Maxwell theory associate two fields with charges and Maxwell equations govern the evolution of these two fields.
I think this could be an interesting question!!! In fact, electromagnetic, be merged two physical concepts (Magnetic and electric field).
- The universe is composed of matter and radiation, which were thought to have very different properties throughout much of human history. Starting from the beginning of the twentieth century, however, there was a dramatic shift of perception. Some physicists became aware that matter and radiation cannot be regarded as two entirely different objects. This changes of thinking was mainly due to the landmark discoveries of: (1) Max Planck, who found that the energy of radiation is quantized, (2) Albert Einstein, who proposed the concept of photon based on the photo-electric effect, and (3) Louis de Broglie, who further proposed that matter also has a wave property such that the wavevector is connected with momentum. These discoveries enabled Bohr to propose the atomic model of hydrogen and inspired Schrödinger to develop the wave mechanics to describe quantum behaviors of the electron.
- With the successful development of quantum mechanics, people became fully aware of the wave-corpuscle duality of a particle and began to treat matter and radiation in a unified view. For example, in his highly popular textbook of quantum mechanics, A. Messiah explicitly stated: “the possibility exists of establishing a unified theory in which matter and radiation are different varieties of the same type of object, having wave-like and corpusclar character. These suppositions, which have guided de Broglie in his theory of matter waves, were found to be entirely justified.
- Yet, according to quantum physics taught today, there are significant differences between matter and radiation. First, radiation, or the motion of photons, is described by the Maxwell’s equations. The motion of electron, on the other hand, is described by the Schrödinger equation in the non-relativistic situation, or by the Dirac’s equation in the relativistic case. Second, while the photon is known to be a wave of the EM field, the wave of a particle described by the quantum mechanics is not regarded as a physical wave; it is thought to be associated with only the probability of finding a particle in a particular location and time. Third, particles that make up matter usually have rest mass, while the radiation does not. Forth, in the quantum field theory of today (i.e., the Standard Model of elementary particle), the particle is regarded as a point-like object . The matter is supposed to be made up of fermions that have spin 1/2, while radiation is carried by bosons that have spin 1.
- Looks, some of these differences might be superficial. If one probes into a deeper layer of the foundation of quantum mechanics, one may be able to find a common root that can truly unify the theories of matter and radiation.
I do not understand the question. What is the aim? As regards electrodynamics, we have already quantum electrodynamics which amounts to the quantum-mechanical description of the classical fields of the Maxwell theory. Incidentally, there is a historical connection between the Maxwell equations and the Schrödinger equation, about which one can read in the biographies of Erwin Schrödinger -- for instance in the one by Walter J. Moore, A Life of Erwin Schrödinger (Cambridge University Press, 1994). This connection becomes manifest in the semi-classical treatment of the Schrödinger equation.
Note added: You may wish to consult the following book, the link to which I attach below: Analogies in Optics and Micro Electronics, edited by W. van Haeringen and D. Lenstra (Springer, 1990).
http://en.wikipedia.org/wiki/Quantum_electrodynamics
http://www.springer.com/materials/book/978-0-7923-0708-2
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