d and f Orbital Shapes as Shadows of 4D Geometry: SO(4) Symmetry, the Fock S³ Mapping, and the Kustaanheimo-Stiefel Transform
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2026
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| _version_ | 1866901462038085632 |
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| author | Kilgore, Brian |
| author_facet | Kilgore, Brian |
| contents | <p>A research note connecting the shapes of atomic orbitals to four-dimensional geometry. The hydrogen atom possesses a hidden SO(4) symmetry (Pauli 1926) arising from conservation of the Runge-Lenz vector. In momentum space, hydrogen wavefunctions are exactly the spherical harmonics on the 3-sphere S³ (Fock 1935 stereographic projection). The familiar d and f orbital shapes are therefore 3D equatorial shadows of 4D hyperspherical harmonics.</p><p>Covers: the Runge-Lenz vector and SO(4) Lie algebra; the Fock stereographic projection theorem; the Kustaanheimo-Stiefel transform (3D hydrogen ≡ 4D harmonic oscillator, exact); the SO(4,2) conformal group encoding the full hydrogen spectrum in a single representation; n-sphere generalization table; and open research directions including an interactive visualization paper (Am. J. Phys. target), angular-momentum dependence of AMEM matrix elements, and lanthanide crystal field analysis in the S³ harmonic basis.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20130358 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
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| spellingShingle | d and f Orbital Shapes as Shadows of 4D Geometry: SO(4) Symmetry, the Fock S³ Mapping, and the Kustaanheimo-Stiefel Transform Kilgore, Brian hydrogen atom SO(4) symmetry Runge-Lenz vector Fock projection S³ hyperspherical harmonics orbital geometry 4D mathematics Kustaanheimo-Stiefel SO(4,2) atomic orbitals quantum mechanics <p>A research note connecting the shapes of atomic orbitals to four-dimensional geometry. The hydrogen atom possesses a hidden SO(4) symmetry (Pauli 1926) arising from conservation of the Runge-Lenz vector. In momentum space, hydrogen wavefunctions are exactly the spherical harmonics on the 3-sphere S³ (Fock 1935 stereographic projection). The familiar d and f orbital shapes are therefore 3D equatorial shadows of 4D hyperspherical harmonics.</p><p>Covers: the Runge-Lenz vector and SO(4) Lie algebra; the Fock stereographic projection theorem; the Kustaanheimo-Stiefel transform (3D hydrogen ≡ 4D harmonic oscillator, exact); the SO(4,2) conformal group encoding the full hydrogen spectrum in a single representation; n-sphere generalization table; and open research directions including an interactive visualization paper (Am. J. Phys. target), angular-momentum dependence of AMEM matrix elements, and lanthanide crystal field analysis in the S³ harmonic basis.</p> |
| title | d and f Orbital Shapes as Shadows of 4D Geometry: SO(4) Symmetry, the Fock S³ Mapping, and the Kustaanheimo-Stiefel Transform |
| topic | hydrogen atom SO(4) symmetry Runge-Lenz vector Fock projection S³ hyperspherical harmonics orbital geometry 4D mathematics Kustaanheimo-Stiefel SO(4,2) atomic orbitals quantum mechanics |
| url | https://doi.org/10.5281/zenodo.20130358 |