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arXiv:2605.31435

Can a spin-half particle ever give more than two spots in a Stern-Gerlach experiment? -- the subtle physics of effective Hamiltonians

Noah Linden, Sandu Popescu, Anthony J. Short

quant-phSubmitted May 29, 2026• Announced Jun 1, 2026• 0 citations
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Abstract

We show that a spin-1/2 particle can behave as if it were spin-$s$, and generate $2s+1$ spots in a Stern Gerlach measurement (albeit with a smaller gyromagnetic ratio). This arises from some subtle properties of effective Hamiltonians and Hamiltonians with constraints. Examples of implications of the effect in condensed matter are discussed. We also give some simple non-perturbative bounds for a system subjected to strong constraints.

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We show that a spin-1/2 particle can behave as if it were spin-sss, and generate 2s+12s+12s+1 spots in a Stern Gerlach measurement (albeit with a smaller gyromagnetic ratio). This arises from some subtle properties of effective Hamiltonians and Hamiltonians with constraints. Examples of implications of the effect in condensed matter are discussed. We also give some simple non-perturbative bounds for a system subjected to strong constraints.
Abstract We analyze idealized sequential Stern–Gerlach (SG) experiments with higher spin particles. This analysis serves at least two purposes: the widely discussed spin-1/2 case leads to some misunderstandings since the probabilities are always evenly distributed for the sequential orthogonal magnets, which does not generalize to higher spins. A detailed discussion of the higher spin case, as is done here, is highly useful. Secondly, the Wigner rotation matrices for generic spins become conceptually more transparent with this physical example. We also give compact formulas for the probabilities in terms of the angle between the sequential SG apparatus for generic spins. We work out the spin-1/2, spin-1 cases explicitly. Since there are some confusing issues regarding the actual experiment, we also compile a ‘facts and fiction’ section on the SG experiments.
It is suggested that a thorough treatment of electron spin has advantages as a method of presenting many of the concepts and techniques of intermediate quantum mechanics. The starting point is the Stern-Gerlach experiment, from which the anticommutation relations for the spin operators may immediately be deduced. The commutation relations, which demonstrate that the spin is an angular momentum, are derived from these and it is shown that the anomalous gyromagnetic ratio is a natural consequence of the properties of the spin operators. The Pauli theory, which is the simplest example of the use of matrices in quantum mechanics, is briefly summarised, and employed in an analysis of the double Stern-Gerlach experiment. This example of a change of representation is used to introduce Dirac's transformation theory which, together with a variant of the double Stern-Gerlach experiment, is made use of in a discussion of the EPR paradox, Bell's inequality and hidden-variable theories. It is emphasised that the quantum mechanical description of these phenomena is consistent and that, from the viewpoint of quantum theory, no action-at-a-distance is required to explain them.
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.
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.