# 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

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 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 like 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 energy states $\pm\mu B$, and as such not valid for individual fermions. The two fermion states $\pm\mu B$ 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, ...

Cited literature [20 references]

https://hal.archives-ouvertes.fr/hal-02882969
Contributor : Gerrit Coddens <>
Submitted on : Saturday, August 29, 2020 - 11:40:36 AM
Last modification on : Thursday, September 3, 2020 - 4:09:40 AM

### Files

Stern-Gerlach-experiment-lates...
Files produced by the author(s)

### Identifiers

• HAL Id : hal-02882969, version 5

### Citation

Gerrit Coddens. 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. 2020. ⟨hal-02882969v5⟩

Record views