Unified Momentum Formulations in Special Relativity and Quantum Mechanics: Bridging Massive and Massless Particle Models

The de Broglie relation serves as a foundational pillar connecting classical linear momentum with quantum mechanical wave properties. For massive particles moving at relativistic speeds, momentum is routinely expressed using the Lorentz factor, yielding a direct identity with the quantum wave parameters. However, when applied to massless particles such as photons traveling at the speed of light, this product encounters an indeterminate mathematical form. This paper demonstrates how the general relativistic energy momentum dispersion relation resolves this breakdown. Through two detailed numerical case studies involving a relativistic electron moving at sixty percent of light speed and a green photon, it shows that while the algebraic expression relying on rest mass fails for massless entities, the underlying physical equivalence between momentum and wavelength remains globally valid across all particle regimes.