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 $$e^(-E/(k_B T)$$ ) 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.
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) ) 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.
: Our goal is to quantize Maxwell's equations, leading to a natural interpretation of light as being composed of photons. The key is that the quantization of harmonic oscillators (or, more prosaically, the mathematics of springs) is critical to understanding light. In particular, we show that the possible energies of light form a discrete set, linked to the classical frequency, giving us an interpretation for the photoelectric effect. Our Approach From Einstein's explanation of the photoelectric effect, light seems to be made up of photons, which in turn are somewhat particle-like. Can we start with the classical description of light as an electromagnetic wave solution to Maxwell's equations when there are no charges or currents, quantize, and then get something that can be identified with photons? That is the goal of this chapter. (As in the previous chapter, while all of this is standard, we will be following the outline given in chapter 2 of [41].) Maxwell's equations are linear. We will concentrate on the ‘monochromatic’ solutions, those with fixed frequency and direction. We will see that the Hamiltonians of these monochromatic solutions will have the same mathematical structure as a Hamiltonian for a harmonic oscillator. We know from the last chapter how to quantize the harmonic oscillator. More importantly, we saw that there are discrete eigenvalues and eigenvectors for the corresponding Hamiltonian operator. When we quantize the Hamiltonians for the monochromatic waves in an analogous fashion, we will again have discrete eigenvalues and eigenvectors. It is these eigenvectors that we can interpret as photons.
Photon-photon Interaction has already been described in 2005 and measured in 2011. By D. Englund e.o. in Ultrafast Photon-Photon Interaction in a Strongly Coupled Quantum Dot-Cavity System (2) . This manuscript explains the origin and the mathematical approach in Maxwell’s theory to explain this quantum mechanical effect. To understand, to describe and to calculate this unique phenomenon, we have to go back in time. Before the famous 1927 Sovay Conference to the time before Maxwelll published his famous set of 4 Electromagnetic Equations. To create a new path in Classical Physics to explain Modern Quantum Physics. The famous 1927 Solvay Conference was considered a turning point in the world of Physics. The scientific realists like Albert Einstein had lost and the instrumentalists like Niels Bohr had won the fundamental conflict. Since then Physics has followed the path of the instrumentalists in which Quantum Physics has been determined by the concept of Elementary Particles and Probability Waves. Until May 2013, when for the first time a photo had been made of the Probability Waves Inside the Hydrogen Atom . It should be impossible to make a photo of a non-existing complex probability wave with mass zero But the fact could not be denied anymore that scientists had succeeded to make the “complex” and the “non-existing” quantum mechanical “Probability Wave” visible. Since the Copenhagen Interpretation there is still no clarity about the physical interpretation of the Quantum Mechanical “Probability Wave Function”This Manuscript describes a third unknown interpretation of the Quantum Mechanical “Probability Wave Function”. A fundamental physical interpretation that will change the generally accepted Copenhagen Interpretation. This Manuscript will answer the fundamental question: Does an alternative theory exist in which the existence of this particular photo, of the Quantum Mechanical Probability Wave within the Hydrogen Atom, could be explained. And that the famous photo of the Hydrogen Atom can be explained in a logical and a simple way. To find the answer, we have to go back in time. Back 300 years in time to the moment when Isaac Newton discovered the beauty and one of the many secrets of the light. The moment when science walked away from light. And we return to the fundamental question: “Is it possible to build Matter out of Light (confined electromagnetic waves)?” To answer the fundamental question: Is it possible to create Matter out of Light?, we first have to answer the question how we have come to this world of Probability Waves and Elementary Particles and Quantum Physics.Louis de Broglie described in 1924 in his PhD thesis the wave properties of matter. Erwin Schrödinger published in 1926 the well-known Schrödinger wave equation with the characteristic spherical and elliptical wave solutions. To describe these material waves, the immediately and first explanation was: “There is only one possibility. These are “electromagnetic waves”. De Broglie waves are ” Light Waves”. Because these were the only waves, known in the world of Physics. But there was a fundamental problem. These material waves, discovered by Louis de Broglie and mathematically described by Erwin Schrödinger could never be solutions of the well-known 4 equations of James Clerk Maxwell. The material waves were spherical and elliptical solutions and that was impossible for the solutions of the Maxwell Equations. According to the Maxwell Equations, light always travels along a straight path. A second problem was that the model of the atom of Bohr could not hold. How could an electron be everywhere at the same time in a kind of a spherical shell surrounding the nucleus. And be a particle at the same time. And Bohr found a compromise to declare the “De Broglie” waves to be the probability waves. Two problems were solved. Bohr’s planetary model could hold. And there was a name for the unknown material waves, discovered by Louis de Broglie and mathematically described by Erwin Schrödinger. The theory of “Probability Waves, Elementary Particles and Quantum Physics” had been created. To find a possibility, to create matter out of light we have to go back to the Equations of James Clerk Maxwell. The only possibility to declare the probability waves, discovered by Louis de Broglie, to be electromagnetic waves, to be light, and to be the building element of our universe, is to find evidence that he Maxwell Equations are not complete and that it is possible to confine light (electromagnetic waves) in spherical and elliptical shapes. A new equation has to be found. In this article it has been claimed that this missing equation is equation (5). Because the Theory of General Relativity has been built on the Einstein-Maxwell Equation, a new equation has to be found that describes in a generalized way the interaction between light and matter. This missing equation is equation (5-a).
The classical or semiclassical theory presented in Chapter 2 gives a satisfactory account of light-matter interactions in some cases. However, a detailed understanding of different processes requires a quantum mechanical theory of radiation. The quantum theory of light, like classical theory, is also based on Maxwell's equations. However, in quantum theory the electric field F and the magnetic field H are treated as operators. This chapter develops the correspondence between the two and outlines the method for converting field vectors F and H into quantum mechanical operators. Topics discussed include wave equation in a rectangular cavity, quantization of the electric field, time-dependent perturbation theory, interaction of an electron with the magnetic field, and second-order perturbation theory. Exercises are provided at the end of the chapter.
Light occupies a special position in our attempts to understand nature both classically and quantum mechanically. We recall that Newton, who made so many fundamental contributions to optics, championed a particle description of light and was not favorably disposed to the wave picture of light. However, the beautiful unification of electricity and magnetism achieved by Maxwell clearly showed that light was properly understood as the wave-like undulations of electric and magnetic fields propagating through space. The central role of light in marking the frontiers of physics continues on into the twentieth century with the ultraviolet catastrophe associated with black-body radiation on the one hand and the photoelectric effect on the other. Indeed, it was here that the era of quantum mechanics was initiated with Planck's introduction of the quantum of action that was necessary to explain the black-body radiation spectrum. The extension of these ideas led Einstein to explain the photoelectric effect, and to introduce the photon concept. It was, however, left to Dirac to combine the wave-and particlelike aspects of light so that the radiation field is capable of explaining all interference phenomena and yet shows the excitation of a specific atom located along a wave front absorbing one photon of energy. In this chapter, following Dirac, we associate each mode of the radiation field with a quantized simple harmonic oscillator, this is the essence of the quantum theory of radiation.
The idea of the photon has stirred the imaginations of physicists ever since 1905 when Einstein originally proposed the use of light quanta to explain the photoelectric effect. This concept is formalized in the quantum theory of radiation, which has had unfailing success in explaining the interaction of electromagnetic radiation with matter, seemingly limited only by the ability of physicists to perform the indicated calculations. Nevertheless, it has its conceptual problems—various infinities and frequent misinterpretations. Consequently an increasing number of workers are asking, “to what extent is the quantized field really necessary and useful?” In fact the experimental results of the photoelectric effect were explained by G. Wentzel in 1927 without the quantum theory of radiation. Similarly most electro-optic phenomena such as stimulated emission, reaction of the emitted field on the emitting atom, resonance fluorescence, and so on, do not require the quantization of the field for their explanation. As we will see, these processes can all be quantitatively explained and physically understood in terms of the semiclassical theory of the matter–field interaction in which the electric field is treated classically while the atoms obey the laws of quantum mechanics. The quantized field is fundamentally required for accurate descriptions of certain processes involving fluctuations in the electromagnetic field: for example, spontaneous emission, the Lamb shift, the anomalous magnetic moment of the electron, and certain aspects of blackbody radiation. (The Compton effect also fits here, but see later under references 8b and c.) Here we will outline how the photon concept originated and developed, where it is not required and is often misused, and finally where it plays an essential role in the understanding of physical phenomena. In our discussion we will attempt to give a logically consistent definition of the word “photon”—a statement far more necessary than one might think, for so many contradictory uses exist of this elusive beast. In particular consider the original coining of the word by G. N. Lewis: “[because it appears to spend] only a minute fraction of its existence as a carrier of radiant energy, while the rest of the time it remains an important structural element within the atom…, I therefore take the liberty of proposing for this hypothetical new atom which is not light but plays an essential part in every process of radiation, the name photon!” (our exclamation point). Clearly the present usage of the word is very different.
Quantum theory of photons based on the first quantization technique, similar to that used by Schrödinger in the formulation of quantum mechanics, is considered. First, scalar quantum mechanics of photons operating with the photon wave functions is discussed. Using the first quantization, the wave equation, the Schrödinger-like equations, and the Dirac equation for photons are derived. Then, vector quantum mechanics of photons is introduced, which defines the electromagnetic vector fields. Using the first quantization, the Maxwell equations for photons in a magneto-dielectric medium are obtained. Because the photon’s electric and magnetic fields satisfy the Maxwell equations, all that is known about the classical optical fields can be directly transferred to photons demonstrating their quantum diversity. Relationships between the scalar and vector quantum mechanics of photons and between the Dirac and Maxwell equations are analyzed. To describe the propagation of photons in dispersive media, modified Maxwell equations are introduced.