‌Deep Analysis of Matter Waves

The connection between Planck's constant and wavelength is the theoretical cornerstone of the wave-particle duality of matter. The essence of Planck's constant lies in the angular momentum of fundamental particles, and the conservation of angular momentum in fundamental particles is what gives rise to Planck's constant. Whether non-fundamental particles exhibit wave-like behavior depends on the ratio of their angular momentum to momentum. If this ratio exceeds the particle's radius, the non-fundamental particle can exhibit wave-like properties; otherwise, the wavelength is concealed by the radius. The de Broglie wavelength formula is: λ = h/p, where Planck's constant represents the angular momentum of fundamental particles. Here, h = mrv, where m is the mass of the fundamental particle, r is its spatial radius, and v is its rotational speed. Since h = mrv and p = mv₁ (where v₁ is the propagation speed of the fundamental particle), it follows that λ = h/p = mrv/mv₁ = rv/v₁‌Conclusion‌: The wave-particle duality of fundamental particles is independent of their mass. In other words, the wave-particle duality of matter is not related to mass, nor is it inherently tied to the microscopic or macroscopic scale. Matter waves are universally present.