2026-08-14
Electron spin as a consequence of the double orbital cycle in the geometric model
Electron spin is usually introduced as an intrinsic quantum property, not reducible to the ordinary rotation of a particle. However, the characteristic features of one-half spin two possible projections, a half-angle, and the restoration of the state only after two full rotations allow for a clear geometric interpretation. In the proposed model, the electron's initial orbit is split into two nearly identical orbits in the same plane, which the electron traverses sequentially. One rotation transfers the electron's motion to the second orbit, and the full internal cycle is completed only after the second rotation. 
\[\tag{9} J(a,0) \longleftrightarrow \uparrow, \qquad J(0,a) \longleftrightarrow \downarrow. \] 
1. Problem Statement
For a particle with spin, one second dimension of the spin projection onto an arbitrary axis yields only two results:
\[\tag{1} S_{\mathbf n}=+\frac{\hbar}{2}, \qquad S_{\mathbf n}=-\frac{\hbar}{2}. \] If the direction of measurement is rotated relative to the original spin direction by an angle \(\theta\), the probabilities of the two outcomes are given by
\[\tag{2} P_{\uparrow}(\theta) = \cos^2\frac{\theta}{2}, \qquad P_{\downarrow}(\theta) = \sin^2\frac{\theta}{2}. \] Another characteristic property is that after a spatial rotation of \(2\pi\), the state changes sign and is fully restored only after a rotation of \(4\pi\):
\[\tag{3} \Psi(2\pi)=-\Psi(0), \qquad \Psi(4\pi)=\Psi(0). \] The goal of this paper is to show how these properties can arise from the two-orbital geometry of \(J\) motion without taking the half-angle as the initial formula.
2. Initial motion J
The motion in the model is given by the expression
\[\tag{4} J(t) = \jmath^{\,a}(-\jmath)^{\,b}, \qquad V(t)=cJ(t), \] where the phase parameter \(a\) is related to time and angular frequency:
\[\tag{5} a=\varpi t, \qquad \varpi\pi=\omega. \] In general, the second parameter can be written in terms of relative velocity:
\[\tag{6} b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] To describe two opposite modes of internal motion, we distinguish two limit states:
\[\tag{7} J_{\uparrow}(a) \equiv J(a,0) = \jmath^{\,a}, \] \[\tag{8} J_{\downarrow}(a) \equiv J(0,a) = (-\jmath)^{\,a}. \] Important: \(J(a,0)\) and \(J(0,a)\) do not denote the two split orbitals separately. Each of these expressions describes one of two possible modes of motion along the entire two-orbital trajectory. The state \(J(a,0)\) is associated with spin up, and the state \(J(0,a)\) is associated with spin down.
3. Orbital Splitting
Let the original orbit \(C\) split into two close contours \(C_1\) and \(C_2\):
\[\tag{10} C \longrightarrow \{C_1,C_2\}. \] Both orbits are in the same geometric plane and have similar radii. They can be represented as
\[\tag{11} r_1 = r-\frac{\Delta r}{2}, \qquad r_2 = r+\frac{\Delta r}{2}, \qquad \Delta r\ll r. \] Splitting does not create two independent particles and does not divide the motion into two simultaneously traversed trajectories. The electron first completes a revolution along the first circuit, then moves to the second circuit and completes another revolution:
\[\tag{12} C_1 \longrightarrow C_2 \longrightarrow C_1. \] The transition points must be part of a continuous geometric trajectory. The specific form of the transitions depends on a more detailed model of the splitting, but for the present conclusion, it is only essential that the orbital number changes after the first revolution and is restored after the second.
4. Two-turn inner cycle
We denote the initial state of the complete trajectory by the pair \((C_1,0)\), where the first element indicates the orbit and the second the angular position. After one revolution, the spatial angle is again zero modulo \(2\pi\), but the electron is now in its second orbit:
\[\tag{13} (C_1,0) \xrightarrow{\ 2\pi\ } (C_2,0). \] Therefore, one revolution does not yet return the complete geometric configuration to its original state. The second revolution returns the electron from \(C_2\) to \(C_1\):
\[\tag{14} (C_2,0) \xrightarrow{\ 2\pi\ } (C_1,0). \] The complete path is as follows
\[\tag{15} (C_1,0)\xrightarrow{\ 2\pi\ } (C_2,0) \xrightarrow{\ 2\pi\ } (C_1,0). \] Therefore, the angular extent of the complete inner cycle is
\[\tag{16} \boxed{ \Theta_J=4\pi }. \] Each ordinary angular position \(\theta\) corresponds to two interior points:
\[\tag{17} (\theta,C_1), \qquad (\theta,C_2). \] Thus, the split orbit forms a two-sheeted covering of the ordinary angular cycle. The spatial position repeats every 2 pi, but the complete internal state repeats only every 4 pi.
5. Geometric Origin of the Half Angle
The phase of any closed cyclic state changes by 2 pi per complete cycle. In the geometry under consideration, a complete cycle has an angular length of 4 pi. Therefore, the internal phase (chi) should be normalized relative to the complete two-orbital path:
[tag{18} chi = 2 pi frac {theta}{Theta_J}. \] Substituting \(\Theta_J=4\pi\), we obtain
\[\tag{19} \boxed{ \chi=\frac{\theta}{2} }. \] Thus, the half-angle is not introduced into the model from the outside. It appears because one complete internal cycle corresponds to two consecutive spatial rotations:
\[\tag{20} \theta:0\longrightarrow4\pi \qquad\Longleftrightarrow\qquad \chi:0\longrightarrow2\pi. \] The main geometric result is the ratio \(2\pi/4\pi=1/2\). The internal phase of a two-orbital state changes half as slowly as the accumulated spatial angle.
6. Two Spin Regimes and Projections
Two opposite modes of motion along the same two-orbital trajectory are described by the expressions \(J(a,0)\) and \(J(0,a)\). Taking into account the half-phase, their transformation upon rotation can be written as
\[\tag{21} \Psi_{\uparrow}(\theta) = e^{-i\theta/2}J(a,0), \] \[\tag{22} \Psi_{\downarrow}(\theta) = e^{+i\theta/2}J(0,a). \] The generator of rotation about the angle \(\theta\) has the form
\[\tag{23} \widehat S_{\mathbf n} = i\hbar\frac{\partial}{\partial\theta}. \] For the mode \(J(a,0)\) we obtain
\[\tag{24} \widehat S_{\mathbf n}\Psi_{\uparrow} = +\frac{\hbar}{2}\Psi_{\uparrow}, \] a for mode \(J(0,a)\)
\[\tag{25} \widehat S_{\mathbf n}\Psi_{\downarrow} = -\frac{\hbar}{2}\Psi_{\downarrow}. \] Therefore, the two regimes of two-orbital motion correspond to the two observed projections:
\[\tag{26} \boxed{ \begin{aligned} J(a,0) &\longleftrightarrow S_{\mathbf n}=+\frac{\hbar}{2}\\[1mm] J(0,a) &\longleftrightarrow S_{\mathbf n}=-\frac{\hbar}{2}. \end{aligned} } \] The one-half factor appears because the derivative of the internal phase with respect to the spatial angle is
\[\tag{27} \frac{\partial\chi}{\partial\theta} = \frac12. \] 7. Rotations by Two Pi and Four Pi
After one spatial rotation
\[\tag{28} \theta=2\pi, \qquad \chi=\pi. \] For both modes, the phase factor becomes equal to minus one:
\[\tag{29} e^{\pm i\pi}=-1. \] Therefore
\[\tag{30} \boxed{ \Psi_J(2\pi)=-\Psi_J(0) }. \] Geometrically, this means a transition from the first split orbit to the second. The spatial angular coordinate has been repeated, but the full two-orbital route is not yet complete.
After the second revolution
\[\tag{31} \theta=4\pi, \qquad \chi=2\pi, \] and the phase factor returns to unity:
\[\tag{32} e^{\pm i2\pi}=1. \] Therefore,
\[\tag{33} \boxed{ \Psi_J(4\pi)=\Psi_J(0) }. \] The overall sign of a state does not affect the probability of an individual outcome, but can manifest itself through interference with a state that has not undergone the corresponding rotation.
8. Changing the Measurement Axis
So far, we have considered phase accumulation during rotation. Now let's choose a new measurement axis that forms an angle of \(\theta\) with the original axis. We denote the two modes of motion as
\[\tag{34} |\uparrow\rangle_J \equiv J(a,0), \qquad |\downarrow\rangle_J \equiv J(0,a). \] The state directed upward relative to the original axis is decomposed in the basis of the new axis as
\[\tag{35} \boxed{ |\Psi(\theta)\rangle_J = \cos\frac{\theta}{2}|\uparrow\rangle_J + \sin\frac{\theta}{2}|\downarrow\rangle_J }. \] For an arbitrary azimuth angle \(\varphi\), the relative phase appears:
\[\tag{36} |\Psi(\theta,\varphi)\rangle_J = \cos\frac{\theta}{2}|\uparrow\rangle_J + e^{i\varphi} \sin\frac{\theta}{2}|\downarrow\rangle_J. \] The coplanarity of the physical orbits \(C_1\) and \(C_2\) does not imply the coincidence of the regimes \(J(a,0)\) and \(J(0,a)\). For a probabilistic description, these two alternative regimes must form independent normalized states:
\[\tag{37} {}_J\langle\uparrow|\uparrow\rangle_J=1, \qquad {}_J\langle\downarrow|\downarrow\rangle_J=1, \qquad {}_J\langle\uparrow|\downarrow\rangle_J=0. \] Then the full norm is preserved:
\[\tag{38} \|\Psi\|^2 = \cos^2\frac{\theta}{2} + \sin^2\frac{\theta}{2} =1. \] If the probability of registering a mode is equal to the fraction of the quadratic norm that falls on this mode, then
\[\tag{39} \boxed{ P_{\uparrow}(\theta) = \cos^2\frac{\theta}{2}, \qquad P_{\downarrow}(\theta) = \sin^2\frac{\theta}{2} }. \] The average spin projection is
\[\tag{40} \begin{aligned} \langle S_{\mathbf n}\rangle &= +\frac{\hbar}{2}P_{\uparrow} - \frac{\hbar}{2}P_{\downarrow}\\ &= \frac{\hbar}{2}\cos\theta. \end{aligned} \] 9. The Stern-Gerlach Experiment
The Stern-Gerlach experiment is a direct demonstration of the discreteness of the magnetic moment projection. The historic experiment used a beam of neutral silver atoms, not free electrons. In the ground state of the silver atom, the outermost electron has zero orbital angular momentum, so the observed splitting is primarily due to its spin. The use of neutral atoms also avoids strong deflection of the entire beam by the usual Lorentz force.
9.1. Non-uniform magnetic field
The potential energy of the magnetic moment in the field is
\[\tag{41} U = -\boldsymbol{\mu}\cdot\mathbf B. \] If the field is predominantly directed along the z-axis and varies along this coordinate, the force is
\[\tag{42} F_z = -\frac{\partial U}{\partial z} \simeq \mu_z\frac{\partial B_z}{\partial z}. \] In a uniform field, opposing moments acquire different energies, but no constant separating force arises. For spatial beam splitting, the gradient \(\partial B_z/\partial z\neq0\) is required.
9.2. The Relationship between Spin and Magnetic Moment
For an electron, the magnetic moment is opposite to the spin due to its negative charge:
\[\tag{43} \boldsymbol{\mu}_e = -g\frac{e}{2m_e}\mathbf S. \] Via the Bohr magneton
\[\tag{44} \mu_B = \frac{e\hbar}{2m_e} \] we obtain for two projections
\[\tag{45} \begin{aligned} J(a,0): \quad S_z&=+\frac{\hbar}{2}, & \mu_z&=-\frac{g}{2}\mu_B\\[1mm] J(0,a): \quad S_z&=-\frac{\hbar}{2}, & \mu_z&=+\frac{g}{2}\mu_B. \end{aligned} \] At \(g\approx2\), the magnetic projections are approximately equal to \(\mp\mu_B\). Therefore, it is necessary to distinguish between the spin sign and the direction of the deviation: the spin-up state of an electron has a downward-directed magnetic moment.
The two-orbital geometry in this work derives the half-phase and spin projections. The relationship between the magnetic moment and spin and the magnitude of the \(g\) factor are currently used as a physical correspondence with standard electrodynamics. Their independent geometric derivation is a separate problem.
9.3. Two Modes in a Magnetic Analyzer
Let the analyzer magnet be oriented along the \(z\) axis. In the two-orbital model, two regimes are possible:
\[\tag{46} J(a,0) \longleftrightarrow S_z=+\frac{\hbar}{2}, \qquad J(0,a) \longleftrightarrow S_z=-\frac{\hbar}{2}. \] The corresponding forces have opposite signs:
\[\tag{47} \begin{aligned} F_z[J(a,0)] &= -\frac{g}{2}\mu_B \frac{\partial B_z}{\partial z}\\[1mm] F_z[J(0,a)] &= +\frac{g}{2}\mu_B \frac{\partial B_z}{\partial z}. \end{aligned} \] Therefore, the first part of the beam is deflected in one direction, and the second in the opposite direction. What appears on the screen is not a continuous strip, but two separate spots.
Within the proposed geometry, the absence of intermediate spots is explained by the fact that the magnet interacts not with an arbitrary classical orientation of a single orbit, but with one of two admissible modes of a complete two-orbital cycle:
\[\tag{48} J(a,0) \quad\text{or}\quad J(0,a). \] The distance between the spots is determined by the field gradient, the time of travel inside the magnet, the beam velocity, and the subsequent free flight. If the atom is in the field for a time of \(\tau\), the acquired transverse momentum is approximately equal to
\[\tag{49} \Delta p_z \simeq F_z\tau. \] The corresponding change in the transverse velocity of an atom of mass \(M\):
\[\tag{50} \Delta v_z \simeq \frac{F_z\tau}{M}. \] After free prFor a flight of duration \(T\), the additional shift is
\[\tag{51} \Delta z \simeq \frac{F_z\tau T}{M}. \] For two opposite projections, the forces have opposite signs, so the distance between the spot centers is approximately equal to twice the absolute value of this shift.
9.4. Unprepared Beam
If the incoming beam is unpolarized, it lacks a preferred spin direction. Relative to the magnet axis, the two projections meet with equal probability:
\[\tag{52} P_{\uparrow}=P_{\downarrow}=\frac12. \] Therefore, the intensities of the two spots are ideally equal. The magnet does not create a third state and does not decompose a single classical orbit into a continuous set of directions. It spatially separates two admissible modes \(J(a,0)\) and \(J(0,a)\) relative to the chosen axis.
9.5. Beam Prepared in the Upward State
Suppose that the first magnet selects only the channel
\[\tag{53} J(a,0) \longleftrightarrow S_z=+\frac{\hbar}{2}. \] If the second analyzer is oriented along the same axis, the two-orbital mode does not change:
\[\tag{54} P_{\uparrow\rightarrow\uparrow}(0)=1, \qquad P_{\uparrow\rightarrow\downarrow}(0)=0. \] The entire transmitted beam reenters the same channel. This means that the first analyzer not only divided the beam spatially, but also prepared a specific two-orbital mode relative to its axis.
9.6. Rotation of the Second Analyzer
Let the axis of the second analyzer be rotated by an angle \(\theta\) relative to the axis of the first. The state \(J(a,0)\) produced by the first magnet relative to the new axis is represented as
\[\tag{55} |J(a,0)\rangle_z = \cos\frac{\theta}{2}|J(a,0)\rangle_{\mathbf n} + e^{i\varphi} \sin\frac{\theta}{2}|J(0,a)\rangle_{\mathbf n}. \] The second magnet links these two amplitudes with two spatially diverging trajectories. Therefore, the probabilities of the outcomes are equal
\[\tag{56} \boxed{ \begin{aligned} P_{\uparrow\rightarrow\uparrow}(\theta) &= \cos^2\frac{\theta}{2}\\[1mm] P_{\uparrow\rightarrow\downarrow}(\theta) &= \sin^2\frac{\theta}{2}. \end{aligned} } \] For perpendicular analyzers \(\theta=\pi/2\):
\[\tag{57} P_{\uparrow\rightarrow\uparrow} = P_{\uparrow\rightarrow\downarrow} = \frac12. \] For oppositely oriented axes \(\theta=\pi\):
\[\tag{58} P_{\uparrow\rightarrow\uparrow}=0, \qquad P_{\uparrow\rightarrow\downarrow}=1. \] Thus, the two-orbital model relates the observed angular dependence of the experiment to the fact that the internal phase angle is equal to half the spatial angle.
9.7. Why is one result measured?
Before entering the magnet, the state relative to its axis can contain both modes:
\[\tag{59} |\Psi\rangle_J = A_{\uparrow}|J(a,0)\rangle + A_{\downarrow}|J(0,a)\rangle. \] An inhomogeneous field associates each mode with its own spatial trajectory:
\[\tag{60} |\Psi\rangle_J\Phi_0 \longrightarrow A_{\uparrow}|J(a,0)\rangle\Phi_{-} + A_{\downarrow}|J(0,a)\rangle\Phi_{+}, \] where \(\Phi_{-}\) and \(\Phi_{+}\) denote wave packets deflected in opposite directions. The signs of the spatial deflection are opposite to the signs of the electron spin, since its magnetic moment is antiparallel to the spin.
On the screen, each atom is registered at a single local point, belonging to one of the two packets. After accumulating a large number of events, the relative intensities of the spots tend to
\[\tag{61} I_{\uparrow}:I_{\downarrow} = |A_{\uparrow}|^2:|A_{\downarrow}|^2. \] The two-orbital model defines two geometric regimes and their half-angle law. However, converting the superposition into a single result requires a separate description of the interaction of the particle, magnet, and screen. In this paper, we use the correspondence between the quadratic norm of a regime and the statistical frequency of its detection.
9.8. Three Sequential Analyzers
A sequence of three analyzers is especially illustrative. The first analyzer along \(z\) selects the state \(J(a,0)\) relative to the \(z\) axis. The second analyzer along \(x\) splits it into two equally probable modes:
\[\tag{62} P_{z\uparrow\rightarrow x\uparrow} = P_{z\uparrow\rightarrow x\downarrow} = \frac12. \] If we then select only the \(x\uparrow\) channel and send it to the third analyzer along \(z\), we again obtain two results:
\[\tag{63} P_{x\uparrow\rightarrow z\uparrow} = P_{x\uparrow\rightarrow z\downarrow} = \frac12. \] 10. Comparison with the Standard Spinor
After geometric derivation, the state can be written in the standard two-component form:
\[\tag{64} \chi_{\mathbf n} = \begin{pmatrix} \cos(\theta/2)\\ e^{i\varphi}\sin(\theta/2) \end{pmatrix}. \] In this notation, the first component corresponds to the \(J(a,0)\) mode, and the second to the \(J(0,a)\) mode:
\[\tag{65} \begin{pmatrix} 1\\ 0 \end{pmatrix} \longleftrightarrow J(a,0), \qquad \begin{pmatrix} 0\\ 1 \end{pmatrix} \longleftrightarrow J(0,a). \] The standard spinor is used here not as the original cause of the half-angle, but as a compact mathematical notation for the structure previously obtained from a complete cycle of two coplanar orbits.
11. What has been obtained and what remains to be investigated
Within the accepted geometric principles, the following sequence was obtained:
\[ \begin{aligned} C &\longrightarrow \{C_1,C_2\}\\ C_1\longrightarrow C_2\longrightarrow C_1 &\longrightarrow \Theta_J=4\pi\\ \Theta_J=4\pi &\longrightarrow \chi=\frac{\theta}{2}\\ J(a,0) &\longrightarrow S_{\mathbf n}=+\frac{\hbar}{2}\\ J(0,a) &\longrightarrow S_{\mathbf n}=-\frac{\hbar}{2}\\ \chi=\frac{\theta}{2} &\longrightarrow \Psi(2\pi)=-\Psi(0)\\ \chi=\frac{\theta}{2} &\longrightarrow \Psi(4\pi)=\Psi(0). \end{aligned} \] In this paper, orbital splitting is adopted as the initial postulate of the model. Further research is required to quantify the value of \(\Delta r\), the geometry of transitions between split contours, the construction of a positive-definite norm and justification of the orthogonality of the \(J(a,0)\) and \(J(0,a)\) regimes, the construction of a general law for mapping model states onto arbitrarily oriented axes of three-dimensional space, and a description of the dynamics of a single dimension.
Conclusion
In the proposed model, spin-half is considered as a manifestation of the sequential motion of an electron along two nearly identical coplanar orbits. After the first revolution, the spatial position is repeated, but the motion shifts to the second contour. The full geometric configuration is restored only after the second revolution, so the full cycle has a length of \(4\pi\), and the internal phase is \(\theta/2\).
Two opposing modes of complete two-orbital motion are described by the quantities \(J(a,0)\) and \(J(0,a)\). The first mode corresponds to the spin projection \(+\hbar/2\), the second to the projection \(-\hbar/2\). The half-phase leads to a change in the sign of the state after \(2\pi\), recovery after \(4\pi\), and angular probabilities \(\cos^2(\theta/2)\) and \(\sin^2(\theta/2)\).
In the Stern-Gerlach experiment, a non-uniform magnetic field links these two regimes with opposite magnetic forces and spatially separates them. As a result, instead of a continuous distribution, two beams emerge. When the analyzer is rotated, the ratio of their intensities is determined by the half-internal angle of the two-orbital state.
The central result of the model: one complete internal cycle of an electron includes two successive revolutions along two split orbits. Therefore, the geometric cycle \(4\pi\) manifests itself in the observable world as a spin of one-half.
Based on this work, a model of the transition of electron and positron spin to photon polarization, which is at the next level of logic, was constructed. This model can be found in the next part.