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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Light quantum from classical electromagnetic theory

Weilong She

Published Aug 26, 2007• 1 citation
Publisher

Abstract

We show in this paper that the quantized energies of light in vacuum can be derived directly from the classical electromagnetic theory through the consideration of statistics based on classical physics, which are just a series of expectation values corresponding to relatively stable statistic distributions of electromagnetic harmonic oscillators. Therefore, the quantization of energy is an intrinsic property of light as a classical electromagnetic wave and has no need of being related to particles. The theory developed in this paper is also used to derive the energy equipartition theorem and the discrete levels of hydrogen atom from classical mechanics.

DOI
10.1117/12.730984

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We show in this paper that the quantized energies of light in vacuum can be derived directly from the classical electromagnetic theory through the consideration of statistics based on classical physics, which are just a series of expectation values corresponding to relatively stable statistic distributions of electromagnetic harmonic oscillators. Therefore, the quantization of energy is an intrinsic property of light as a classical electromagnetic wave and has no need of being related to particles. The theory developed in this paper is also used to derive the energy equipartition theorem and the discrete levels of hydrogen atom from classical mechanics.
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Today it still remains a challenge whether quantum mechanics has an underlying statistical explanation or not. While there are and were a lot of models trying to explain quantum phenomena with statistical methods these all failed on certain levels. In this paper a method is proposed that is not only based on a classical statistical framework but it has an underlying physical model behind it and it can explain some of the basic characteristics of quantum mechanics. It will be shown that if look at the properties of a charged particle in a random electric field one can obtain the discrete energy values of the harmonic oscillator and the infinite potential well, and also gives a good qualitative description of the double-slit experiment and measurement theory. Also the side-effect of the model is the emergence of a constant with an action dimension.
James Maxwell is one of the most important physicists of all time. He did landmark work in unifying electricity and magnetism and also describing the two with a coherent mathematical framework. In the process, he discovered that light is an electromagnetic wave that can oscillate in a vacuum. He also played critical roles in the development of statistical mechanics, which is why we call e(−E/(kBT)e^(-E/(k_B T)e(−E/(kB​T) ) the Maxwell-Boltzmann distribution. Unfortunately, to fully work with his mathematical formulation of electromagnetism, via the so-called Maxwell equations, requires one to master vector calculus. So, in this chapter, we do not cover the complete derivation of his work, but instead show how it applies directly to light and from that we are then be able to jump into quantum optics. The goal of this chapter is to precisely explain and illustrate for you what a photon really is. Not just a particle of light, nor just a wavepacket, but a specific quantum structure that has the property that it can be measured once and only once. Then we put together all of these new insights to explain how one measures gravitational waves with quantum mechanics.
A photon quantization volume is defined, an intrinsic physical geometrical property of a single photon state, showing that photons are not point particles and constitute a particular case in the standard particle model. Based on the quantized vector potential amplitude a photon wave function is established satisfying Schrödinger’s equation and normalized with respect to the quantization volume. The single photon electric and magnetic fields in vacuum are calculated independently on any external cavity parameter and they are proportional to the square of the frequency. The electromagnetic vacuum derives straightforward from the photon vector potential function and has both classical and quantum representations. The photon wave function is directly related to the electromagnetic vacuum states setting the fundamentals for understanding entanglement.
A photon quantization volume is defined, an intrinsic physical geometrical property of a single photon state, showing that photons are not point particles and constitute a particular case in the standard particle model. Based on the quantized vector potential amplitude a photon wave function is established satisfying Schrödinger’s equation and normalized with respect to the quantization volume. The single photon electric and magnetic fields in vacuum are calculated independently on any external cavity parameter and they are proportional to the square of the frequency. The electromagnetic vacuum derives straightforward from the photon vector potential function and has both classical and quantum representations. The photon wave function is directly related to the electromagnetic vacuum states setting the fundamentals for understanding entanglement.
Atoms and ions containing residual bound electrons do not quite resemble the simple harmonic oscillator model used so successfully in the classical theory of radiation. However, replacing the atoms or ions with sets of harmonic oscillators of a great number of discrete resonance frequencies and having various amplitudes, together with the results of classical radiation theory, go a long way toward a quantitative description of emission or absorption spectra. The set of resonance frequencies is obtained from measured or calculated energy levels using Ritz's combination principle. The amplitudes are associated with matrix elements of appropriate quantum mechanical operators between wave functions of the two energy eigenstates involved at a given frequency. In other words, quantities of the emitters, absorbers, or scatterers are described quantum-mechanically, whereas the electromagnetic field is treated classically. Such semi-classical description of matter-electromagnetic field interactions became unnecessary very early in the development of quantum theory. It will therefore not be discussed in any detail. Instead, we will begin immediately with the combined theory of matter and radiation (Heitler 1954, Dirac 1958, Loudon 1983). Quantum theory of particles and fields There are various ways to also quantize the electromagnetic fields (Cohen-Tannoudji, DuPont-Roc and Grynberg 1989), of which that performed on the combined Hamiltonian equations of motion for the field-matter system is followed here.