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.
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.
Abstract The quantum mechanical description of the interaction of light with atoms treats absorption, gain, and recombination as upward and downward induced transitions of electrons between two states by perturbation due to the optical electric field. This provides the basis for a quantitative theory of diode lasers. The chapter begins with Einstein’s description of absorption, stimulated emission, and spontaneous emission, setting the scene for calculation of the transition probability by solving the time-dependent Schrödinger equation for a coherent superposition of the wavefunctions of two states. The resulting absorption spectrum has a Lorentzian lineshape due to dephasing of these wavefunctions. Expressions for the optical absorption cross section and Einstein coefficients are derived in terms of the dipole matrix element, and the dipole is illustrated using a hypothetical one-dimensional atom. Fermi’s Golden Rule for transitions to a continuum of states (as in quantum wells) is introduced in terms of the momentum matrix element.
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 To describe the interaction of a two-level atom with radiation we shall use a semiclassical treatment, i.e. the radiation is treated as a classical electric field but we use quantum mechanics to treat the atom. From the theory of the interaction with radiation, we will be able to find the conditions for which the equations reduce to a set of rate equations that describe the populations of the atomic energy levels (with a steady-state solution). In particular, for an atom illuminated by broad band radiation, this approach allows us to make a connection with Einstein’s treatment of radiation that was presented in Chapter 1; we shall find the Einstein B coefficient in terms of the matrix element for the transition.
An attempt is made to present a consistent detailed theory of optical phenomena, based on the suggestion recently made by the author in collaboration with Bohr and Kramers. This suggestion is that atoms are radiating or absorbing continuously, during their stationary states, and that transitions influence radiation only in terminating the radiation characteristic of the old state, and substituting that of the new. The atoms, on the other hand, are supposed not to change their energy while radiating, but to change it discontinuously on transition. This necessitates giving up the detailed application of the conservation of energy in interaction between atoms and radiation. In the present paper, an additional suggestion is made, namely that resonance radiation is to be identified with the radiation carried out by the spherical wavelets which, by their interference with the external field, also produce absorption. This is practically a return to the classical picture of resonance radiation. Statement of the theory consists formally of two parts: the description of the behavior of atoms, and of radiation, when the two interact. The first involves the statement of the probabilities of transition of atoms, and is taken without change from Einstein. The probability of interruption of coherent vibrations is also discussed. The second requires the specification of the fields emitted by atoms in any steady state under the influence of any external field. These consist of spherical wavelets of the frequencies of the various quantum lines which the atom can emit or absorb in the stationary state in which it is. For emission frequencies, wavelets are emitted even in the absence of an external field; for absorption frequencies, external fields induce wavelets, similar to the wavelets of linearly bound electrons in electron theory, which on the one hand interfere with the external field to produce the phenomena of dispersion, etc., and on the other carry out the resonance radiation in all directions. Detailed descriptions of these fields are given. In a discussion of the theory, it is first shown that the assumptions which have been made about the field satisfy the correspondence principle, in that for large quantum numbers the field aproaches that which would be emitted classically on account of the interaction of an external field and a multiply periodic atom. On the other hand, it is shown that the field has a character essentially like that of classical electron theory, which is known to be in general agreement with experiments. Next, the energy and momentum relations at interaction of atoms and light are investigated, and it is shown that, although these quantities are not precisely conserved, still the assumptions as to atomic and radiation processes have been so made that they are conserved on an average over a great number of atomic processes. Finally the spectral resolution of emitted and absorbed radiation is considered; the theory gives a minimum breadth for the lines, depending on the finite length of wave trains resulting from the finite life in stationary states, and it is shown that Kirchhoff's law is obeyed, the emission and absorption lines being of the same breadth. By way of illustration, specific application of the theory is made to the simple examples of emission of light by bombardment of electrons at the resonance potential, resonance radiation, its quenching by presence of foreign gases, absorption, scattering, and disperison.
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.
Introduction The preceding chapters have focused on atomic transitions induced by applied radiation. That is, there is an optical field illuminating the atoms, and their response is calculated. It is implicitly assumed that an excited atom left alone will spontaneously return to the ground state, and conserve energy by emitting light that satisfies E 2 − E 1 = ħ ω 21 in accordance with the Planck and Bohr pictures. But this spontaneous emission process is not addressed quantitatively in these earlier chapters. Spontaneous emission is one of the most pervasive phenomena of atomic physics, and yet its origins remain among the least well-understood. The number of misconceptions is enormous, and they are to be found in textbooks, journal articles, and lecture notes. Because the natural decay of an excited atom adds energy to what is usually an empty mode of the electromagnetic field, the process is fundamentally quantum mechanical. Classical or even semi-classical descriptions cannot be assured to give correct answers. This is not to say that such discussion is to be avoided, but only that caution is needed in interpreting the results. Toward the end of the nineteenth century, many experimental results made it clear that classical mechanics needed to be supplemented by new theories. The first venture into the new physics was Planck's hypothesis that the energy of classical radiating oscillators like those discussed in Chap. 1 would have to be quantized into integer multiples of ħ ω. This notion led to his famous formula for the spectrum of thermal radiation from a non-reflective object (black body), and this formula describes a radiation field in thermodynamic equilibrium with its environment. The result of Planck's hypothesis in the context of thermal equilibrium plays a vital role in the discussion of spontaneous emission. Einstein A - and B -coefficients Soon after Bohr proposed a model of atomic structure consisting of discrete energy levels, connected by radiation whose energy satisfied Δ E = ħω when atoms made transition between them, Einstein published the first serious attempt to connect the absorption and emission processes. Absorption, he reasoned, could occur only in the presence of applied radiation, whereas spontaneous decay occurred without the application of any field.