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Symmetry

The Exact Theory of the Stern–Gerlach Experiment and Why it Does Not Imply that a Fermion Can Only Have Its Spin Up or Down

G. Coddens

Published Jan 14, 2021• 5 citations
Publisher

Abstract

The Stern–Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. However, that theory is based on mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry. Contrary to what Pauli has been reading into the Stern–Gerlach experiment, the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured. Surprisingly, this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin-down. Indeed, despite the presence of the many different possible angles θ between the spin axis s and the magnetic field B, the fermions can only have two possible energies m0c2±μB. The values ±μB thus do not correspond to the continuum of values −μ·B Einstein and Ehrenfest had conjectured. The energy term V=−μ·B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies ±μB, and as such not valid for individual fermions. The two fermion states with energy ±μB are not potential-energy states. We also explain the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.

DOI
10.3390/sym13010134

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The Stern-Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. But that theory is based on mathematical errors in the way it (mis)treats spinors and group theory.We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry.Contrary to what Pauli has been reading into the Stern-Gerlach experiment,the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured.Surprizingly this leads to only two energy states, which should be qualified as precession-up and precession-down rather thanspin-up and spin down.Indeed, despite the presence of the many different possible angles $\theta$ betweenthe spin axis ${\mathbf{s}}$ and the magnetic field ${\mathbf{B}}$, the fermions can only have two possible energies $m_{0}c^{2}\pm\mu B$. The values $\pm\mu B$ do thus not correspond to the continuum of values $-{\boldsymbol{\mu\cdot}}{\mathbf{B}}$ Einstein and Ehrenfest had conjectured. The energy term $V= -{\boldsymbol{\mu\cdot}}{\mathbf{B}}$ is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies $\pm\mu B$, and as such not valid for individual fermions. The two fermion states with energy $\pm\mu B$ are not potential-energy states. We also explain themathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clearand understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.

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The exact theory of the Stern-Gerlach experiment and why it does not imply that a fermion can only have its spin up or down

G. Coddens

Appendix added in order to explain the geometrical meaning of the spin-up and spin-down states as right-haded and left-handed representations. References added. Typos corrected.

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The Stern–Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. However, that theory is based on mathematical errors in the way it (mis)treats spinors and group theory. We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry. Contrary to what Pauli has been reading into the Stern–Gerlach experiment, the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured. Surprisingly, this leads to only two energy states, which should be qualified as precession-up and precession-down rather than spin-up and spin-down. Indeed, despite the presence of the many different possible angles θ between the spin axis s and the magnetic field B, the fermions can only have two possible energies m0c2±μB. The values ±μB thus do not correspond to the continuum of values −μ·B Einstein and Ehrenfest had conjectured. The energy term V=−μ·B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies ±μB, and as such not valid for individual fermions. The two fermion states with energy ±μB are not potential-energy states. We also explain the mathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clear and understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.
The Stern-Gerlach experiment is notoriously counter-intuitive. The official theory is that the spin of a fermion remains always aligned with the magnetic field. Its directions are thus quantized: It can only be spin-up or spin-down. But that theory is based on mathematical errors in the way it (mis)treats spinors and group theory.We present here a mathematically rigorous theory for a fermion in a magnetic field, which is no longer counter-intuitive. It is based on an understanding of spinors in SU(2) which is only Euclidean geometry.Contrary to what Pauli has been reading into the Stern-Gerlach experiment,the spin directions are not quantized. The new corrected paradigm, which solves all conceptual problems, is that the fermions precess around the magnetic-field just as Einstein and Ehrenfest had conjectured.Surprizingly this leads to only two energy states, which should be qualified as precession-up and precession-down rather thanspin-up and spin down.Indeed, despite the presence of the many different possible angles θ\thetaθ betweenthe spin axis s{\mathbf{s}}s and the magnetic field B{\mathbf{B}}B, the fermions can only have two possible energies m0c2±μBm_{0}c^{2}\pm\mu Bm0​c2±μB. The values ±μB\pm\mu B±μB do thus not correspond to the continuum of values −μ⋅B-{\boldsymbol{\mu\cdot}}{\mathbf{B}}−μ⋅B Einstein and Ehrenfest had conjectured. The energy term V=−μ⋅BV= -{\boldsymbol{\mu\cdot}}{\mathbf{B}}V=−μ⋅B is a macroscopic quantity. It is a statistical average over a large ensemble of fermions distributed over the two microscopic states with energies ±μB\pm\mu B±μB, and as such not valid for individual fermions. The two fermion states with energy ±μB\pm\mu B±μB are not potential-energy states. We also explain themathematically rigorous meaning of the up and down spinors. They represent left-handed and right-handed reference frames, such that now everything is intuitively clearand understandable in simple geometrical terms. The paradigm shift does not affect the Pauli principle.
A detailed analysis of the classic Stern–Gerlach experiment is presented. An analytical simple solution is presented for the quantum description of the translational and spin dynamics of a silver atom in a magnetic field with a gradient along a single z-direction. This description is then used to obtain an approximate quantum description of the more realistic case with a magnetic field gradient also in a second y-direction. An explicit relation is derived for how an initial off center deviation in the y-direction affects the final result observed at the detector. This shows that the “mouth shape” pattern at the detector observed in the original Stern–Gerlach experiment is a generic consequence of the gradient in the y-direction. This is followed by a discussion of the spin dynamics during the entry of the silver atom into the magnet. An analytical relation is derived for a simplified case of a field only along the z-direction. A central question for the conceptual understanding of the Stern–Gerlach experiment has been how an initially unpolarized spin ends up in a polarized state at the detector. It is argued that this can be understood with the use of the adiabatic approximation. When the atoms first experience the magnetic field outside the magnet, there is in general a change in the spin state, which transforms from a degenerate eigenstate in the absence of a field into one of two possible non-degenerate states in the field. If the direction of the field changes during the passage through the device, there is a corresponding adiabatic change of the spin state. It is shown that an application of the adiabatic approximation in this way is consistent with the previously derived exact relations.
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 origins of quantum theory lie in a failure to explain certain phenomena using classical theory. But despite its adoption, quantum theory abounds with concepts described as “weird” or “spooky.” Even Feynman said that no one understands quantum theory. We revisit the Stern-Gerlach experiment that gave rise to the concept of spin and its quantization from a classical point of view. We present a simplified model of the classical electron which has two stable states which separate into two beams in the presence of an inhomogenous magnetic field.