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.
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.
Abstract In 1921, Otto Stern conceived the idea for an experiment that would decide between a classical and a quantum description of atomic behavior, as epitomized by the Bohr–Sommerfeld–Debye model of the atom. This model entailed not only the quantization of the magnitude of the orbital electronic angular momentum but also of the projection of the angular momentum on an external magnetic field – the so‐called space quantization. Stern recognized that space quantization would have observable consequences: namely, that the magnetic dipole moment due to the orbital angular momentum would be space quantized as well, taking two opposite values for atoms whose only unpaired electron has just one quantum of orbital angular momentum. When acted upon by a suitable inhomogeneous magnetic field, a beam of such atoms would be split into two beams consisting of deflected atoms with opposite projections of the orbital angular momentum on the magnetic field. In contradistinction, if atoms behaved classically, the atomic beam would only broaden along the field gradient and have maximum intensity at zero deflection, i. e., where there would be a minimum or no intensity for a beam split due to space quantization. Stern anticipated that, although simple in principle, the experiment would be difficult to carry out – and invited Walther Gerlach to team up with him. Gerlach's realism and experimental skills together with his sometimes stubborn determination to make things work proved invaluable for the success of the Stern–Gerlach experiment (SGE). After a long struggle, Gerlach finally saw, on 8 February 1922, the splitting of a beam of silver atoms in a magnetic field. The absence of the concept of electron spin confused and confounded the interpretation of the SGE, as the silver atoms were, in fact, in a 2S state, with zero orbital and spin angular momentum. However, a key quantum feature whose existence the SGE was designed to test – namely space quantization of electronic angular momentum – was robust enough to transpire independent of whether the electronic angular momentum was orbital or due to spin. The SGE entails other key aspects of quantum mechanics such as quantum measurement, state preparation, coherence, and entanglement. Confronted with the outcome of the SGE, Stern noted: “I still have objections to the idea of beauty of quantum mechanics. But she is correct.”
Spin and helicityTraditionally, in textbooks on quantum mechanics, spin is introduced via an idealized Stern-Gerlach experiment in which a non-relativistic beam of silver atoms passes through an inhomogeneous magnetic field.Each atom is treated as a single valence electron of charge -e in an s-state.The subsequent splitting of the beam into two indicates the two-valuedness of Sz, which is related to the value 1/2 for s, and the magnitude of the splitting shows that the magnetic moment /l is related to s by e /l = -~s, me the proportionality factor (the gyromagnetic ratio) being twice as big as the factor that classically gives the magnetic moment due to the orbital angular momentum of a point charge.Historically, however, it seems that the early Stern-Gerlach experiments, begun in 1922, had no influence at all upon the discovery of spin, simply because they were too imprecise.Rather, the concept of spin appeared after a long and tedious battle to understand the splitting patterns and separations in line spectra.Several people had for various reasons discussed classical models of rotating charge distributions but Kronig, in 1924, was the first to show that an electron with spin 1/2 would explain the pattern of what we would today call L • S splitting, as well as anomalies in the Zeeman effect.He realized, though, that the gyromagnetic ratio ( -ejmc) needed for the latter would give L • S splittings twice as big as those observed.It is said that Pauli expressed his negative reaction to Kronig's idea with such vehemence that Kronig never published his work (Mehra and Rechenberg, 1982).Soon thereafter, in 1925, the same idea occurred to Uhlenbeck and Goudsmit (1925), who proceeded to a detailed analysis of the splittings, concluding at first that everything worked beautifully, but then becoming aware, as a consequence of a comment by Heisenberg, of the factor-of-2 inconsistency mentioned above.
It is well known that electron spin is quantized, and is measured to be either spin up or spin down in a magnetic field, as was first demonstrated in the classic Stern-Gerlach experiment almost 100 years ago. However, it is also believed that a quantum spin may be indeterminate until it is measured, being in a quantum superposition of the two quantum states. On the contrary, I propose (based on a locally realistic spin-quantized picture of quantum waves) that an electron quantum state is always either spin up or spin down, but is never in a superposition of the two. This concept should be directly testable using a two-stage Stern-Gerlach experiment, similar to that presented in standard quantum textbooks, but apparently never carried out experimentally. This experiment should be straightforward using modern atomic beam laboratory equipment. If successful, this could lead to a serious review of quantum foundations, as well as a new set of practical applications. In particular, a binary storage or logic element based on quantum spin should continue to work down to the atomic scale. This should enable computer memories with orders of magnitude greater density than those based on conventional magnetic memories. 2
It is proved that, according to Classical Mechanics and Electrodynamics, the trajectory of the center of mass of a neutral system of electrical charges can be deflected by an inhomogeneous magnetic field, even if its internal angular momentum is zero. This challenges the common view about the function of the Stern-Gerlach apparatus, as resolving the eigen-states of an intrinsic angular momentum. Doubts are cast also on the supposed failure of Schrodinger's theory to explain the properties of atoms in presence of magnetic fields without introducing spin variables.
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 (SG) experiment (1922) was one of the earliest experiments for which the result could not be explained using classical physics by any means. Hence it demonstrated unambiguously the importance of quantum effects. The SG experiment can be explained using the quantum theory for the spin-½ particle, which is a quantum system that can be represented simply by 2 × 2 matrices. We will use the spin-½ particle to introduce some of the basic concepts of quantum mechanics, such as state space, operators, measurement, the uncertainty principle, and the evolution equation. Furthermore, the theory for the spin-½ particle can be readily generalized to finite dimensional quantum systems, i.e., systems that can be represented by finite dimensional matrices.