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CRC Press eBooks

Quantum Theory of Radiation

Pradip Narayan Ghosh

Published Feb 19, 2018• 103 citations
Publisher

Abstract

The quantum theory of radiation was formulated by Dirac in 1927, when he combined the wave nature of light with the particle behavior and could explain the interference phenomena. He also explained the atomic behavior of absorbing a discrete quantum of radiation for excitation from one level to another. The Michelson-Morley experiment on light led to the formulation of special theory of relativity, while Rayleigh-Jeans formulation on Black Body radiation led Planck to propose the quantum of action that finally marked the beginning of quantum mechanics. The energy of the radiation field of the classical wave has been expressed as the sum of the energies of individual modes in the cavity. The interaction of atom with radiation field was explained by Einstein in terms of stimulated emission, absorption and spontaneous emission. After the development of quantum mechanics, perturbation theory could be used to describe the Einstein B-coefficients.

DOI
10.1201/9780429492396-6

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The quantum theory of radiation was formulated by Dirac in 1927, when he combined the wave nature of light with the particle behavior and could explain the interference phenomena. He also explained the atomic behavior of absorbing a discrete quantum of radiation for excitation from one level to another. The Michelson-Morley experiment on light led to the formulation of special theory of relativity, while Rayleigh-Jeans formulation on Black Body radiation led Planck to propose the quantum of action that finally marked the beginning of quantum mechanics. The energy of the radiation field of the classical wave has been expressed as the sum of the energies of individual modes in the cavity. The interaction of atom with radiation field was explained by Einstein in terms of stimulated emission, absorption and spontaneous emission. After the development of quantum mechanics, perturbation theory could be used to describe the Einstein B-coefficients.
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.
Abstract The present chapter is devoted to a description of the procedure by which quantum mechanics is applied to the electromagnetic field. We obtain expressions for the operators that represent the field observables and we examine some of their properties. The quantization introduces characteristic quantum-mechanical effects into the properties of the radiation field, for example field operators that do not commute with each other and thus represent incompatible measured quantities. The Hamiltonian of the coupled radiation-atomic system is expressed wholly in terms of quantum-mechanical operators. We show that the full quantum theory of the interaction between the radiation field and atoms produces the same transition rates for absorption and stimulated emission as does the semiclassical theory. The rate of spontaneous emission confirms that obtained from the phenomenological Einstein theory. The treatment of absorption of radiation by atoms also identifies the quantum-mechanical operator that represents a measurement of the energy or intensity of a light beam, and this has important consequences for the descriptions of the experiments that can be performed in practice.
This chapter presents the basic physical ideas that underpin the quantum theory. Historians place the beginnings of the quantum theory at 1900. That year Max Planck published his famous formula for the distribution of energy as a function of frequency in black-body radiation. Curiously, although radiation theory gave birth to quantum mechanics, the interaction of matter and electromagnetic radiation is a rather advanced topic in quantum mechanics and it took almost fifty years for an acceptable formulation to be achieved. One reason for this is that photons are quanta of the electromagnetic field, and the quantum theory of fields requires a relativistic treatment to be entirely successful. The corpuscular nature of light was only half of the input that gave birth to quantum mechanics. The other half concerns matter. Given that the discreteness of photons manifests itself during the emission and absorption process, it is not surprising to find that quantum oddities afflict the behaviour of material particles too.
This work examines the physical and conceptual evolution of the light quantum from Planck's blackbody theory to the theoretical and experimental developments that led to the quantization of the electromagnetic field. The present study shows that the decisive transition occurred in Einstein's quantum theory of radiation (1916-1917). Absorption, stimulated emission, and spontaneous emission were formulated as elementary probabilistic mechanisms whose statistical balance alone reproduces the blackbody spectrum. In particular, spontaneous emission requires the emission of a light quantum, thereby implicitly proving its physical necessity before its theoretical status was clarified. At the same time, the already existing term photon began to acquire a stable usage following Lewis's 1926 proposal and became increasingly associated with Einstein's light quantum. By the mid-1920s, the central problem had shifted from whether light quanta were physically required to how radiation could be incorporated into the emerging quantum-mechanical formalism. This transition marks a key stage, illustrating how initial debates about the existence of light quanta gave way to their integration into a comprehensive theoretical structure. The resulting asymmetry between the novel quantum description of matter and the still-classical description of radiation, called into question by the phenomenon of spontaneous emission, identifies the physical problem that led to the quantization of the electromagnetic field.
Abstract Another main stream in the initial formulation of the quantum theory was concerned with the interpretation of atomic spectral lines. The interaction of electromagnetic radiation with atoms was discussed by Einstein f4] in 1917. His theory of the absorption and emission of light by an atom depends upon simple phenomenological considerations but it leads to predictions that are reproduced by the more formal quantum mechanics developed later. The use of quantum theory is, of course, essential for even a gross description of the nature of the states of an atom and the theory is outstandingly successful in its ability to predict the finest details of atomic energy-level structures.
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.