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

Anthony J. Short; Noah Linden; Sandu Popescu

arxiv: 2605.31435 · v1 · pith:TFZZDVOSnew · submitted 2026-05-29 · 🪐 quant-ph

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 This is my paper

Pith reviewed 2026-06-28 22:10 UTC · model grok-4.3

classification 🪐 quant-ph

keywords spin-1/2 particleStern-Gerlach experimenteffective Hamiltoniansconstraintsgyromagnetic ratiocondensed matterquantum measurements

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The pith

A spin-1/2 particle can behave as if it were spin-s and produce 2s+1 spots in a Stern-Gerlach measurement.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that effective Hamiltonians with constraints can make a spin-1/2 particle generate the measurement outcomes expected from a higher spin-s particle. This occurs even though the underlying particle remains spin-1/2, at the cost of a reduced gyromagnetic ratio. The result follows from the way constraints reshape the allowed dynamics without adding new degrees of freedom. Readers would care because it shows how standard spin intuitions can be altered in effective descriptions that appear in many physical systems.

Core claim

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.

What carries the argument

Effective Hamiltonians with constraints, which reshape the dynamics so that spin-1/2 evolution produces the deflection pattern of higher spin while preserving the two-level Hilbert space.

If this is right

Condensed matter systems can exhibit effective higher-spin behavior through constrained Hamiltonians.
The gyromagnetic ratio scales down as the effective spin value increases.
Non-perturbative bounds exist for the energy or response of any system under strong constraints.
The usual two-spot Stern-Gerlach pattern is replaced by 2s+1 spots only while the effective description holds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same constraint mechanism could alter expected outcomes in other spin-dependent measurements beyond Stern-Gerlach.
Engineered constraints in quantum devices might be used to access higher effective spin responses from two-level systems.
The effect suggests a general route by which effective theories hide or modify discrete quantum numbers in constrained settings.

Load-bearing premise

The effective Hamiltonian with constraints remains a valid description of the dynamics and does not introduce additional degrees of freedom or decoherence that would restore the standard two-spot pattern.

What would settle it

Perform a Stern-Gerlach measurement on a spin-1/2 system prepared under the stated constraints and check whether the observed spot count matches 2s+1 or reverts to exactly two.

Figures

Figures reproduced from arXiv: 2605.31435 by Anthony J. Short, Noah Linden, Sandu Popescu.

**Figure 2.** Figure 2: a) shows a hypothetical three-layer material composed of spin [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims constraints let a spin-1/2 system mimic higher spin in Stern-Gerlach via effective Hamiltonians, but the inhomogeneous field may break that description.

read the letter

The main takeaway is that strong constraints on a spin-1/2 particle can produce an effective Hamiltonian whose eigenvalues allow 2s+1 distinct deflections in a Stern-Gerlach gradient, though with reduced gyromagnetic ratio. The authors present this as a consequence of how effective descriptions work under constraints and supply non-perturbative bounds for the strongly constrained regime.

The bounds are the clearest concrete contribution; they are simple and do not rely on perturbation theory, which is useful when thinking about condensed-matter models where similar constraints appear. The discussion of implications for effective spin in materials is also straightforward and points to places where the effect might be checked.

The soft spot is exactly the one flagged in the stress test. The Stern-Gerlach Hamiltonian has a position-dependent magnetic field, so the coupling between position and spin is not uniform. It is not obvious that the constrained effective dynamics survive this inhomogeneity without leakage, additional decoherence, or restoration of the original two-spot pattern. The abstract states the result but does not show an explicit example Hamiltonian or a derivation that keeps the effective description intact through the measurement. If the full paper contains a concrete model where the extra spots remain visible after the gradient is applied, that would address the concern directly.

The work is aimed at people who use effective Hamiltonians for constrained quantum systems, especially in condensed matter. It is coherent on its own terms and engages the relevant literature, so it deserves a serious referee even if the measurement part needs more detail.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that a spin-1/2 particle described by an effective Hamiltonian subject to constraints can mimic the deflection pattern of a higher spin-s particle in a Stern-Gerlach experiment, producing 2s+1 spots (with reduced gyromagnetic ratio). It discusses condensed-matter implications and supplies non-perturbative bounds for strongly constrained systems.

Significance. If the central claim is substantiated, the result would illustrate non-trivial consequences of effective descriptions and constraints in quantum mechanics, with possible relevance to spin-like behavior in condensed-matter systems. The provision of explicit non-perturbative bounds constitutes a concrete, falsifiable contribution.

major comments (2)

The central claim requires that the constrained effective Hamiltonian continues to govern the dynamics under the inhomogeneous magnetic field of the Stern-Gerlach apparatus. No explicit derivation or bound is supplied showing that the position-dependent coupling preserves the constraints without inducing decoherence, leakage, or additional degrees of freedom that would collapse the pattern to two spots.
The non-perturbative bounds for strongly constrained systems are stated in the abstract but their applicability to the inhomogeneous-field SG measurement Hamiltonian is not demonstrated; without this step the bounds do not directly support the 2s+1-spot claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for identifying points that strengthen the presentation of our results on effective Hamiltonians with constraints. We respond to the major comments below.

read point-by-point responses

Referee: The central claim requires that the constrained effective Hamiltonian continues to govern the dynamics under the inhomogeneous magnetic field of the Stern-Gerlach apparatus. No explicit derivation or bound is supplied showing that the position-dependent coupling preserves the constraints without inducing decoherence, leakage, or additional degrees of freedom that would collapse the pattern to two spots.

Authors: We agree that an explicit demonstration is needed. The manuscript relies on the general validity of the constrained effective description but does not derive how the position-dependent SG coupling preserves the constraints. We will add a derivation in the revised manuscript showing that, for sufficiently strong constraints, leakage and decoherence remain suppressed, using the non-perturbative bounds to quantify the effect. revision: yes
Referee: The non-perturbative bounds for strongly constrained systems are stated in the abstract but their applicability to the inhomogeneous-field SG measurement Hamiltonian is not demonstrated; without this step the bounds do not directly support the 2s+1-spot claim.

Authors: The bounds are formulated for arbitrary constrained Hamiltonians and therefore encompass the SG case. The manuscript does not, however, explicitly apply them to the inhomogeneous-field Hamiltonian. We will revise to include this explicit application, thereby linking the bounds directly to the 2s+1-spot result. revision: yes

Circularity Check

0 steps flagged

No circularity identified; derivation chain not inspectable from given text

full rationale

The provided document contains only the abstract and a high-level reader's summary with no equations, no explicit derivation steps, and no self-citations or fitting procedures visible. The central claim is stated as arising from 'subtle properties of effective Hamiltonians and Hamiltonians with constraints,' but without any quoted equations or reduction shown, no instance of self-definitional construction, fitted input renamed as prediction, or load-bearing self-citation can be exhibited. The derivation is therefore treated as self-contained for the purpose of this pass; a full manuscript would be required to perform the requested walk-through.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated validity of the effective-Hamiltonian approximation under constraints.

pith-pipeline@v0.9.1-grok · 5608 in / 995 out tokens · 16983 ms · 2026-06-28T22:10:06.045486+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references

[1]

Das magnetische Moment des Silberatoms,

W. Gerlach and O. Stern, “Das magnetische Moment des Silberatoms,”Zeitschrift f¨ ur Physik, vol. 9, pp. 353–355, Dec 1922

1922
[2]

Robust quantification of spectral transitions in perturbed quantum systems,

Z. Szab´ o, S. Gehr, P. Facchi, K. Yuasa, D. Burgarth, and D. Lonigro, “Robust quantification of spectral transitions in perturbed quantum systems,”Phys. Rev. A, vol. 112, p. 032202, Sep 2025

2025
[3]

Sur la th´ eorie des perturbations des ´ etats li´ es,

C. Bloch, “Sur la th´ eorie des perturbations des ´ etats li´ es,”Nuclear Physics, vol. 6, pp. 329–347, 1958

1958
[4]

Relation between the Anderson and Kondo Hamiltonians,

J. R. Schrieffer and P. A. Wolff, “Relation between the Anderson and Kondo Hamiltonians,” Phys. Rev., vol. 149, pp. 491–492, Sep 1966

1966
[5]

Schrieffer–Wolff transformation for quantum many- body systems,

S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer–Wolff transformation for quantum many- body systems,”Annals of Physics, vol. 326, no. 10, pp. 2793–2826, 2011. 7

2011
[6]

Universal Error Bound for Constrained Quantum Dynamics,

Z. Gong, N. Yoshioka, N. Shibata, and R. Hamazaki, “Universal Error Bound for Constrained Quantum Dynamics,”Phys. Rev. Lett., vol. 124, p. 210606, May 2020

2020
[7]

Error bounds for constrained dynamics in gapped quantum systems: Rigorous results and generalizations,

Z. Gong, N. Yoshioka, N. Shibata, and R. Hamazaki, “Error bounds for constrained dynamics in gapped quantum systems: Rigorous results and generalizations,”Phys. Rev. A, vol. 101, p. 052122, May 2020

2020
[8]

J. J. Sakurai and J. J. Napolitano,Modern Quantum Mechanics. New York: Pearson, 2nd ed., 2014

2014
[9]

Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model,

F. Haldane, “Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model,”Physics Letters A, vol. 93, no. 9, pp. 464–468, 1983

1983
[10]

Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semi- classically Quantized Solitons of the One-Dimensional Easy-Axis N´ eel State,

F. D. M. Haldane, “Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semi- classically Quantized Solitons of the One-Dimensional Easy-Axis N´ eel State,”Phys. Rev. Lett., vol. 50, pp. 1153–1156, Apr 1983

1983
[11]

Quantum spin chains and the Haldane gap,

I. Affleck, “Quantum spin chains and the Haldane gap,”Journal of Physics: Condensed Matter, vol. 1, p. 3047, may 1989

1989
[12]

E. U. Condon and G. H. Shortley,The Theory of Atomic Spectra. Cambridge: Cambridge University Press, 1935. Appendices A Coupled spins Note that we can rewrite the initial Hamiltonian as H0 =−βS 1 ·S 2 =− β 2 J2 −S 2 1 −S 2 2 ,(18) where we have denoted the total spin byJ =S 1 +S 2. Following standard procedures for combining angular momentum, a complete s...

1935
[13]

Prob(outside subspace, t) =⟨Ψ 1(t)|I−Π|Ψ 1(t)⟩ ≤ D(V) E0 2 (29) where |Ψ1(t)⟩=U 1(t)|Ψ 1(0)⟩=e −iH1t/ℏ |Ψ1(0)⟩

The state of the system has only a very small component outside the +1 eigenspace of Π for all time. Prob(outside subspace, t) =⟨Ψ 1(t)|I−Π|Ψ 1(t)⟩ ≤ D(V) E0 2 (29) where |Ψ1(t)⟩=U 1(t)|Ψ 1(0)⟩=e −iH1t/ℏ |Ψ1(0)⟩. 9 Proof ⟨Ψ1(t)|I−Π|Ψ 1(t)⟩=⟨Ψ 1(t)|(I−Π) 2 |Ψ1(t)⟩ =⟨Ψ 1(t)| I− H1 E0 + V E0 2 |Ψ1(t)⟩ =⟨Ψ 1(t)| I− H1 E0 2 + I− H1 E0 V E0 + V E0 I− H1 E0 + V ...
[14]

And let 6 6Note that the trace distance between two pure states is given byd(|Ψ 1⟩,|Ψ 2⟩) = p 1− |⟨Ψ 2|Ψ1⟩|2

Consider now a second Hamiltonian H2 =E 0Π + ΠVΠ,(31) and let|Ψ 2(t)⟩be the state that evolves underH 2, starting with the same initial state, |Ψ2(0)⟩=|Ψ 1(0)⟩. And let 6 6Note that the trace distance between two pure states is given byd(|Ψ 1⟩,|Ψ 2⟩) = p 1− |⟨Ψ 2|Ψ1⟩|2. Hence, we can also use this result to bound the trace distance viad(|Ψ 1(t)⟩,|Ψ 2(t)⟩)...

[1] [1]

Das magnetische Moment des Silberatoms,

W. Gerlach and O. Stern, “Das magnetische Moment des Silberatoms,”Zeitschrift f¨ ur Physik, vol. 9, pp. 353–355, Dec 1922

1922

[2] [2]

Robust quantification of spectral transitions in perturbed quantum systems,

Z. Szab´ o, S. Gehr, P. Facchi, K. Yuasa, D. Burgarth, and D. Lonigro, “Robust quantification of spectral transitions in perturbed quantum systems,”Phys. Rev. A, vol. 112, p. 032202, Sep 2025

2025

[3] [3]

Sur la th´ eorie des perturbations des ´ etats li´ es,

C. Bloch, “Sur la th´ eorie des perturbations des ´ etats li´ es,”Nuclear Physics, vol. 6, pp. 329–347, 1958

1958

[4] [4]

Relation between the Anderson and Kondo Hamiltonians,

J. R. Schrieffer and P. A. Wolff, “Relation between the Anderson and Kondo Hamiltonians,” Phys. Rev., vol. 149, pp. 491–492, Sep 1966

1966

[5] [5]

Schrieffer–Wolff transformation for quantum many- body systems,

S. Bravyi, D. P. DiVincenzo, and D. Loss, “Schrieffer–Wolff transformation for quantum many- body systems,”Annals of Physics, vol. 326, no. 10, pp. 2793–2826, 2011. 7

2011

[6] [6]

Universal Error Bound for Constrained Quantum Dynamics,

Z. Gong, N. Yoshioka, N. Shibata, and R. Hamazaki, “Universal Error Bound for Constrained Quantum Dynamics,”Phys. Rev. Lett., vol. 124, p. 210606, May 2020

2020

[7] [7]

Error bounds for constrained dynamics in gapped quantum systems: Rigorous results and generalizations,

Z. Gong, N. Yoshioka, N. Shibata, and R. Hamazaki, “Error bounds for constrained dynamics in gapped quantum systems: Rigorous results and generalizations,”Phys. Rev. A, vol. 101, p. 052122, May 2020

2020

[8] [8]

J. J. Sakurai and J. J. Napolitano,Modern Quantum Mechanics. New York: Pearson, 2nd ed., 2014

2014

[9] [9]

Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model,

F. Haldane, “Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model,”Physics Letters A, vol. 93, no. 9, pp. 464–468, 1983

1983

[10] [10]

Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semi- classically Quantized Solitons of the One-Dimensional Easy-Axis N´ eel State,

F. D. M. Haldane, “Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semi- classically Quantized Solitons of the One-Dimensional Easy-Axis N´ eel State,”Phys. Rev. Lett., vol. 50, pp. 1153–1156, Apr 1983

1983

[11] [11]

Quantum spin chains and the Haldane gap,

I. Affleck, “Quantum spin chains and the Haldane gap,”Journal of Physics: Condensed Matter, vol. 1, p. 3047, may 1989

1989

[12] [12]

E. U. Condon and G. H. Shortley,The Theory of Atomic Spectra. Cambridge: Cambridge University Press, 1935. Appendices A Coupled spins Note that we can rewrite the initial Hamiltonian as H0 =−βS 1 ·S 2 =− β 2 J2 −S 2 1 −S 2 2 ,(18) where we have denoted the total spin byJ =S 1 +S 2. Following standard procedures for combining angular momentum, a complete s...

1935

[13] [13]

Prob(outside subspace, t) =⟨Ψ 1(t)|I−Π|Ψ 1(t)⟩ ≤ D(V) E0 2 (29) where |Ψ1(t)⟩=U 1(t)|Ψ 1(0)⟩=e −iH1t/ℏ |Ψ1(0)⟩

The state of the system has only a very small component outside the +1 eigenspace of Π for all time. Prob(outside subspace, t) =⟨Ψ 1(t)|I−Π|Ψ 1(t)⟩ ≤ D(V) E0 2 (29) where |Ψ1(t)⟩=U 1(t)|Ψ 1(0)⟩=e −iH1t/ℏ |Ψ1(0)⟩. 9 Proof ⟨Ψ1(t)|I−Π|Ψ 1(t)⟩=⟨Ψ 1(t)|(I−Π) 2 |Ψ1(t)⟩ =⟨Ψ 1(t)| I− H1 E0 + V E0 2 |Ψ1(t)⟩ =⟨Ψ 1(t)| I− H1 E0 2 + I− H1 E0 V E0 + V E0 I− H1 E0 + V ...

[14] [14]

And let 6 6Note that the trace distance between two pure states is given byd(|Ψ 1⟩,|Ψ 2⟩) = p 1− |⟨Ψ 2|Ψ1⟩|2

Consider now a second Hamiltonian H2 =E 0Π + ΠVΠ,(31) and let|Ψ 2(t)⟩be the state that evolves underH 2, starting with the same initial state, |Ψ2(0)⟩=|Ψ 1(0)⟩. And let 6 6Note that the trace distance between two pure states is given byd(|Ψ 1⟩,|Ψ 2⟩) = p 1− |⟨Ψ 2|Ψ1⟩|2. Hence, we can also use this result to bound the trace distance viad(|Ψ 1(t)⟩,|Ψ 2(t)⟩)...