# Exact theory of the Stern-Gerlach experiment - extended version

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. 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 than spin-up and spin down. Indeed, despite the presence of the many different possible angles $\theta$ between the 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 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.
Keywords :
Document type :
Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-03084916
Contributor : Gerrit Coddens <>
Submitted on : Tuesday, January 12, 2021 - 7:08:31 AM
Last modification on : Friday, January 15, 2021 - 3:33:00 AM

### Files

Stern-Gerlach-extension2.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-03084916, version 2

### Citation

Gerrit Coddens. Exact theory of the Stern-Gerlach experiment - extended version. 2021. ⟨hal-03084916v2⟩

Record views