and the timescale of the environmental fluctuations. We predict the output is initially coherent and grows gradually into a mixed state. Moreover, we also estimate how the magnetic-field gradient helps broaden the beam waist. Moreover, our treatment implies that the wave function of an atom passing through a Stern-Gerlach device does not undergo an instantaneous collapse; rather, a gradual decay of coherences results in a mixed state, resulting in an irreversible process.
Abstract The discussion of the recently derived quantum Hamilton equations for a spinning particle is extended to spin measurement in a Stern–Gerlach experiment. We show that this theory predicts a continuously changing orientation of the particles magnetic moment over the course of its motion across the Stern–Gerlach apparatus. The final measurement results agree with experiment and with predictions of the Pauli equation. Furthermore, the Einstein–Podolsky–Rosen–Bohm thought experiment is investigated, and the violation of Bells’s inequalities is reproduced within this stochastic mechanics approach. The origin of the violation of Bell’s inequalities is traced to the the non-local nature of the velocity fields for an entangled state in the stochastic formalism, which is a result of a non-separable probability distribution of the considered particles.
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.
Abstract The Stern–Gerlach experiment reveals the quantum mechanical nature of spin through the force experienced by a non-relativistic neutral atom in a static magnetic field gradient. A key open question concerns when, along the atom’s trajectory, the spin state collapses into an eigenstate and by what mechanism. We analyze the influence of quantum electrodynamic (QED) vacuum fluctuations on spin dynamics and show that they induce spin self-interactions, leading to decoherence and memory effects. However, the magnitude of these effects is shown to be too small to account for the deterministic collapse of the spin wavefunction.