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Planck’s radiation law and the Einstein coefficients

R. Loudon

Published Sep 7, 2000• 58 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780198501770.003.0002

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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.
Abstract The calculations of the preceding chapter display some of the inadequacies of the Einstein theory of interaction of light with atoms. The theory itself gives no prescription for computing the values of the A and B coefficients appropriate to a given atomic transition. For this one must look to the quantum-mechanical theory of transition probabilities. As light beams are generated by emission from atoms and measurements on light beams ultimately depend on the absorption of light by atoms, it is essential to cover the theory of the atomic interaction processes, even though the main interest of the book is the quantum nature of the light itself. More seriously, the Einstein theory is applicable only for broad-band illumination of the atom, where the energy in the incident light is distributed smoothly across the frequencies within the linewidth of the atomic transition. By contrast, many experiments with laser sources use light beams whose frequency distributions are much narrower than atomic transition linewidths.
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.
Abstract The new quantum mechanics could at first be used to answer questions concerning radiation only through analogies with the classical theory. In Heisenberg’s original matrix theory, for instance, it is assumed that the matrix elements of the polarisation of an atom determine the emission and absorption of radiation analogously to the Fourier components in the classical theory. In more recent theories a certain expression for the electric density obtained from the quantum mechanics is used to determine the emitted radiation by the same formulæ as in the classical theory. These methods give satisfactory results in many cases, but cannot even be applied to problems where the classical analogies are obscure or non-existent, such as resonance radiation and the breadths of spectral lines. A theory of radiation has been given by the author which rests on a more definite basis. It appears that one can treat a field of radiation as a dynamical system, whose interaction with an ordinary atomic system may be described by a Hamiltonian function. The dynamical variables specifying the field are the energies and phases of its various harmonic conrponents, each of which is effectively a simple harmonic oscillator. One must, of course, in the quantum theory take these variables to be q-numbers satisfying the proper quantum conditions. One finds then that the Hamiltonian for the interaction of the field with an atom is of the same form as that for the interaction of an assembly of light-quanta with the atom. There is thus a complete formal reconciliation between the wave and light-quantum points of view.
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.
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.
Abstract The events that led to the formulation of the laws of quantum mechanics are presented. Quantum revolution took place in two phases. In the first phase, three important developments took place. Max Planck explained the radiation emitted by heated objects via a quantum postulate, Albert Einstein explained the emission of electrons when light shines on metals by treating light as consisting of photons, and Niels Bohr developed a model for the hydrogen atom by invoking a quantum postulate. In the second phase, Heisenberg and Schrödinger formulated the laws of quantum mechanics that could not only explain all the known phenomena but predict many others.