First-Principles-Based Grand Unified Theory (GUT) for Micro–Macro Modal Quantization (MQ)—Part V: Planck–Einstein–de Broglie Eigen-Relations, Act 1

As a continuation of our previous papers, the Planck-Einstein eigen-relation (or alternatively called Lagrange-Hamilton eigen-relation) and de Broglie eigen-relation in macroscopic electrodynamics are revealed in this paper. Using the macroscopic Planck-Einstein eigen-relation, the generalized quality factor defined in our previous papers can be derived naturally and rigorously. These above provide a solid physical and mathematical foundation for defining the concept of generalized quality factor, whose physical meaning is the reciprocal of normalized free-photon number. Using the above these, we also derive a novel and effective calculation formulation for the generalized quality factor. Based on the macroscopic Planck-Einstein eigen-relation and generalized quality factor, a series of conclusions regarding macroscopic modal quantization (MMQ) are also derived, for example: the physical meaning of eigen-value is further clarified from several different points of view; a natural normalization method for eigenmodal quanta is obtained; a generalized Parseval identity is formulated; the lower bound of generalized quality factor is revealed.