Abstract For the first time we discover four space dimensions in a sphere, namely two periphery points, two semicircular arcs, two hemispherical surfaces, and two spherical solids. When these four geometric elements are compared with electronic orbitals in a neon shell, we draw a conclusion that while 2s orbitals are spherical, three types of 2p orbitals in the X, Y, and Z directions are different in the number of space dimensions. These anisotropic 2p orbitals prompt us to modify Schrödinger’s equation to reflect these fine space gradients. Within the atomic sphere, electrons are transforming from one dimension to another following the rule of calculus implemented by circular functions rather than position movement. An electron may exist in the physical shape of a point, an arc, a surface, or a solid. Eight electrons in the neon shell are undergoing various harmonic oscillations that form a continuous octet cycle. This motion feature is derived from a four-dimensional space harmonic oscillation equation coupled with a one-dimensional time harmonic oscillation equation in synthesis of the modified Schrödinger’s equation. Substantial amount of evidence can be found in organic chemistry where a carbon atom has an anisotropic electron configuration of 2s2px2py2pz. By furnishing a fresh and fundamental spacetime worldview, the quaternity theory may change the direction of quantum mechanics from discrete linear algebra toward continuous dynamic calculus in the future.
Abstract For the first time we discover four space dimensions in a sphere, namely two periphery points, two semicircular arcs, two hemispherical surfaces, and two spherical solids. When these four geometric elements are compared with electronic orbitals in a neon shell, we draw a conclusion that while 2s orbitals are spherical, three types of 2p orbitals in the X, Y, and Z directions are different in the number of space dimensions. These anisotropic 2p orbitals prompt us to modify Schrödinger’s equation to reflect these fine space gradients. Within the atomic sphere, electrons are transforming from one dimension to another following the rule of calculus implemented by circular functions rather than position movement. An electron may exist in the physical shape of a point, an arc, a surface, or a solid. Eight electrons in the neon shell are undergoing various harmonic oscillations that form a continuous octet cycle. This motion feature is derived from a four-dimensional space harmonic oscillation equation coupled with a one-dimensional time harmonic oscillation equation in synthesis of the modified Schrödinger’s equation. Substantial amount of evidence can be found in organic chemistry where a carbon atom has an anisotropic electron configuration of 2s2px2py2pz. By furnishing a fresh and fundamental spacetime worldview, the quaternity theory may change the direction of quantum mechanics from discrete linear algebra toward continuous dynamic calculus in the future.
Euclidean geometry does not characterize dynamic electronic orbitals satisfactorily for even a single electron in a hydrogen atom is a formidable mathematical task with the Schrodinger equation. Here the author puts forward a new spacetime concept that regards space and time as two orthogonal, symmetric and complementary quantities. They are inherent physical quantities that cannot be divorced from physical objects themselves. In two-dimensional helium shell, space and time are instantiated by two interactive 1s electrons; in four-dimensional neon shell, space and time dimensions blend into four types of curvilinear vectors represented by 2s, 2px, 2py, and 2pz electronic orbitals. The description of electronic orbitals constitutes an explanation of canonical spacetime properties such as harmonic oscillation, electromagnetism, and wave propagation. Through differential and integral operations, the author formulates a precise wavefunction for every electron in an inert neon atom where spacetime, as dimensional graduated by ten electrons, is continuous, and trigonometric function is the mechanism for dimension curling up. This fresh spacetime view based on dimensional interpretation of complex functions is an extension of classical mechanics and is compatible with relativity and quantum physics. It brings sharp insight into the geometries of 2p-orbitals and has broad support from chemistry.
In the context of two-dimensional spacetime within a helium atom, both 1s electrons are characterized by wave functions that observe duality equation. They are symmetric, orthogonal and interwoven, forming a dynamic rope structure at any moment. Instead of elliptical orbit of planets around the sun, electronic orbitals take the form of matter state transformation cycle. While the kinematic movement of planets is governed by Kepler's first law, electronic transformation obeys Pythagorean theorem, both being equivalent in physical principle. The atomic spacetime is a continuous medium of electron clouds in synchronized differential and integral processes that are implemented by smooth trigonometry. In order to integrate this new approach with conventional physics, the author translates the pattern of electronic motion in the atomic spacetime into spherical volume undulation in Euclidean geometry and calculates the probability density of an electron within the sphere from the classical perspective. From the primary wave function of a 1s electron, the author also tries to derive the mathematical expression of central force that guides the surrounding bodies along the orbits. The result is exciting and surprising that questions the exactness of the venerable Coulomb's law.
Euclidean space and linear algebra do not characterize dynamic electronic orbitals satisfactorily for even the motion of both electrons in an inert helium atom cannot be defined in reasonable details. Here the author puts forward a novel two-dimensional spacetime model from scratch in the context of defining both electrons in a helium atom. Space and time are treated as two orthogonal, symmetric and complementary quantities under the atomic spacetime. Electronic motion observed the rule of differential and integral operations that were implemented by dynamic trigonometric functions. It is demonstrated that the atomic spacetime is not a linear vector space with Newtonian time, and within which calculus has non-classical definition, and complex wave functions have fresh physical significances. This alternative approach is original, informative and refreshing but still compatible with quantum mechanics in the formulation. The description of electronic resonance in helium is also comparable with classical mechanics such as an oscillating pendulum and with classical electromagnetism such as an LC oscillator. The study has effectively unified complex function, calculus, and trigonometry in mathematics, and provided a prospect for unifying particle physics with classical physics on the novel spacetime platform.
The present approach is an advancement of the author’s former attempts to develop an atom model of Helium with well-defined electron trajectories. Thus it calls in question the traditional quantum mechanics which assume—in contrast and as a consequence of Heisenberg’s uncertainty principle—electronic probabilities of presence. Its basic idea consists of the assumption that the motions of the two electrons are influenced by their spins exhibiting the value h/2π, but in two different ways: on the one hand, one spin induces a rotation; and on the other hand, the other spin induces a harmonic oscillation. A second important relation is given by the fact that the retroactive force of the oscillation process is due to the centrifugal force when the process runs along the surface of a sphere, whereas in usual oscillation processes—such as the one of a spring pendulum—it is due to a permanent shift between potential and kinetic energy. Therefore, in the present case, the potential energy remains constant since the distance between the nucleus and the—diametrically positioned—electrons remains constant. Considering these two conditions and the usual physical relations such as Coulomb attraction, centrifugal force, and the conservation laws of the angular momentum and of the energy, it was possible to compute the respective key values. Thereby, the deflection of the oscillation angle ψ = 45˚ is remarkable. Finally, the process is described using a Cartesian coordinate system with z as the rotation axis, a variable oscillation distance d and variable rotation velocities rrot. Thereby, the projections onto the x-axis and on the y-axis are not identically equal, leading to an elliptic projection shape. Thus this system is anisotropic, in contrast to the isotropic array of the conventional quantum mechanics according to Schrödinger, where the 1s-orbital is spherically symmetrical. This anisotropy explains the existence of interatomic Van der Waals forces, which enable the condensation of Helium, even though the condensation temperature is very low. But in particular, it exhibits well-defined electron waves, thus finally delivering the explanation of the hypothesis of Louis de Broglie, which has been established 100 years ago.
The Geometric Algebra formalism allows for the development of a theory to replace conventional quantum mechanics. Generalizations resulting from the replacement of complex numbers with geometrically feasible three-dimensional objects, followed by unambiguous definitions of states, observables, and measurements, bring into reality clear explanations of strange quantum mechanical features, such as primitively considering atoms as a type of solar system. The three-sphere S3 is transformed into a playground for torsion kind states that do not require abstract Hilbert space vectors. The points S3 evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators.
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. 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.