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 betweenthe spin axis
s and the magnetic field
B, the fermions can only have two possible energies
m0c2±μB. The values
±μB do thus 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 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.
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 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.
Abstract. We study the experimental signatures of several states of a Bose-Einstein condensate of spin-1 atoms by quantum trajectory simulations of Stern-Gerlach experiments. The measurement process itself creates an apparent random alignment for the spins, so that it is difficult to distinguish between a condensate that initially has an alignment in an unknown direction and one with no alignment at all. Nonetheless, in repeated experiments with identically prepared condensates, it is possible to discriminate between certain ground states of the condensate proposed in the literature. PACS numbers: 03.75.Fi,03.65.BzMeasurement theory for spinor condensates 2 1.
We study the experimental signatures of several states of a Bose-Einstein condensate of spin-1 atoms by quantum trajectory simulations of Stern-Gerlach experiments. The measurement process itself creates an apparent random alignment for the spins, so that it is difficult to distinguish between a condensate that initially has an alignment in an unknown direction and one with no alignment at all. Nonetheless, in repeated experiments with identically prepared condensates, it is possible to discriminate between certain ground states of the condensate proposed in the literature.
We present an analytical solution to the decoherence time for the spin measurement and the diagonalization of the density matrix of spin variables in the Stern-Gerlach experiment.This solution requires the calculation of the Pauli spinor with a spatial extension, which is not found in quantum mechanics textbooks.With this full spinor and the measured position of the particle we demonstrate the three postulates of quantum measurement: quantization, spectral decomposition and wave function reduction.The transition from a quantum superposition to a statistical mixture is well explained in this way, but not the single result that always emerges from a particular experiment.The spinor spatial extension also allows the introduction of the de Broglie-Bohm trajectories which give a very simple explanation of the particles' impact.