The Temporal Wave Function of Elementary Particles and the Discrete Inverse Square Law ——From Wave Function Substantialization to a Unified Description of Strong Nuclear Potential Steps
**Translation of the Abstract:**
Previous works [1,2] established a phenomenological framework of “observer-compiled” dynamics, revealing a cross-scale scaling relation \( L = C \cdot (m_{\mathrm{P}}/m_{\mathrm{ref}}) \cdot \ell_{\mathrm{P}} \) and deriving a discrete distance sequence \( R_n = n \times 0.21\ \mathrm{fm} \) for the strong force window. However, two recurring mathematical features in these relations—the \( 1/n^2 \) decay law of energy levels and the frequent appearance of the geometric factor \( \pi \)—have not yet received a unified geometric explanation. Starting from the core picture that “elementary particles are the materialization of wavefunctions”, this paper establishes a formal system of three-dimensional temporal wavefunctions for elementary particles. From the normalization condition of discrete summation on a ninefold grid, we define the squared modulus of the wavefunction as the “existence weight” of the particle in the six-dimensional intrinsic spacetime. From this, we derive: the particle’s magnetic pole is a topological necessity of the wavefunction deviating from the equilibrium baseline direction; the particle radius corresponds to a half-period (phase change of \( \pi \)); the amplitude of a single particle decays as \( 1/R \) with distance; and the coupling strength between two particles decays as \( 1/R^2 \). On a discrete lattice \( R = n \ell_{\mathrm{P}} \), the inverse-square law naturally yields a discrete energy level structure of \( 1/n^2 \), providing a unified explanation for the common mathematical origin of the hydrogen atom energy levels and the stepwise strong nuclear force potential.
Based on this framework, we introduce a three-quark semi-fusion model—whose quark sharing rules are strictly derived from the interference dynamics of wavefunctions (Appendix A)—and systematically compute the neutron-neutron interaction potential energy at discrete lattice points from \( n = 1 \) to \( n = 9 \). In the fully fused region (\( n = 1 \)), the potential is strongly repulsive at \( +1880\ \mathrm{MeV} \); the hard repulsive core (\( n = 2 \)) is \( +940\ \mathrm{MeV} \); the force equilibrium point (\( n = 3 \)) is zero; in the medium-range attractive region (\( n \ge 4 \)), the absolute value of the potential energy obeys a strict inverse-square law \( |V_n| = 940 / n^2\ \mathrm{MeV} \), in good agreement with phenomenological potentials such as Argonne \( v_{18} \) at the corresponding distances.
We propose a falsifiable prediction: at center-of-mass energies around 188, 157, 134, and 118 MeV, the neutron-proton scattering differential cross-section or the \( ^1S_0 \) partial-wave phase shift should exhibit discontinuous inflection points, which can be directly tested by high-precision scattering experiments with an energy step size \( \le 1\ \mathrm{MeV} \).
temporal wavefunction/ three-dimensional time/ normalization on a ninefold grid/ existence weight/ inverse-square law/ discrete energy levels/ strong nuclear force potential/ three-quark semi-fusion/ falsifiable prediction
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