TITLE:
A Link Merges Classical Mechanics to Quantum Theory (Part III)
AUTHORS:
Alaa S. Bayoumi
KEYWORDS:
Maxwell’s Wave Equation, Internal & External Vacuum, Photon’s Kinetic Energy, Kinetic Theory, Kinetic Temperature, Wien’s Displacement Law, Cosmic Microwave Background, Hydrogen Line 21, Radio Long Wavelength, Fine Structure Constant, Universal Photon Gas Modulus, Photon Thermal Volume, Free Electron Region & X-Ray
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.7,
July
31,
2026
ABSTRACT: This part of the series “a link merges classical mechanics to quantum theory” is dedicated to finding the mathematical equations that govern the relationships between electromagnetic waves, thermal radiation waves, and photons, based on the results obtained in the first two parts of this series. The physical implications of these relationships are then presented within a coherent framework. The process begins by finding a solution to maxwell’s equation in both external and internal vacuum. From this solution, the kinetic energy of the photon is extracted. To relate these results to thermal radiation, Wien’s displacement law is derived from kinetic theory. However, kinetic theory, in its current form, cannot be used to derive Wien’s displacement law. Therefore, kinetic theory is re-derived using an innovative mathematical method specifically designed to solve complex mathematical problems. This method has not been published in any scientific journal, periodical, or conference. As a first consequence, it became possible to define the kinetic temperature of the particle and the temperature of the system, and to find the relationship between them. As a second consequence, a natural wave with an electromagnetic wavelength similar to that observed by radio telescopes was discovered. This wave is believed to originate from deep space and is accompanied by two long waves in the normal radio frequency range, thus providing an explanation for the hiss of radio waves. The generation of the discovered electromagnetic waves is attributed to the exchange of energy between the vacuum and the particles, and vice versa. Then, a connection was made between two of the most famous laws in physics: the ideal gas law and coulomb’s law of electrostatic attraction. This connection led to the discovery of a new physical constant related to the fine-structure constant. The relationship of this constant to Planck’s constant and Boltzmann’s constant was then presented, leading to the concept of the thermal volume of the photon and its relationship to the photon’s spatial volume, as proposed by Heisenberg. From this relationship, the wavelength of the photon was calculated. Therefore, the relationship between electromagnetic waves, thermal radiation waves, and the photon can be mathematically understood, allowing for easy transitions between the mathematical relationships of these three concepts. At the end of this article, a mathematical and physical explanation was presented for the phenomenon of the relationship between x-ray and free electron region. The content of this article is based on the concepts, equations, mathematical relationships, and mathematical formulas that were introduced, developed, and deduced in the previous parts, part one and part two. If you are not quite familiar with their contents, you will not be able to follow the content of this article.