Electron Orbitals as 4-D Rotor Projections: A Geometric Basis for Hydrogenic Structure and Chemical Bonding

Electron Orbitals as 4-D Rotor Projections: A Geometric Basis for Hydrogenic Structure and Chemical Bonding | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 6 October 2025 V1 Latest version Share on Electron Orbitals as 4-D Rotor Projections: A Geometric Basis for Hydrogenic Structure and Chemical Bonding Author : Stephen Euin Cobb 0009-0001-2971-0984 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175973067.76205913/v1 236 views 117 downloads Contents Abstract Abstract 1. Introduction 2. The 4-D Rotor Model 3. Mapping Rotor States to Orbital Quantum Numbers 4. Radial Equation and Laguerre Structure 5. Geometry of Orbitals and Exclusion 6. From Orbitals to Bonds 7. Predictions and Deviations from QM 8. Discussion 9. Conclusion References Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract The hydrogen atom exhibits an exact SO(4) symmetry that has long been recognized but seldom explained. Conventional quantum mechanics reproduces the spectrum through algebraic methods, yet leaves unanswered why spin-½, orbital degeneracies, and the Pauli exclusion principle exist. In this work, we show that these features arise naturally if the electron is modeled as a four-dimensional rotor constrained to the three-sphere, S 3 . The two independent spin modes of the rotor generate the SU(2) + × SU(2) - structure, whose balanced irreducible representations project into three-dimensional orbitals with the familiar quantum numbers n, l, and m. The radial equation reduces to the associated Laguerre form, yielding exact hydrogenic wavefunctions and energies. Orbital filling, exclusion, and hybridization emerge as geometric necessities rather than axioms. The model also predicts small but definite deviations from ideal quantum mechanics, including corrections from finite rotor inertia, spin–spin couplings, and symmetry-breaking effects. These deviations are accessible to precision spectroscopy and condensed matter experiments, providing falsifiable tests. By grounding quantum mechanics in four-dimensional geometry, the rotor framework preserves all known successes while offering new explanatory clarity and predictive power. Electron Orbitals as 4-D Rotor Projections: A Geometric Basis for Hydrogenic Structure and Chemical Bonding Author: Stephen Euin Cobb Address: 901 Kerr Drive SW, Aiken, SC, USA ORCID: 0009-0001-2971-0984 Date: September 2025 Abstract The hydrogen atom exhibits an exact SO(4) symmetry that has long been recognized but seldom explained. Conventional quantum mechanics reproduces the spectrum through algebraic methods, yet leaves unanswered why spin-½, orbital degeneracies, and the Pauli exclusion principle exist. In this work, we show that these features arise naturally if the electron is modeled as a four-dimensional rotor constrained to the three-sphere, S³. The two independent spin modes of the rotor generate the SU(2)₊ × SU(2)₋ structure, whose balanced irreducible representations project into three-dimensional orbitals with the familiar quantum numbers n, l, and m. The radial equation reduces to the associated Laguerre form, yielding exact hydrogenic wavefunctions and energies. Orbital filling, exclusion, and hybridization emerge as geometric necessities rather than axioms. The model also predicts small but definite deviations from ideal quantum mechanics, including corrections from finite rotor inertia, spin–spin couplings, and symmetry-breaking effects. These deviations are accessible to precision spectroscopy and condensed matter experiments, providing falsifiable tests. By grounding quantum mechanics in four-dimensional geometry, the rotor framework preserves all known successes while offering new explanatory clarity and predictive power. 1. Introduction The electronic structure of hydrogen has been a proving ground for every new theory of physics. In nonrelativistic quantum mechanics, the Coulomb potential admits an exact solution for bound states [6,7]. These are described by a principal quantum number n, an orbital angular momentum quantum number l, and a magnetic quantum number m. Their wavefunctions are products of radial functions, expressed in terms of associated Laguerre polynomials, and spherical harmonics Yₗₘ(θ,φ) [6]. Their energies follow the familiar Balmer formula, scaling as 1/n². Beyond this standard result, there is a less celebrated fact: the hydrogen atom possesses a hidden SO(4) symmetry. This arises because, in addition to angular momentum L, the quantum Runge–Lenz vector A is conserved for the Coulomb problem [2,3]. The algebra generated by {L, A} closes on SO(4), and the entire spectrum can be understood as unitary representations of SO(4) [4,5]. This symmetry explains the large degeneracy of hydrogen levels: for each principal quantum number n, all orbital angular momenta from l = 0 to l = n − 1 are admitted with equal energy. The full degeneracy at fixed n is n². Although the algebra is well known, its geometric meaning remains opaque. Why should the states of an electron bound to a nucleus be organized by SO(4), a symmetry of four-dimensional rotations? Why do quantum numbers arise as if there were two independent spins rather than one? In conventional pedagogy, these features are presented as curiosities, consequences of hidden symmetries that just happen to exist. They are not explained. In this paper, we propose that the SO(4) symmetry is not accidental but a reflection of the true internal geometry of the electron. We model the electron as a four-dimensional rotor, constrained to move on the three-sphere S³, possessing two independent spin degrees of freedom. The hydrogen spectrum then arises naturally as the projection of this 4-D rotor onto ordinary 3-D space. In this picture, the familiar quantum numbers n, l, m are not abstract labels but the shadows of genuine four-dimensional rotational states. The well-known wavefunctions, node structures, and degeneracies are recovered exactly, but with an underlying geometry that renders them inevitable rather than mysterious. This interpretation offers a natural origin for several otherwise axiomatic features of quantum mechanics: the existence of spin-½ [13], the necessity of antisymmetry [1], and the peculiar filling of orbitals [7]. Moreover, because the electron is treated as a physical rotor with finite inertia in four dimensions, the model admits small but definite corrections to the ideal spectrum. These corrections, though tightly constrained by precision spectroscopy [10,11], provide testable avenues for falsification or refinement. The geometric rotor view therefore preserves the exact successes of quantum mechanics while opening a window onto new physics. 2. The 4-D Rotor Model We begin by specifying the internal configuration space of the electron. Instead of treating the electron as a pointlike particle with an abstract spinor degree of freedom, we posit that it is an extended dynamical object whose internal motion is a rigid rotation on the three-sphere, S³. This three-sphere is the natural manifold of SU(2), the double cover of SO(3), and it possesses two independent planes of rotation. In group-theoretic terms, these are represented by SU(2)₊ and SU(2)₋. Each electron state is therefore labeled by a pair of spin quantum numbers (j₊, j₋), with associated magnetic projections (m₊, m₋). The total symmetry group of the internal rotor is SO(4), which is isomorphic to SU(2)₊ × SU(2)₋. A rotation on S³ can be decomposed into simultaneous rotations in two orthogonal planes. Physically, this means the electron possesses not a single intrinsic spin, but two independent rotational modes, each quantized in half-integer steps. When observed in three dimensions, only the combination of these two modes is accessible, giving the appearance of a single spin-½ degree of freedom. The need for a 720° rotation to restore the identity of a spinor follows immediately: each SU(2) factor requires a double rotation for closure, and their product inherits this property. The mapping from the 4-D rotor to observable 3-D states is achieved through the Hopf projection, which maps the three-sphere S³ onto the two-sphere S². Under this projection, the pair (j₊, j₋) determines the allowed orbital angular momentum quantum numbers. The full decomposition is l = |j₊ − j₋|, |j₊ − j₋|+1, …, (j₊ + j₋), with each l containing the usual m = −l,…,l multiplet. The principal quantum number is set by the total spin magnitude, n = j₊ + j₋ + 1. Thus, for each n, all orbital angular momenta from l=0 to l=n−1 appear exactly once, in agreement with hydrogenic degeneracies. The n² degeneracy at fixed energy is explained not as an accident of the Coulomb potential, but as a geometric inevitability of the 4-D rotor’s balanced representation with j₊ = j₋ = (n−1)/2. In this construction, the abstract structure of quantum mechanics gains a direct physical interpretation. Orbital angular momentum is the visible projection of two hidden spins. The hierarchy of orbitals is a bookkeeping of how these spins combine. The Pauli exclusion principle reflects the impossibility of two electrons occupying the same complete (j₊, j₋, m₊, m₋) state. What appears in 3-D as algebraic rules emerges in 4-D as geometry. 3. Mapping Rotor States to Orbital Quantum Numbers The correspondence between 4-D rotor states and 3-D orbital quantum numbers follows directly from the representation theory of SO(4). Each bound-state multiplet corresponds to a balanced irreducible representation with j₊ = j₋ = (n − 1)/2. Within this irrep, the projection onto 3-D yields every orbital angular momentum l from 0 to n−1, each accompanied by its full set of magnetic sublevels m. The mapping is therefore: • Principal quantum number: n = j₊ + j₋ + 1 • Orbital angular momentum: l = 0, 1, …, n−1 • Magnetic quantum number: m = −l,…,l • Degeneracy: n² Table 1 below lists the explicit correspondence for the first four principal levels. Each case illustrates how the pair (j₊, j₋) generates the entire familiar set of orbitals. Table 1. Mapping of rotor states to orbital quantum numbers (n = 1…4). n (j₊, j₋) Allowed l values Magnetic m values Degeneracy (total states) 1 (0, 0) {0} {0} 1 = 1² 2 (½, ½) {0, 1} l=0 → {0}; l=1 → {−1,0,+1} 4 = 2² 3 (1, 1) {0, 1, 2} l=0 → {0}; l=1 → {−1,0,+1}; l=2 → {−2,−1,0,+1,+2} 9 = 3² 4 (3/2, 3/2) {0, 1, 2, 3} l=0 → {0}; l=1 → {−1,0,+1}; l=2 → {−2…+2}; l=3 → {−3…+3} 16 = 4² This table reproduces exactly the orbital ladder structure of hydrogen as normally derived from the Schrödinger equation. The difference is interpretive: the degeneracies are not accidents of the Coulomb potential but consequences of the balanced SO(4) representation enforced by the electron’s 4-D rotor structure. This perspective elevates the familiar quantum numbers from abstract symbols to geometric projections of real rotational states. 4. Radial Equation and Laguerre Structure The hydrogen atom’s radial structure emerges directly when the 4-D rotor is projected into 3-D spherical coordinates. Separation of variables proceeds as usual: Ψ(r,θ,φ) = R(r)·Yₗₘ(θ,φ), with the angular part supplied by the Hopf projection. Writing u(r) ≔ rR(r), the radial Schrödinger equation is −(ħ²/2μ) d²u/dr² + [ l(l+1)ħ²/(2μ r²) − κ/r ] u = E u [6,7]. with κ = e²/(4πϵ₀) and μ the reduced mass. For bound states E < 0, define the Bohr radius a₀ = ħ²/(μκ) and substitute ρ = 2r/(n a₀). With u(r) = ρ^{l+1} e^{−ρ/2} w(ρ), the equation reduces to the associated Laguerre form [6]. ρ w″ + (2l+2 − ρ) w′ + (n − l − 1) w = 0. Normalizability requires termination of the series, so w(ρ) ∝ L^{(2l+1)}_{n−l−1}(ρ). Thus the radial functions are R_{n l}(r) ∝ e^{−ρ/2} · ρ^l · L^{(2l+1)}_{n−l−1}(ρ), with ρ = 2r/(n a₀). The energy spectrum is identical to that of quantum mechanics [6,7]: Eₙ = − μκ² / (2ħ² n²). In this rotor picture, the standard hydrogenic results appear automatically. The Laguerre structure, degeneracies, and 1/n² energy law are consequences of balanced SO(4) irreps rather than postulates. This demonstrates that the electron’s 4-D geometry naturally enforces all familiar quantum features. Standard hydrogenic orbitals reappear in the rotor framework, but not as postulates of the Schrödinger equation. Each radial node and most-probable radius is the direct projection of two hidden SU(2) spin modes interfering in 4-D space. What were once algebraic Laguerre polynomials are here revealed as geometric interference fringes of a real rotor. 5. Geometry of Orbitals and Exclusion The 4-D rotor model provides a natural geometric basis for orbital structure and the Pauli exclusion principle. Each orbital is not an arbitrary mathematical solution but a projection of a distinct 4-D rotational state. The familiar nodal patterns of s, p, d, and higher orbitals arise from interference between the two independent spin modes, their phase relationships dictating where constructive and destructive amplitudes appear in 3-D space. The exclusion principle follows directly. Two electrons cannot occupy the same complete state labeled by (j₊, j₋, m₊, m₋), even if their 3-D projections differ only subtly. In 3-D quantum mechanics this appears as a rule enforced by antisymmetric wavefunctions. In the rotor view, it is simply the geometric impossibility of two rotors occupying the identical orientation and spin configuration on S³. Orbital filling, the ladder of shells and subshells across the periodic table, is likewise clarified. Each electron occupies a unique 4-D rotor state. The progression n=1,2,3,… corresponds to successively larger balanced representations of SO(4). Within each representation, the range l = 0…n−1 appears once. The apparent “slots” available in each subshell (2, 6, 10, 14, …) correspond exactly to the number of distinct (m₊, m₋) projections consistent with a given l. Thus the periodic table is a direct reflection of how 4-D rotor states project into 3-D. This picture demystifies the peculiarities of quantum mechanics. Spin-½ behavior, orbital multiplicities, and exclusion are not axioms imposed upon nature, but necessary consequences of four-dimensional geometry. Chemistry, in turn, becomes the study of how these projected rotor states combine and overlap to form stable arrangements. In later sections we will show how bond directionality and hybridization follow naturally from this framework. Allowed and forbidden electron pair states form a simple lattice: off-diagonal combinations are permitted, the diagonal is forbidden. This visualization shows Pauli exclusion not as an imposed antisymmetry rule, but as a geometric impossibility of two electrons occupying the same 4-D rotor orientation. The “forbidden diagonal” is the shadow of S³ geometry in 3-D space. 6. From Orbitals to Bonds Chemical bonding can now be reinterpreted as the mutual alignment of 4-D rotor states. Whereas conventional quantum mechanics describes bonds as overlaps of wavefunctions constrained by antisymmetry, the rotor model explains bonding in terms of how two electrons’ spin modes combine before projection into 3-D. Bond directionality arises from fixed phase relationships between SU(2)₊ and SU(2)₋ amplitudes. When these are recombined, the resulting 3-D projection produces the familiar hybrid orbitals: sp, sp², and sp³. Each hybrid corresponds to a stable interference pattern of two underlying 4-D spins. In this view, the tetrahedral geometry of carbon’s sp³ bonds is not an arbitrary rule, but the natural projection of a symmetric configuration in S³. Bond polarity follows from asymmetries in the combined rotor states. If one electron’s spin mode dominates in a given direction of projection, electron density is shifted toward one nucleus, reproducing polar covalent character. Complete dominance yields ionic-like transfer of density. Thus, the continuum from covalent to ionic bonding is explained geometrically as varying degrees of asymmetry in rotor phase-locking. Multiple bonds and resonance structures can also be interpreted in this language. Double and triple bonds correspond to rotor states whose projections produce two or three stable overlap directions, while resonance arises when rotor phases oscillate between competing configurations. These pictures are faithful to quantum chemistry’s predictive successes while supplying a deeper geometrical rationale. Superconductivity and metallic bonding may likewise be revisited. Collective rotor states can align across a lattice, producing shared electron density not by delocalized wavefunctions alone but by coherent synchronization of 4-D spin modes. This offers a new perspective on why electrons form Cooper pairs and why lattice vibrations can stabilize their coherence. The tetrahedral lobes familiar from chemistry textbooks arise automatically from equal-phase combinations of the electron’s two SU(2) spin modes. In the rotor view, sp³ hybridization is not an empirical rule but the inevitable projection of a balanced interference state on S³. Bond directionality is therefore a geometric necessity, not a chemical heuristic. An asymmetric combination of rotor states projects into unequal density around two nuclei, reproducing the familiar skew of polar covalent bonds. In conventional MO theory, this asymmetry is a fitted parameter; in the rotor model, it is the natural outcome of phase imbalance between SU(2)₊ and SU(2)₋. Polarity thus reflects geometry in four dimensions rather than arbitrary coefficients. 7. Predictions and Deviations from QM The 4-D rotor model exactly reproduces the gross features of the hydrogen atom: the 1/n² energy spectrum, the n² degeneracy, the Laguerre radial structure, and the spherical harmonics [6,7]. However, because the rotor is treated as a real dynamical object with finite inertia and couplings between its two spin modes, the model naturally introduces small deviations from ideal quantum mechanics. These deviations are highly constrained by existing data, but they define a concrete experimental program for testing the model. 7.1 Finite Rotor Inertia A true rotor possesses a finite moment of inertia I₄D. This introduces an additional energy contribution proportional to the total Casimir C₊ + C₋ = j₊(j₊+1) + j₋(j₋+1). For balanced irreps with j₊ = j₋ = (n−1)/2, this adds a small correction of order (n²−1)/2. The correction would appear as a tiny departure from exact 1/n² scaling. Precision spectroscopy of Rydberg states can constrain I₄D or reveal such deviations. 7.2 Spin–Spin Coupling The two spin modes may couple via a weak interaction j₊·j₋. This coupling splits levels of different l within the same n, breaking the perfect degeneracy predicted by SO(4) [2,3]. In standard hydrogen, this degeneracy is already lifted by fine structure and Lamb shift effects [14]. The rotor model predicts an additional, structured splitting that scales with n and l in a distinct pattern. Comparison with existing Lamb shift measurements provides an immediate upper bound [10]. 7.3 Symmetry-Breaking Effects External perturbations that act differently on SU(2)₊ and SU(2)₋ could lead to small violations of conventional selection rules. Transitions usually forbidden by angular momentum conservation may acquire suppressed but nonzero amplitudes. Observation of ultra-weak spectral lines or anomalous transition strengths could serve as a signature of rotor geometry. 7.4 Multi-Electron Atoms In many-electron systems, exclusion is enforced by 4-D state uniqueness. However, residual couplings between rotor modes of different electrons could produce subtle shifts in exchange and correlation energies. This may manifest as small corrections to Hund’s rules or to the ordering of subshell energies, especially in transition metals where near-degeneracy occurs. These effects are potentially observable in high-resolution atomic spectroscopy. 7.5 Condensed Matter Consequences If electrons carry genuine 4-D rotor structure, collective phenomena may be altered. Superconductivity, magnetism, and band structure could contain signatures of rotor synchronization. For example, Cooper pairing may be interpreted as phase-locking of two electrons’ SU(2)₊ × SU(2)₋ spins, offering a geometric perspective on pairing symmetry and anisotropy. While speculative, this opens paths for novel condensed-matter tests. Table 2. Predicted deviations from standard QM. Source Expected Effect Observable Signature Current Constraint Finite rotor inertia Departure from 1/n² law Tiny shifts in Rydberg energies <10⁻¹² eV bounds Spin–spin coupling l-dependent splitting within fixed n Extra fine-structure terms Lamb shift data Symmetry breaking Weakly allowed forbidden transitions Ultra-faint spectral lines Not yet detected Multi-electron rotor Corrections to Hund’s rules, subshell ordering Spectroscopy of transition metals To be constrained Collective effects Modified pairing or band anisotropy Superconducting gap structure, magnetism Open question These deviations are not arbitrary. They are calculable once I₄D and coupling strengths are specified. The model is therefore falsifiable: if all such deviations are constrained below plausible values, the rotor picture is ruled out. Conversely, detection of any consistent anomaly could signal the geometric reality of the electron’s 4-D structure. 8. Discussion The 4-D rotor interpretation recasts the familiar landscape of atomic physics in geometric terms. Instead of treating electron orbitals, spin, and exclusion as axioms of quantum mechanics, the model presents them as shadows of higher-dimensional rotation. This shift transforms features long regarded as mysterious coincidences into necessary consequences of geometry. 8.1 Demystifying Quantum Features Spin-½ behavior emerges because each SU(2) factor requires a 720° rotation for closure. The exclusion principle is no longer an imposed antisymmetry, but the impossibility of duplicate occupancy of identical rotor states. Orbital multiplicities, hybridization, and bonding directionality are all understood as interference patterns of two independent spin modes projected into 3-D. This geometric interpretation thus provides an explanatory foundation beneath the formalism of quantum mechanics. 8.2 Relation to Known Symmetries That hydrogen bound states possess SO(4) symmetry is a long-established algebraic result. The rotor picture gives this algebra a physical home: SO(4) is simply the natural symmetry of S³, the electron’s internal configuration space. Balanced irreps correspond to bound states because they represent closed, stable rotations on S³. The long-noted degeneracy of hydrogenic energy levels thus reflects a deeper geometry rather than a mathematical curiosity. 8.3 Implications for Chemistry Chemistry depends critically on orbital filling and bonding rules. These are normally presented as empirical facts explained by the Pauli principle and Schrödinger’s solutions. In the rotor view, chemistry is the study of how 4-D states overlap and stabilize when projected into 3-D. Hybrid orbitals, bond polarity, resonance, and even metallic delocalization gain a simple geometrical explanation. This promises to unify the abstract rules of quantum chemistry with an underlying mechanistic picture. 8.4 Context Among Theories Other geometric approaches to quantum mechanics have explored hidden variables, higher dimensions, and twistor frameworks. The rotor model differs by its minimalism: it introduces only one assumption—that the electron is a rotor on S³—and from this recovers the exact hydrogenic structure. It is therefore falsifiable, predictive, and conservative in scope. It does not discard the machinery of quantum mechanics, but rather grounds it in a deeper geometry. 8.5 Future Directions The natural next step is to extend the model beyond hydrogen: helium fine structure, multi-electron atoms, and chemical bond lengths provide rich opportunities for comparison. Precision spectroscopy and condensed matter experiments may offer tests of rotor corrections. The possibility of rotor synchronization in superconductors, or altered exchange behavior in correlated electron systems, provides avenues for condensed matter research. On the theoretical side, embedding the rotor picture within relativistic Dirac theory and quantum field theory remains an open frontier. In summary, the rotor framework bridges the gap between algebraic formalism and geometric intuition. It explains why quantum mechanics works as it does, while also predicting where its limits may lie. If supported by experiment, it offers a path toward a deeper synthesis of physics and chemistry. 9. Conclusion The electron’s behavior in atoms and molecules has long been described by the formal rules of quantum mechanics, yet those rules have remained mysterious in origin. The 4-D rotor model presented here demonstrates that these rules arise inevitably from the geometry of rotation on the three-sphere, S³. The hydrogen atom’s SO(4) symmetry, its n² degeneracy, its radial Laguerre structure, and the existence of spin-½ all follow naturally when the electron is treated as a genuine rotor with two independent spin modes. This perspective recasts the Pauli exclusion principle, orbital filling, and chemical bonding as geometric consequences of 4-D structure rather than arbitrary axioms. Hybridization, bond polarity, and resonance appear as stable projection patterns of rotor states. The entire edifice of atomic and chemical behavior gains a deeper foundation. At the same time, the rotor model makes specific predictions. Finite rotor inertia, spin–spin coupling, and symmetry-breaking interactions introduce small but measurable corrections to energy levels and transition rules. These provide clear targets for experimental tests in atomic spectroscopy, Rydberg state measurements, and condensed matter phenomena. The model is therefore falsifiable: its success or failure will be determined by experiment. The central claim of this work is simple: the electron is a four-dimensional rotor, and what we observe in three dimensions are the shadows of its higher-dimensional rotations. If this claim is correct, the mysteries of quantum mechanics reduce to the geometry of S³, and the strangeness of orbital structure becomes a natural and inevitable feature of reality. This interpretation preserves every success of quantum mechanics while offering a new level of explanatory clarity and predictive power. The path forward is clear. Systematic comparison with high-precision data, extension to multi-electron atoms, and integration with relativistic field theory will determine the viability of the rotor picture. Should it succeed, the model offers not only a new understanding of atomic structure but also a bridge between physics and chemistry, and perhaps a step toward unifying the foundations of natural law. Keywords: 4-D rotor model, hydrogen atom spectrum, SO(4) symmetry, electron spin, Pauli exclusion principle, orbital hybridization, quantum geometry, geometric quantum mechanics References 1. [1] Pauli, W. (1926). On the hydrogen spectrum from the standpoint of the new quantum mechanics. Zeitschrift für Physik, 36, 336–363. Google Scholar 2. [2] Fock, V. (1935). Zur Theorie des Wasserstoffatoms. Zeitschrift für Physik, 98, 145–154. Google Scholar 3. [3] Bargmann, V. (1936). Zur Theorie des Wasserstoffatoms. Zeitschrift für Physik, 99, 576–582. Google Scholar 4. [4] Biedenharn, L. C., & Louck, J. D. (1981). Angular Momentum in Quantum Physics: Theory and Application. Addison–Wesley. Google Scholar 5. [5] Wybourne, B. G. (1974). Classical Groups for Physicists. Wiley. Google Scholar 6. 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[14] Schwinger, J. (1951). On gauge invariance and vacuum polarization. Physical Review, 82(5), 664–679. Google Scholar 15. [15] Cobb, S. E. (forthcoming / preprints). Rotor papers R1–R12. Google Scholar Information & Authors Information Version history V1 Version 1 06 October 2025 Copyright This work is licensed under a Non Exclusive No Reuse License. Keywords chemical bonding four-dimensional rotor projections hydrogenic orbitals runge–lenz vector so(4) symmetry Authors Affiliations Stephen Euin Cobb 0009-0001-2971-0984 [email protected] no affiliation View all articles by this author Metrics & Citations Metrics Article Usage 236 views 117 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Stephen Euin Cobb. Electron Orbitals as 4-D Rotor Projections: A Geometric Basis for Hydrogenic Structure and Chemical Bonding. Authorea . 06 October 2025. 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