The usual method of calculating the probability of a switch between stationary states by the Quantum Mechanics is really equivalent to finding the intensity of the dipole radiation. Other switches of the system not involved in this radiation may be possible but are not taken account of by the calculation. A method of calculating the probability of switches due to radiation by the quadripole moment seems to be supplied by Dirac's recent theory of the interaction between matter and radiation in his paper “The Quantum Theory of Dispersion.” Equations from this paper will be denoted by a D. The results are probably of a purely theoretical interest and the intensities found probably too faint to be observed; but the present considerations do appear to emphasise the need for including the radiation in the exact theoretical treatment of a Quantum Mechanical system.
The usual method of calculating the probability of a switch between stationary states by the Quantum Mechanics is really equivalent to finding the intensity of the dipole radiation. Other switches of the system not involved in this radiation may be possible but are not taken account of by the calculation. A method of calculating the probability of switches due to radiation by the quadripole moment seems to be supplied by Dirac's recent theory of the interaction between matter and radiation in his paper “The Quantum Theory of Dispersion.” Equations from this paper will be denoted by a D. The results are probably of a purely theoretical interest and the intensities found probably too faint to be observed; but the present considerations do appear to emphasise the need for including the radiation in the exact theoretical treatment of a Quantum Mechanical system.
Abstract 1. The procedure usually adopted in the Quantum Theory for the estimation of the intensities of a spectrum line includes two stages : (1) the electric moment, p, of the distribution responsible for the spectrum line is calculated by means of Schrodinger’s integrals or some equivalent method ; (2) the rate of loss of energy due to radiation is calculated by the classical formula due to H. Hertz, viz., 2p2/3c3, just as if the distribution were concentrated in a dipole of precisely the same moment p. The first stage involves no difficulties beyond those met with in the evaluation of the integrals or matrices with which we are concerned, the only principles needed being those of the particular quantum theory adopted. But the second stage is open to grave objections and can only be justified on the ground of necessity owing to the absence of any completely satisfactory quantum theory of the electromagnetic field. Sometimes an appeal is made to Bohr’s Correspondence Principle, but this itself involves a somewhat hazardous extrapolation from infinitely large down to small quantum numbers and cannot take the place of a rigorous calculation. A logically consistent method is possible, based on Maxwell’s theory as represented by his electromagnetic equations and their solutions in terms of the classical retarded potential integrals. It may be objected that we do not know how far Maxwell’s equations can be applied to the interiors of atoms, in particular to distributions like those of Schrodinger, but precisely the same objection applies to the customary use of the classical expression for the radiation from an oscillating dipole. Moreover, the attempts made by Heisenberg and Pauli, Dirac and others to formulate a quantum theory of the electromagnetic field have all been based on the assumption that such a theory must be consistent with Maxwell’s equations. The direct application of the methods of classical electromagnetic theory to Schrodinger’s distributions can be justified at least as a rigorous attempt to determine their electromagnetic field and the radiation from them, and the validity of the procedure will have to be judged by the agreement, or otherwise, of the results obtained with experience. In this way we avoid the assumption that a spatially extended distribution can be treated as if its electric moment were concentrated in a point; in fact, the results obtained in this paper for the more important spectrum line series show definitely that the assumption is false, because it leads to results in many cases quite inconsistent with a rigorous application of Maxwell’s theory.
Annalen der Physik , Ser. 5, 9: 338–46 (1931). Received 25 February 1931. The Dirac theory of radiation gives a satisfactory account of the absorption, emission, and dispersion of radiation by atoms. Beyond this, quantum electrodynamics also provides a satisfactory treatment of interference phenomena. The results of the theory in most cases agree with what is expected from the correspondence principle, but the calculations which lead to this objective have hitherto been rather complicated, and the simplest consequences of classical radiation theory can only be derived along the circuitous route through a quite nontransparent Schrödinger equation in an infinite-dimensional space. Below we shall describe a method for treating radiation problems which connects, much more closely than the previous ones, the intuitive conceptions of classical theory with those of wave mechanics and which therefore, in most cases, and without detours, yields the result expected in terms of the correspondence principle. In this method, the starting point shall not be the Hamiltonian function or the corresponding Schrödinger equation in configuration space but rather the equations of motion, i.e. the Maxwell and Dirac wave equations. These differential equations are integrated explicitly, after the value of the wave functions ( E, H , and ψ σ ) at time t = 0 are regarded as given; here it should be noted that the wave functions at time t = 0 are noncommuting quantities. This noncommutability of the initial values of E, H , and ψ σ does not disturb the integration, however, as long as the interaction between matter and radiation is assumed to be small and only linear terms in this interaction are considered.
Abstract In a recent paper Schott has criticized the quantum mechanical method of finding the intensities of spectral lines, and in particular the assumption that the intensity may be derived by treating the atom as a dipole, radiating classically. The electric moment of this dipole is taken as p = e-2πivt ∫ Ψ*frΨidτ + Conjugate complex, (1A) where Ψi and Ψf are the wave functions of the initial and final states of the atom respectively, and in the Quantum Theory the usual assumption is that the energy radiated per unit time is given by R = 2 |p¨¯|2/3c2, (1B) where p¨¯ is the time average of p¨. A more consistent method is suggested in which the electric density ρ and the current j, corresponding to the transition, are found, and the electromagnetic field due to these two is examined.
Abstract In a recent paper under the same title, G. A. Schott finds that the usual method of calculating the intensities of spectrum lines is liable to lead to gross errors. Although one would expect from general reasons that any improvement of the usual method would only lead to small corrections and that, therefore, Schott’s conclusion cannot be free from error, it may not be devoid of interest to show that a correct calculation starting with Schott’s formulæ leads, in contradiction to Schott’s conclusion, to the generally accepted classical result. The aim of Schott’s calculations is to obtain a quantum mechanical expression for the energy radiated by an atom, which is classically given by R=2/3C3{р(t-r/c)}2, (1) p(t) being the electric moment of a dipole representing the atom. Schott considers the special case of a hydrogen atom emitting a radiation corresponding to a transition from the centrosymmetrical state 1=Af(r) (2)
In our preceding treatments of the interaction of radiation with matter, we assumed that the electric field is classical. In most cases, this assumption is validated by experiment. There are several instances, however, for which a classical field fails to give experimentally observed results, whereas a quantized field succeeds. This is true, for example, of spontaneous emission, a phenom enon which was described phenomenologically in our earlier work (see Chaps. 2 and 7). Another case involves the 2 and 2p \ states in the hydrogen atom, which are predicted to have the same energy not only by the simple theory of Chap. 1, but also by the more sophisticated Dirac theory. In both cases, a classical electric field was assumed. Experimentally, the two levels differ by approximately the Lamb shift, 1057 MHz. A fully quantized treatment of the field and atom system gives impressive agreement with the experimentally observed shift. Derivation of the fluctuations in intensity of a laser near threshold requires the quantum theory o f radiation. In addition there are other problems for which the concept of a “particle,” the photon, is either necessary or convenient. Despite a popular impression to the contrary, however the explanation of the photoelectric effect is not one of these (see Prob. 2-10).
The formal procedure used to predict the results of experiments by wave mechanical methods is closely related to the corresponding classical calculation and in many cases the result for a particular quantum state is essentially an average of the appropriate classical variable over the particle density function. The author presents a discussion of the way this concept can be applied to an elementary treatment of the emission of radiation from a system of charges and it also includes an account of an apparatus capable of demonstrating the conditions necessary for the emission of radiation.