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The Prince Wave Equation: A Complete Derivation from de Broglie's Fundamental Principle Without Hamiltonian Formalism

Mustafa Khan

Published Jun 6, 2026• 0 citations
Publisher

Abstract

We derive a complete wave equation framework from de Broglie’s fundamental relation λ = h/p, expressed in differential form as δp = hδψ where ψ = 1/λ represents spatial compaction. Without invoking Hamiltonian formalism or particleconcepts, we obtain a general nonlinear, anisotropic wave equation that reduces to both Maxwell’s electromagnetic equations and d’Alembert’s universal wave equation in appropriate limits. We further develop the multiplicative superpositionprinciple for combining multiple Prince waves, showing that ψ_n+1 = (Product of ψ_i)^1/n is the unique formulation maintaining dimensional consistency and physical unity. Remarkably, Planck’s constant h cancels from all final wave equations, revealing the scale-independent nature of wave propagation. All results emerge from the single principle δp = hδψ.

DOI
10.5281/zenodo.20573484

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We derive a non-linear wave equation directly from de Broglie’s fundamental relation λ= h/p without invoking the Hamiltonian formalism or probability interpretations. By treating ψ = 1/λ as an ontological spatial compaction field, we obtain an exact, general non-linear field equation. The nonlinear term is written as |Ψ|^2 Ψ to maintain phase coherence between the linear propagation and the self-interaction. Both isotropic and anisotropic forms are presented. Planck’s constant cancels out completely, revealing the geometric nature of the propagation.
We present a mathematically finalized, non-linear wave equation framework derived directly from de Broglie’s fundamental relation λ = h/p[cite: 15, 225]. By treating ψ = 1/λ as an ontological spatial compaction field [cite: 15, 226], we eliminate arbitrary te xtbook global coordinate grids and unphysical Hamiltonian scaffolding[cite: 16, 225]. We demonstrate that when the spatial derivative operator is constrained strictly by the Prince Razor (δp = hδψ) along a local, physics-determined 1D radial axis[cite: 19, 163], all coordinate artifacts and mixed space-time cross-derivatives naturally vanish. Incorporating a phase coherent non-linear self-interaction term |Ψ|^2 Ψ, the master equation reduces to a pristine structural identity (1 = 1) when evaluated against the Euler Complex Plane (ECP) baseline wave, flawlessly vindicating de Broglie’s original vision of a localized wave-particle soliton.
We present a mathematically finalized, non-linear wave equation framework derived directly from de Broglie’s fundamental relation λ = h/p, via the Prince Razor (named in honor of Louis de Broglie, Prince de Broglie). By treating ψ = 1/λ as an ontological spatial compaction field, we eliminate arbitrary textbook global coordinate grids and unphysical Hamiltonian scaffolding. We demonstrate that when the spatial derivative operator is constrained strictly by the Prince Razor (δp = h δψ)along a local, physics-determined 1D radial axis, all coordinate artifacts and mixed space-time cross-derivatives naturally vanish. Incorporating a phase-coherent nonlinear self-interaction term |Ψ|^2 Ψ, the master equation reduces to a pristine structuralidentity (1 = 1) when evaluated against the Euler Complex Plane (ECP)baseline wave, flawlessly vindicating de Broglie’s original vision of a localized waveparticle soliton. The ECP is the geometric representation of the Euler identity; its two faces and absence of a center are essential for obtaining the correct coefficients and for avoiding Cartesian artifacts.
We present a mathematically finalized, non-linear wave equation framework derived directly from de Broglie’s fundamental relation λ= h/p[cite: 15, 225]. By treating ψ = 1/λ as an ontological spatial compaction field [cite: 15, 226], we eliminate arbitrary textbook global coordinate grids and unphysical Hamiltonian scaffolding[cite: 16, 225]. We demonstrate that when the spatial derivative operator is constrained strictly by the Prince Razor (δp= hδψ) along a local, physics-determined 1D radial axis[cite: 19, 163], all coordinate artifacts and mixed space-time cross-derivatives naturally vanish. Incorporating a phase-coherent non-linear self-interaction term |Ψ|^2 Ψ, the master equation reduces to a pristine structural identity (1 = 1) when evaluated against the Euler Complex Plane (ECP) baseline wave, flawlessly vindicating de Broglie’s original vision of a localized wave-particle soliton.
We derive the nonlinear de Broglie wave equation inthe Prince Frame from the work equation, the deBroglie relation λp = h, and the geometry of theEuler Complex Plane (ECP). The separation of spatialand temporal components of the wavefunctionarises naturally from the ECP, where the spatial compactionfield ψ(λi) is multiplied by the Euler oscillatore±iωt. The Prince Frame provides the coupledspatial Laplacian, while the ECP provides the nonlinearself-interaction term 8π^2{v(g)}^2|ψ|^2 ψ. The resultis a fully derived geometric wave equation with noempirical inputs.