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Seeing Spin: Geometry from a Schoolbag

Okupski Arkadiusz

Published Dec 23, 2025• 0 citations
Publisher

Abstract

We've been describing spin with equations. What if we could *see* it instead? For a century, electron spin has been a mathematical abstraction: a two-valued quantum property with no classical picture, described by SU(2) algebra and $4\pi$ rotations. This work turns the problem inside out. It asks: *What minimal geometry could generate such behavior?* The answer is deceptively simple: a Möbius strip coupling the electron's internal cycle to its observable boundary. But not just one strip—two. A loop ($G_0$) and a figure-eight ($G_8$), energetically degenerate yet topologically distinct. Their choice corresponds to spin $+\tfrac{1}{2}$ or $-\tfrac{1}{2}$. From this single geometric postulate, the model *mechanically* explains: - **Why there are exactly two spin states** (no continuum)- **Why spin "remembers" its state** after a magnetic field is turned off (topological memory)- **How the $g \simeq 2$ relation** finds a striking qualitative parallel in a fundamental $2{:}1$ geometric ratio between the electron's internal cycle and its boundary- **How spinorial $4\pi$ rotation** emerges from Möbius topology This is **"pencil-case physics"**: no Lagrangians, no operators, just visualizable construction. It's a *sonar*—a conceptual probe mapping possible shapes beneath quantum formalism. The model doesn't replace quantum mechanics; it asks what geometry might be whispering beneath its equations. For physicists tired of "because the math says so," this paper offers a "here's how it *could* work." It's an invitation to imagine spin not as a vector in Hilbert space, but as a twist in the fabric of the electron itself.

DOI
10.5281/zenodo.18039596

Related papers

Using Möbius strip to describe non-integral spin of electrons

Arindam Raj

Abstract This article presents a simple demonstration that can be used in high school/introductory college level courses to explain the meaning of ½-spin of fermions like electrons. One can use a simple-to-construct Möbius strip to distinguish 2π and 4π rotations of a point moving on the strip. This idea can then be extended to a pointer/arrow (in classroom settings, say a pen representing a classical analogue to the intrinsic spin vector), to show how a 4π rotation is needed to return the pointer, to its initial configuration with respect to the strip.

Physics Education•Jun 18, 2020•0 citations

The Physical origin of Spin and the Wrong Turn of 1925

D. J. Hallman

In 1925 the concept of electron spin was introduced to explain two experimental observations: the splitting of silver atom beams in an inhomogeneous magnetic field (Stern–Gerlach, 1922), and the fine structure of atomic spectra. The physical picture proposed — an electron spinning on its own axis — was immediately shown by Hendrik Lorentz to require surface velocities exceeding the speed of light. The paper was published anyway. Dirac subsequently rescued the two-valuedness mathematically, whereupon the physical question was declared irrelevant. For one hundred years the field has been unable to state what spin physically is. This paper traces the sequence of observations, inferences, and decisions that produced this situation and identifies two distinct wrong turns. The first wrong turn was publishing a model known to be physically impossible. Lorentz's calculation was correct — at the classical electron radius (2.818 fm), the required surface velocity is 274 times the speed of light. But the classical radius is not the electron's physical rotation radius. The correct rotation radius, derived from the Sagnac closure condition, is 285.5 fm — 101 times larger. At that radius, the rotation velocity is 0.822c, the bare magnetic moment is 0.608 μB, and Lorentz's objection does not arise. Both values are computed explicitly with no free parameters. The second wrong turn was a conflation. The closure geometry of the electron involves two independent numerical facts, both related to the number ½: (1) the least-work oscillation path that closes the Sagnac phase condition is a two-lobe ellipse, giving a winding number of n = ½ in the mass equation; (2) the electron's S³ closure surface presents two poles to the external field, giving a topological factor of 2 in the g-factor. In 1925, both facts were numerically present in the spectroscopic data, but they were collapsed into a single postulate: intrinsic angular momentum ℏ/2. That collapse created a half-integer angular momentum with no classical analogue, forced Dirac to construct the SU(2) spinor formalism to accommodate it, and caused the spinor to appear as a new kind of mathematical object requiring new physics — when it was always a description of closure surface topology. Every experimental result attributed to spin-½ as an intrinsic quantum property — two spots in Stern–Gerlach, the electron magnetic moment, Pauli exclusion, and fine structure — is reproduced by real rotation with two stable orientations, without superluminal motion, without abstract mathematical structure disconnected from physical content, and without the hundred-year admission of ignorance that has accompanied the standard treatment. The mathematical apparatus of quantum mechanics — the Dirac equation, the Pauli matrices, the SU(2) structure — remains valid. What changes is the interpretation. The spinor is not a mysterious new object. It is the natural representation of the two-pole topology of the S³ closure surface. The conflation of 1925 prevented this from being seen. Related work: The Sagnac closure condition and the γcause invariant used in this paper are derived in: D. J. Hallman, "The Sagnac Formula Inverted Reveals Mass, Gravity, and Particle Structure," Zenodo (2026). doi:10.5281/zenodo.20193279

•May 18, 2026•0 citations

The Wonder of Quantum Spin

Indubala I. Satija

Abstract The Wonder of Quantum Spin is a confection of the history and the science of quantum spin covering two centuries of the golden era in mathematics and physics. First glimpses of spin appeared nearly 200 years ago in the study of rotations where spinors emerged as a new entity that changes sign after a 360 degree rotation, reminiscent of the Mobius geometry. A century later, quantum spin described by the spinor was discovered in physics. Among many other things, it led to the discovery of antimatter, raised the possibility of parity violation, and gave the first warning that protons and neutrons are not elementary. The MRI—one of the most important medical advances of the twentieth century—is an application of quantum spin to humanity. As we approach the centennial of the discovery of spin in Nature, the spin mystique prevails as we have no clue of what exactly spinning is. On the other hand, the theoretical framework underlying the spin determines why we exist, explains the mysteries of the 3000 years-old phenomena of magnetism, is behind the precision measurements of electron and muon spin magnetic moments displaying a mindboggling confrontation between theory and experiment that serves as an ambassador to quantum physics. Some of the other frontiers of physics that revolve around spin-1/2 include exotic states of matter known as topological insulators and the qubit—the quantum bits for tomorrow’s computers.

•Jun 25, 2024•0 citations

The Square Root of the Hyperspherical Laplacian: A Geometric Foundation for Spin Two-Valuedness on S³ in the 0-Sphere Model

Satoshi Hanamura

[64 Foundational] The 0-Sphere Model treats the electron not as a structureless point but as a small internal geometric region whose phase trembles in time. This paper asks where one of the most stubborn features of that electron comes from: the fact that its spin has exactly two values, and that the particle must be turned through two full revolutions — not one — before it returns to itself. The proposed answer is that it comes from a single, very ordinary mathematical operation: taking a square root. — The idea in one breath — Picture the electron’s internal world as the surface of a four-dimensional ball — a closed, finite space with no edges. On such a closed space there is a natural “averaging” operation, and its allowed values come out evenly spaced and labelled by whole numbers (0, 1, 2, …). These whole numbers describe the orbital, direction-like part of the internal motion. Nothing surprising yet. The surprise appears when you take the square root of that averaging operation. A square root is always defined only up to a sign — √4 is both +2 and −2 — so its allowed values arrive in matched positive–negative pairs. Those two signs are exactly the two spin states: up and down. The same square root, applied historically by Dirac in 1928 to the equation of a relativistic particle, is what gave physics the electron’s spin in the first place. This paper’s claim is that one and the same operation shows up at two stages of the model and reduces the two-valuedness of spin to plain geometry. — Why this matters — Two structural facts fall out for free, where before they had to be put in by hand: Why the values are discrete. Because the internal space is closed and finite (compact), the allowed values can only be separated, whole-number-labelled steps — never a continuous smear. Why there are exactly two of them. Because the square-root step inevitably splits each value into a + and a − partner. Two signs, two spin sheets. The two sheets join smoothly into the double cover of the rotation group — the precise mathematical reason a spin one-half object needs two full turns to come home. In plain terms: walk once around the rotations and you land on the other sheet; walk around twice and you finally return to where you started. — What is new, and what is not — The square-root operation itself is classical and is claimed as no one’s discovery — it has been textbook material since 1928. What is new is the recognition that the very same operation appears at two stages of the model, and that together they reduce the two-valuedness of spin to the geometry of a closed surface. This replaces a picture of two states being secretly present at once with a cleaner, time-sequential one: the electron visits up, then down, in turn, rather than holding both simultaneously. — Key Contributions — Spin’s two values as a square root. The two spin sheets are the matched plus/minus values that any square root must take — here, the square root of the natural averaging operator on the closed internal space. Closedness gives discreteness. The fact that the allowed values are separated, whole-number steps follows from the internal space being compact (finite and edgeless). One operation across two stages. The same square root acts both on the geometric internal stage (giving a clean, discrete set of states) and on the historical relativistic stage (the one Dirac used in 1928), the two being linked by a standard mathematical bridge. The double cover as a branch structure. The two sheets are the two faces of the familiar “two-storey” surface that the square-root function naturally lives on; their joining is the rotation group’s double cover — the two-valuedness of the square root made visible. A firmer footing for earlier results. The half-angle and g = 2 results obtained earlier from the fiber-bundle picture are placed on an operator-theoretic footing. A correction that keeps the good part. The earliest spin paper’s reading of four simultaneous states is replaced by a time-sequential one, while its elegant two-storey (Riemann-surface) picture is kept — now on firmer ground. — Series Position — This paper continues the half-angle internal-energy programme and gives an operator-theoretic foundation to the geometric g = 2 result of Paper #48, consolidates the threads of Paper #50 and Paper #57, and corrects the four-state reading of Paper #3 while retaining its two-sheeted Riemann surface. Through a standard mathematical bridge it points toward the metric-emergence programme of Paper #51 and Paper #63 as the destination it does not itself develop. It belongs to the series founded by Paper #1. — References (Hanamura papers) — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 3 Distinguishing Electron Spin via Riemann Surface Guidance 10.5281/zenodo.17759634 10 Redefining Electron Spin and the Anomalous Magnetic Moment 10.5281/zenodo.17764997 48 Geometric Origin of g=2: U(1) Fiber Bundle 10.5281/zenodo.19227518 50 Rotation from Scalar Oscillation: Photon-Sphere Angular Momentum 10.5281/zenodo.19482145 51 Helical Trajectory, Line Integrals, Emergence of the Metric (v2) 10.5281/zenodo.20388056 57 Spherical Harmonic Imprinting and Gravitational Coupling 10.5281/zenodo.19932176 59 Proton-Trapped Electron and Spin-2 Gravitational Radiation 10.5281/zenodo.19932907 60 The Berry–Synge–Bonnet–Myers Triangle 10.5281/zenodo.19932922 63 From Line Integral to Covariant Derivative: A Reader's Map 10.5281/zenodo.20767589 — Series Context — The 0-Sphere Model is an ongoing research programme (2018–present) that derives spin, anomalous magnetic moment, Zitterbewegung, and emergent spacetime from the geometry and thermodynamics of a two-kernel electron model. All papers in the series are archived on Zenodo: Zenodo search: Hanamura, Satoshi

•Jun 27, 2026•0 citations

Modelling spin

Stefan Heusler, Malte S. Ubben

Abstract Based on a geometric view on the famous Dirac belt trick in terms of a Möbius strip, we propose a simple haptic model for spin states as standing waves in non-trivial geometry. We compare two representations of the model: the spinor representation in SU (2), which geometrically is described by ‘Dirac belt’ states on S 3 , and the well-known Bloch-sphere representation on S 2 . We show that after Hopf-mapping S 3 → S 2 , the position and number of nodal points are sufficient to describe the spin state and make the relation between the representations on S 2 and on S 3 explicit in simple haptic models. Our approach is well-suited for graduate physics courses, and provides a thorough understanding of the complex geometry of spin states beyond the usual Bloch-sphere visualisation. It turns out that an even number 2 l of nodal points can be related to nodal lines of spherical harmonics Y lm , an odd number can be traced back to spin states. In this sense, our model provides a unified view for orbital and spin states based on a simple picture of standing waves.

European Journal of Physics•Sep 24, 2018•5 citations

Early History of Spin

Most students of science and engineering know that every elementary particle, such as electrons, neutrons, photons, neutrinos, etc., has a quantum mechanical property called “spin” which can be measured (perhaps not easily, but at least in principle) and has a quantized value, including zero. The vast majority of these students mentally visualize spin as the angular momentum associated with the elementary particle spinning or rotating about its own axis (like a top or a planetary object). This mental picture, although convenient and comforting, is actually somewhat crude and certainly incomplete. Landau and Lifshitz, in their classic textbook on quantum mechanics [1], wrote “[the spin] property of elementary particles is peculiar to quantum theory. [It] has no classical interpretation... It would be wholly meaningless to imagine the ‘intrinsic’ angular momentum of an elementary particle as being the result of its rotation about its own axis.” The simplistic notion of self-rotation about an axis, shown in Fig. 1.1, can- not explain many features of spin, such as why its magnitude cannot assume continuous values and why it is quantized to certain specific values. It also causes serious problems if taken too literally. As we will see later (Problem 1.2), if we think of an electron as a solid sphere of radius equal to the Lorentz radius e2/(4πǫ0m0c 2) (where e is the electron’s charge, m0 is the mass, c is the speed of light in vacuum, and ǫ0 is the dielectric constant of vacuum), then the velocity on the surface of a rotating electron would have to be many times the velocity of light in vacuum if such a rotation were to generate an angular momentum equal to the electron’s spin. Obviously that would not be permitted by the theory of relativity. Indeed, a deep understanding of quantum mechanics is required to understand how the spin property comes about. Its origin is in relativistic quantum mechanics and really was first appreciated by Paul Andrew Maurice Dirac when he derived the Dirac equation which is the cornerstone of relativistic quantum mechanics. Richard Feynman, noted for his distaste for mysticism in physics, wrote about the notion of “spin”: “It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is down deep in relativistic quantum mechanics.

CRC Press eBooks•Sep 18, 2015•0 citations
We've been describing spin with equations. What if we could see it instead? For a century, electron spin has been a mathematical abstraction: a two-valued quantum property with no classical picture, described by SU(2) algebra and 4π4\pi4π rotations. This work turns the problem inside out. It asks: What minimal geometry could generate such behavior? The answer is deceptively simple: a Möbius strip coupling the electron's internal cycle to its observable boundary. But not just one strip—two. A loop (G0G_0G0​) and a figure-eight (G8G_8G8​), energetically degenerate yet topologically distinct. Their choice corresponds to spin +12+\tfrac{1}{2}+21​ or −12-\tfrac{1}{2}−21​. From this single geometric postulate, the model mechanically explains: - Why there are exactly two spin states (no continuum)- Why spin "remembers" its state after a magnetic field is turned off (topological memory)- How the g≃2g \simeq 2g≃2 relation finds a striking qualitative parallel in a fundamental 2:12{:}12:1 geometric ratio between the electron's internal cycle and its boundary- How spinorial 4π4\pi4π rotation emerges from Möbius topology This is "pencil-case physics": no Lagrangians, no operators, just visualizable construction. It's a sonar—a conceptual probe mapping possible shapes beneath quantum formalism. The model doesn't replace quantum mechanics; it asks what geometry might be whispering beneath its equations. For physicists tired of "because the math says so," this paper offers a "here's how it could work." It's an invitation to imagine spin not as a vector in Hilbert space, but as a twist in the fabric of the electron itself.
Abstract This article presents a simple demonstration that can be used in high school/introductory college level courses to explain the meaning of ½-spin of fermions like electrons. One can use a simple-to-construct Möbius strip to distinguish 2π and 4π rotations of a point moving on the strip. This idea can then be extended to a pointer/arrow (in classroom settings, say a pen representing a classical analogue to the intrinsic spin vector), to show how a 4π rotation is needed to return the pointer, to its initial configuration with respect to the strip.
In 1925 the concept of electron spin was introduced to explain two experimental observations: the splitting of silver atom beams in an inhomogeneous magnetic field (Stern–Gerlach, 1922), and the fine structure of atomic spectra. The physical picture proposed — an electron spinning on its own axis — was immediately shown by Hendrik Lorentz to require surface velocities exceeding the speed of light. The paper was published anyway. Dirac subsequently rescued the two-valuedness mathematically, whereupon the physical question was declared irrelevant. For one hundred years the field has been unable to state what spin physically is. This paper traces the sequence of observations, inferences, and decisions that produced this situation and identifies two distinct wrong turns. The first wrong turn was publishing a model known to be physically impossible. Lorentz's calculation was correct — at the classical electron radius (2.818 fm), the required surface velocity is 274 times the speed of light. But the classical radius is not the electron's physical rotation radius. The correct rotation radius, derived from the Sagnac closure condition, is 285.5 fm — 101 times larger. At that radius, the rotation velocity is 0.822c, the bare magnetic moment is 0.608 μB, and Lorentz's objection does not arise. Both values are computed explicitly with no free parameters. The second wrong turn was a conflation. The closure geometry of the electron involves two independent numerical facts, both related to the number ½: (1) the least-work oscillation path that closes the Sagnac phase condition is a two-lobe ellipse, giving a winding number of n = ½ in the mass equation; (2) the electron's S³ closure surface presents two poles to the external field, giving a topological factor of 2 in the g-factor. In 1925, both facts were numerically present in the spectroscopic data, but they were collapsed into a single postulate: intrinsic angular momentum ℏ/2. That collapse created a half-integer angular momentum with no classical analogue, forced Dirac to construct the SU(2) spinor formalism to accommodate it, and caused the spinor to appear as a new kind of mathematical object requiring new physics — when it was always a description of closure surface topology. Every experimental result attributed to spin-½ as an intrinsic quantum property — two spots in Stern–Gerlach, the electron magnetic moment, Pauli exclusion, and fine structure — is reproduced by real rotation with two stable orientations, without superluminal motion, without abstract mathematical structure disconnected from physical content, and without the hundred-year admission of ignorance that has accompanied the standard treatment. The mathematical apparatus of quantum mechanics — the Dirac equation, the Pauli matrices, the SU(2) structure — remains valid. What changes is the interpretation. The spinor is not a mysterious new object. It is the natural representation of the two-pole topology of the S³ closure surface. The conflation of 1925 prevented this from being seen. Related work: The Sagnac closure condition and the γcause invariant used in this paper are derived in: D. J. Hallman, "The Sagnac Formula Inverted Reveals Mass, Gravity, and Particle Structure," Zenodo (2026). doi:10.5281/zenodo.20193279
Abstract The Wonder of Quantum Spin is a confection of the history and the science of quantum spin covering two centuries of the golden era in mathematics and physics. First glimpses of spin appeared nearly 200 years ago in the study of rotations where spinors emerged as a new entity that changes sign after a 360 degree rotation, reminiscent of the Mobius geometry. A century later, quantum spin described by the spinor was discovered in physics. Among many other things, it led to the discovery of antimatter, raised the possibility of parity violation, and gave the first warning that protons and neutrons are not elementary. The MRI—one of the most important medical advances of the twentieth century—is an application of quantum spin to humanity. As we approach the centennial of the discovery of spin in Nature, the spin mystique prevails as we have no clue of what exactly spinning is. On the other hand, the theoretical framework underlying the spin determines why we exist, explains the mysteries of the 3000 years-old phenomena of magnetism, is behind the precision measurements of electron and muon spin magnetic moments displaying a mindboggling confrontation between theory and experiment that serves as an ambassador to quantum physics. Some of the other frontiers of physics that revolve around spin-1/2 include exotic states of matter known as topological insulators and the qubit—the quantum bits for tomorrow’s computers.
[64 Foundational] The 0-Sphere Model treats the electron not as a structureless point but as a small internal geometric region whose phase trembles in time. This paper asks where one of the most stubborn features of that electron comes from: the fact that its spin has exactly two values, and that the particle must be turned through two full revolutions — not one — before it returns to itself. The proposed answer is that it comes from a single, very ordinary mathematical operation: taking a square root. — The idea in one breath — Picture the electron’s internal world as the surface of a four-dimensional ball — a closed, finite space with no edges. On such a closed space there is a natural “averaging” operation, and its allowed values come out evenly spaced and labelled by whole numbers (0, 1, 2, …). These whole numbers describe the orbital, direction-like part of the internal motion. Nothing surprising yet. The surprise appears when you take the square root of that averaging operation. A square root is always defined only up to a sign — √4 is both +2 and −2 — so its allowed values arrive in matched positive–negative pairs. Those two signs are exactly the two spin states: up and down. The same square root, applied historically by Dirac in 1928 to the equation of a relativistic particle, is what gave physics the electron’s spin in the first place. This paper’s claim is that one and the same operation shows up at two stages of the model and reduces the two-valuedness of spin to plain geometry. — Why this matters — Two structural facts fall out for free, where before they had to be put in by hand: Why the values are discrete. Because the internal space is closed and finite (compact), the allowed values can only be separated, whole-number-labelled steps — never a continuous smear. Why there are exactly two of them. Because the square-root step inevitably splits each value into a + and a − partner. Two signs, two spin sheets. The two sheets join smoothly into the double cover of the rotation group — the precise mathematical reason a spin one-half object needs two full turns to come home. In plain terms: walk once around the rotations and you land on the other sheet; walk around twice and you finally return to where you started. — What is new, and what is not — The square-root operation itself is classical and is claimed as no one’s discovery — it has been textbook material since 1928. What is new is the recognition that the very same operation appears at two stages of the model, and that together they reduce the two-valuedness of spin to the geometry of a closed surface. This replaces a picture of two states being secretly present at once with a cleaner, time-sequential one: the electron visits up, then down, in turn, rather than holding both simultaneously. — Key Contributions — Spin’s two values as a square root. The two spin sheets are the matched plus/minus values that any square root must take — here, the square root of the natural averaging operator on the closed internal space. Closedness gives discreteness. The fact that the allowed values are separated, whole-number steps follows from the internal space being compact (finite and edgeless). One operation across two stages. The same square root acts both on the geometric internal stage (giving a clean, discrete set of states) and on the historical relativistic stage (the one Dirac used in 1928), the two being linked by a standard mathematical bridge. The double cover as a branch structure. The two sheets are the two faces of the familiar “two-storey” surface that the square-root function naturally lives on; their joining is the rotation group’s double cover — the two-valuedness of the square root made visible. A firmer footing for earlier results. The half-angle and g = 2 results obtained earlier from the fiber-bundle picture are placed on an operator-theoretic footing. A correction that keeps the good part. The earliest spin paper’s reading of four simultaneous states is replaced by a time-sequential one, while its elegant two-storey (Riemann-surface) picture is kept — now on firmer ground. — Series Position — This paper continues the half-angle internal-energy programme and gives an operator-theoretic foundation to the geometric g = 2 result of Paper #48, consolidates the threads of Paper #50 and Paper #57, and corrects the four-state reading of Paper #3 while retaining its two-sheeted Riemann surface. Through a standard mathematical bridge it points toward the metric-emergence programme of Paper #51 and Paper #63 as the destination it does not itself develop. It belongs to the series founded by Paper #1. — References (Hanamura papers) — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 3 Distinguishing Electron Spin via Riemann Surface Guidance 10.5281/zenodo.17759634 10 Redefining Electron Spin and the Anomalous Magnetic Moment 10.5281/zenodo.17764997 48 Geometric Origin of g=2: U(1) Fiber Bundle 10.5281/zenodo.19227518 50 Rotation from Scalar Oscillation: Photon-Sphere Angular Momentum 10.5281/zenodo.19482145 51 Helical Trajectory, Line Integrals, Emergence of the Metric (v2) 10.5281/zenodo.20388056 57 Spherical Harmonic Imprinting and Gravitational Coupling 10.5281/zenodo.19932176 59 Proton-Trapped Electron and Spin-2 Gravitational Radiation 10.5281/zenodo.19932907 60 The Berry–Synge–Bonnet–Myers Triangle 10.5281/zenodo.19932922 63 From Line Integral to Covariant Derivative: A Reader's Map 10.5281/zenodo.20767589 — Series Context — The 0-Sphere Model is an ongoing research programme (2018–present) that derives spin, anomalous magnetic moment, Zitterbewegung, and emergent spacetime from the geometry and thermodynamics of a two-kernel electron model. All papers in the series are archived on Zenodo: Zenodo search: Hanamura, Satoshi
Abstract Based on a geometric view on the famous Dirac belt trick in terms of a Möbius strip, we propose a simple haptic model for spin states as standing waves in non-trivial geometry. We compare two representations of the model: the spinor representation in SU (2), which geometrically is described by ‘Dirac belt’ states on S 3 , and the well-known Bloch-sphere representation on S 2 . We show that after Hopf-mapping S 3 → S 2 , the position and number of nodal points are sufficient to describe the spin state and make the relation between the representations on S 2 and on S 3 explicit in simple haptic models. Our approach is well-suited for graduate physics courses, and provides a thorough understanding of the complex geometry of spin states beyond the usual Bloch-sphere visualisation. It turns out that an even number 2 l of nodal points can be related to nodal lines of spherical harmonics Y lm , an odd number can be traced back to spin states. In this sense, our model provides a unified view for orbital and spin states based on a simple picture of standing waves.
Most students of science and engineering know that every elementary particle, such as electrons, neutrons, photons, neutrinos, etc., has a quantum mechanical property called “spin” which can be measured (perhaps not easily, but at least in principle) and has a quantized value, including zero. The vast majority of these students mentally visualize spin as the angular momentum associated with the elementary particle spinning or rotating about its own axis (like a top or a planetary object). This mental picture, although convenient and comforting, is actually somewhat crude and certainly incomplete. Landau and Lifshitz, in their classic textbook on quantum mechanics [1], wrote “[the spin] property of elementary particles is peculiar to quantum theory. [It] has no classical interpretation... It would be wholly meaningless to imagine the ‘intrinsic’ angular momentum of an elementary particle as being the result of its rotation about its own axis.” The simplistic notion of self-rotation about an axis, shown in Fig. 1.1, can- not explain many features of spin, such as why its magnitude cannot assume continuous values and why it is quantized to certain specific values. It also causes serious problems if taken too literally. As we will see later (Problem 1.2), if we think of an electron as a solid sphere of radius equal to the Lorentz radius e2/(4πǫ0m0c 2) (where e is the electron’s charge, m0 is the mass, c is the speed of light in vacuum, and ǫ0 is the dielectric constant of vacuum), then the velocity on the surface of a rotating electron would have to be many times the velocity of light in vacuum if such a rotation were to generate an angular momentum equal to the electron’s spin. Obviously that would not be permitted by the theory of relativity. Indeed, a deep understanding of quantum mechanics is required to understand how the spin property comes about. Its origin is in relativistic quantum mechanics and really was first appreciated by Paul Andrew Maurice Dirac when he derived the Dirac equation which is the cornerstone of relativistic quantum mechanics. Richard Feynman, noted for his distaste for mysticism in physics, wrote about the notion of “spin”: “It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is down deep in relativistic quantum mechanics.