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How Natural Origins Achieves the Algebraization of the Schrödinger Equation

Xiangqian Zhang, Yuchuan Xu

Published Aug 17, 2026• 0 citations
Publisher

Abstract

This paper develops the algebraization of the Schrodinger equation by reexamining the bottom layer meanings of the wave function, complex phase, kinetic term, potential term, quantum numbers, probability square, interference, uncertainty, and entanglement in modern quantum mechanics. The paper argues that the greatness of the Schrodinger equation lies in its successful compression of a large body of microscopic experimental regularities, while its limitation lies in compressing electron ontology, spatial propagation, path delay, phase relation, boundary action, and material manifestation into one wave function. This compression easily causes the electron itself to be confused with the probability wave. In the Natural Origins framework, the electron is defined as a real closed loop ontology, while the wave function is repositioned as the continuous algorithmic expression of an open ledger. Space unit refresh explains the upper limit of propagation speed, right handed orthogonal refresh explains the complex structure, time element unfolding forms the time potential terrain, the moving unit circle explains the square structure of the kinetic term, orbital volume hierarchy explains principal quantum numbers and energy levels, and elemental cell restoration work explains the material manifestation source of the probability square. In this way, the Schrodinger equation is no longer understood as a mysterious equation of electron probability wave motion. It is reduced to a continuous algorithmic compression describing how, after a complete electron enters a time potential terrain, the open ledger is refreshed through space units, constrained by time potential, reissued through boundaries, and finally squared into manifestation at the elemental cell endpoint. The paper further states that interference is not the splitting of the electron ontology, but the spatial settlement pattern formed after an open ledger is reissued by boundary elemental cells. Uncertainty is not the absence of a definite state of the electron ontology, but the minimum exchange residual between path process ledger and material manifestation ledger. Entanglement is not remote notification, but a double endpoint shared relational ledger opened from the same source event. The whole paper finally forms a unified main line: space units are responsible for propagation, right handed refresh is responsible for orthogonality, time elements are responsible for potential terrain, open ledgers are responsible for unsealed relations, boundaries are responsible for reissuance, and elemental cells are responsible for final manifestation. The Schrodinger equation is thereby transformed from a quantum mystical image into a clear algebraic ontological chain.

DOI
10.5281/zenodo.21992682

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This paper develops the algebraization of the Schrodinger equation by reexamining the bottom layer meanings of the wave function, complex phase, kinetic term, potential term, quantum numbers, probability square, interference, uncertainty, and entanglement in modern quantum mechanics. The paper argues that the greatness of the Schrodinger equation lies in its successful compression of a large body of microscopic experimental regularities, while its limitation lies in compressing electron ontology, spatial propagation, path delay, phase relation, boundary action, and material manifestation into one wave function. This compression easily causes the electron itself to be confused with the probability wave. In the Natural Origins framework, the electron is defined as a real closed loop ontology, while the wave function is repositioned as the continuous algorithmic expression of an open ledger. Space unit refresh explains the upper limit of propagation speed, right handed orthogonal refresh explains the complex structure, time element unfolding forms the time potential terrain, the moving unit circle explains the square structure of the kinetic term, orbital volume hierarchy explains principal quantum numbers and energy levels, and elemental cell restoration work explains the material manifestation source of the probability square. In this way, the Schrodinger equation is no longer understood as a mysterious equation of electron probability wave motion. It is reduced to a continuous algorithmic compression describing how, after a complete electron enters a time potential terrain, the open ledger is refreshed through space units, constrained by time potential, reissued through boundaries, and finally squared into manifestation at the elemental cell endpoint. The paper further states that interference is not the splitting of the electron ontology, but the spatial settlement pattern formed after an open ledger is reissued by boundary elemental cells. Uncertainty is not the absence of a definite state of the electron ontology, but the minimum exchange residual between path process ledger and material manifestation ledger. Entanglement is not remote notification, but a double endpoint shared relational ledger opened from the same source event. The whole paper finally forms a unified main line: space units are responsible for propagation, right handed refresh is responsible for orthogonality, time elements are responsible for potential terrain, open ledgers are responsible for unsealed relations, boundaries are responsible for reissuance, and elemental cells are responsible for final manifestation. The Schrodinger equation is thereby transformed from a quantum mystical image into a clear algebraic ontological chain.
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The solution to a Schrodinger equation is the basis of a broad theory. The Dirac equation superseded the Schrodinger equation because it is relativistic and has 4 components that accurately represent half spin particles like the electron and quarks. This paper presents a way of making the Schrodinger equation relativistic. It has many components and is complex enough to represent the neutron, proton and electron but is general enough to help us understand nature. Our goal is to find clues related to creation. The Schrodinger equation described by MIT as unitary evolution [20] has a simple solution: Probability P=1 in the left hand side (LHS) of the Schrodinger equation is equal to the multiple of complex conjugate terms exp(iEt/H)*exp(-iEt/H) in the right hand side (RHS) where i is the imaginary number, H=Planck’s constant, E is field energy and time t is the time around a quantum circle at velocity C. (exp(iEt/H) stands for the natural number e to power iEt/H). Observation is fundamental to quantum mechanics and the Copenhagen interpretation indicates that we can only describe the probability of an event within certain limits. Shannon’s definition [9] of information (information = -natural logarithm(probability) will be used to explore the relationship between information and energy. Has something been separated in the multiple exp(iEt/H)*exp(-iEt/H)=1? The number 1 has been separated into two expressions that represent waves, but it is a dynamic separation; it repeatedly comes back to 1 as time moves forward. The idea that nature originates as a series of separations is an old idea, for example, recall that Genesis contains the words “So God made the expanse and separated the water under the expanse from the ... as Genesis 1:7 ends with the phrase 'from the water above it [the expanse]'. Another phrase from Genesis is “in the beginning was the word”. Consider a beginning with zero energy. This avoids the endless argument that things are made of other things, ad infinitum. Meaningful equal and opposite energy pairs come into existence at the same time but represent zero overall. Also consider probability one as a beginning condition. It simply means the universe exists.
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The Schrödinger equation has been the central mathematical instrument of quantum mechanics for a century. This paper argues that it is not a quantum mechanical equation at all. It is classical energy conservation applied inside a sphere, with a classical wave form assumed rather than derived. The wavefunction ψ is not a physical object but a lossy geometric encoding. The Born rule—the requirement to square ψ to obtain probabilities—is not a law of nature but a decompression artifact produced by discarding the imaginary phase. The Heisenberg uncertainty principle is a measurement limit, not an ontological statement about particles, and its use to derive the Bohr radius is circular. All distances obtained from this framework are ratios pretending to be absolutes in a universe where only ratios persist. Relativistic Orthogonal Selection Theory (ROST) dissolves each of these confusions by returning to the geometric substrate the equation was always implicitly describing.
The mathematical and physical foundations of the Born rule remain open questions, largely because the wave function lacks a mathematical identity and a physical origin. Here we uncover a mathematical structure of the wave function by treating the phase as a circular variable. This structure reveals that the wave function is the pullback of the phase probability-density amplitude to position space via the phase field. Because the phase probability distribution forms an equal-area contour in the complex plane, its probability-area element yields the modulus squared as the position probability density, revealing the Born rule as a geometric relation rooted in the circular topology of the phase variable. The same structure governs the phase-energy distribution of electromagnetic waves. By quantizing phase energy into energy quanta, in the single-quantum limit the energy-quantum density reduces to the probability density, and its amplitude becomes the probability-density amplitude. This therefore identifies the energy-quantum density amplitude in the single-quantum limit as the physical origin of the wave function. To verify this, we derive the time-evolution equation for the energy-quantum density amplitude from five physical principles. This yields the Gross-Pitaevskii equation for many-body systems and, in the single-quantum limit, the Schrödinger equation. The framework also recovers Maxwell's equations in the classical limit. Overall, this study introduces the energy-quantum density amplitude from the classical phase-energy distribution, and derives the three core postulates of quantum mechanics: the wave function, the Born rule, and the Schrödinger equation, thereby identifying the position probability density as the energy-quantum density in the single-quantum limit.
This paper proposes a physical interpretation of the imaginary unit i in the Schrödinger equation, arguing that its complex structure is not a mathematical artifact but a direct signature of the zero field’s internal phase dynamics. In standard quantum mechanics, i enables oscillatory, unitary time evolution—yet its physical meaning remains unaddressed. Within the framework of R-theory (Resonance Theory), the universe is understood as a coherent resonant field with two inseparable components: a real, space-time-manifested aspect and an imaginary, non-local phase domain. This paper identifies i as the mathematical projection of this hidden imaginal axis. The wavefunction’s complex nature—traditionally viewed as formal—is reinterpreted as a physical necessity. The real amplitude corresponds to manifested energy, while the imaginary component expresses the latent coherence through which all energy oscillates between potential and presence. In this view, the Schrödinger equation becomes a dynamical law describing rotations within the zero field, where i functions as the operator of coherence rather than an abstract algebraic device. The implications are significant: Quantum oscillation reflects the continuous exchange between real and imaginal components of the zero field. Complexity is elevated from a mathematical convenience to an ontological principle. Consciousness emerges as the experiential side of the same imaginal phase that maintains physical coherence. Reality itself is intrinsically complex—built from orthogonal dimensions of manifestation and potential. This paper reframes the imaginary dimension as the fundamental axis through which the universe sustains resonance. What physics calls “imaginary” is revealed as the inner structure of reality, the phase without space that allows energy, matter, and awareness to exist as coherent oscillations within the zero field.