The Necessity of the Spectral Method from the Perspective of the Classical Atomic Model
——The Spectralized World from the View of Natural Quantum Theory
I. Introduction: The Complete Picture of the Classical Atomic Model
Within the framework of Natural Quantum Theory (NQT), the image of the atom is restored to a real resonant system.
However, in the attempt to find the eigenoscillation frequencies of electrons by solving the radial oscillation model, it is found that pure dynamic methods are infeasible. This makes perfect sense, because dynamics is a spatiotemporally local and instantaneous process, while the boundary conditions for electromagnetic oscillations (which thus determine the eigenstates) require global information. Therefore, it is ultimately necessary to use the spectral method—that is, the Schrödinger equation—to obtain the eigenoscillation spectrum, which in turn constrains the behavior of electrons, thereby obtaining a comprehensive physical image including the eigen electromagnetic oscillation spectrum and electron trajectories.
Electrons are not mysterious probability clouds, but local electromagnetic-momentum flow structures constrained by the electric field of the atomic nucleus. They form stable rotational and flux-closed circular currents in orbits at approximately the speed of light, and these circular currents correspond to the atomic magnetic moment, energy levels, and radiation frequencies.
The key characteristics of this model are:
Thus, in this image, the atom is essentially a spectral system rather than one described solely by particle dynamics.This compels us to re-evaluate the status of the spectral method in atomic theory.
II. The Necessity of the Spectral Method: Determination of Global Eigenstates
In classical dynamics, we can obtain the orbital motion of electrons through Newton's equations or Lorentz force equations, but such methods only provide instantaneous local solutions.
Since the system is nonlinear, periodic, and has coupling terms, simple time integration is difficult to ensure global stability. To find truly permanently existing stable states, it is necessary to use the spectral method—to solve for the eigenfrequencies and eigenfunctions of the system.
The Schrödinger equation is a typical form of this spectral method.It does not directly describe how electrons "move," but rather provides standing wave solutions that allow the system to exist self-consistently through boundary conditions and constraints.
These standing wave solutions form an energy level spectrum, where each energy level corresponds to a stable frequency-locked orbit. The significance of this mathematical process lies in its revelation of the global constraint structure of the system.
Therefore, although Natural Quantum Theory points out that the spectral method has been misinterpreted in physical interpretation (concepts beyond classical physics such as the uncertainty principle and operator commutativity should not be introduced), the mathematical function of the spectral method remains irreplaceable.It is the only means to find the globally stable states of the system.
III. The Limitations of Dynamic Methods: Lack of Global Information
Dynamic methods (such as directly solving Newton's equations or Lorentz force equations) focus on the time evolution of the system under a given initial condition.
However, for a self-consistent resonant system like the atom, whether the system can form stable orbits depends not on initial conditions, but on the overall spectral structure and boundary resonance conditions.
In other words, dynamic methods can only see "locally changing trajectories" but not "whether the trajectories can be closed";they can describe the acceleration of electrons but cannot predict which energy states are allowed to exist.
Only the spectral method can answer this question:Which frequencies and which spatial modes can form stable self-consistent waves under global boundary conditions?
As pointed out in Natural Quantum Theory:"Dynamic methods cannot obtain global information; what the spectral method reveals is precisely the result of global resonant frequency locking."
IV. The Universality of the Spectral Method: A Main Line from Fluids to Quantum
The spectral method is not a patent of quantum mechanics, but a universal tool for dealing with waves and stable structures in nature.It reveals the overall self-consistent state of the system. Whether in macroscopic fluids, plasmas, or atoms and light fields, the spectral method holds an equally fundamental position.
(1) Fluid Mechanics: Mode Structure and Stability
In fluid systems such as Rayleigh-Bénard convection, the spectral method reveals the formation of critical unstable modes and striped convection cells.Dynamic equations can only tell us how the flow velocity changes, while spectral analysis tells us under what conditions the system will spontaneously organize into periodic structures.The Kolmogorov energy spectrum of turbulence, E(k) ∝ k⁻⁵ᐟ³, is a manifestation of the global spectral distribution.
(2) Plasma Physics: Dispersion Relations and Mode Coupling
In plasmas, all wave phenomena—from ion acoustic waves to Alfvén waves—derive from spectral analysis:linearized equations, Fourier expansion, and solving the eigenequation for ω(k).Only in this way can we know which waves can propagate and which modes will become unstable.Dynamic integration fails here because the coupling and resonance of the system are global cooperative effects.
(3) Electromagnetic Fields and Acoustics: Cavity Modes and Resonance
In the study of electromagnetic resonant cavities, optical fiber modes, and acoustic waveguides, the spectral method reveals the resonance frequencies and spatial distribution modes of the system.These TE/TM modes have the same mathematical structure as the eigen solutions of the Schrödinger equation—both are "standing waves satisfying boundary conditions."This indicates that the quantum spectral method is not a mysterious quantum phenomenon, but a natural extension of classical wave resonance.
V. Conclusion: Spectralization is the Global Language of Nature
In summary, the irreplaceability of the spectral method lies in the fact that:it directly gives the global eigenstructure of the system, while dynamic methods can only give local evolution paths. From fluid mechanics to plasmas, and then to atomic structures, the spectral method has always been the common language of nature for revealing stable states and quantization characteristics.
From the perspective of Natural Quantum Theory, the spectral method is not an "auxiliary tool of quantum mechanics,"but a universal framework for revealing the resonance-frequency locking-stability mechanism of nature.Quantization is spectralization; energy levels are resonant frequency locking;the atom is a natural spectral system, and the spectral method is the key to understanding it.