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.
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.
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)
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 It will probably be agreed that among all the recent developments of the quantum theory, one of the least satisfactory is the theory of radiation. The present paper is intended as a preliminary to a new line of attack on the subject. It was begun some time ago, but owing to lack of success in carrying it to a conclusion, its publication has been much delayed. In the meantime other papers have appeared, which in some respects follow the same train of thought. The authors of these works have carried their methods further in some directions than I have attempted, but there is still perhaps room for the discussion of a number of questions from the rather different point of view adopted here. 1. The main principle of the present work is the idea that, since matter and light both possess the dual characters of particle and wave, a similar mathematical treatment ought to be applied to both, and that this has not yet bee done as fully as should be possible. Whereas we have a fairly complete calculus for dealing with the behaviour of any number of electrons or atoms, for photons the existing processes are much less satisfactory. The central difficulty, which makes it hard to apply the ordinary methods of wave mechanics to light, is the fact that (at least according to our present ideas) photons can be created and annihilated, and to represent this in a wave system we have to be able to think of a medium suddenly coming into existence and then going out again, when the light that it was carrying is absorbed. Such behaviour is a grave difficulty in the way of allowing us to think of the photon as a wave, and tends to make us think with more favour of its particle aspect, until we recall that after all it is quite unlike any known particle to come into existence and later to disappear without trace. The theories at present current, such as that of Heisenberg and Pauli, avoid these difficulties because they are mainly formal generalisations of the classical theory; this frees them from the above difficulties, but they pay for it in being highly abstract, and, as it has turned out, rather unsuccessful.
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 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.
Bohr’s correspondence principle is that associated with each observable in classical mechanics is an observable in quantum mechanics that follows certain quantization axioms which can be used to explain the energy spectra of atoms and quantum oscillators. In Schrodinger’s wave mechanics, the state of the system at any time is described by a density operator in a Hilbert space. The chapter describes how the Pauli equation can be used to calculate the average electric dipole moment and average magnetic dipole moment of an atom in an external electromagnetic field. This calculation enables to calculate the electric polarization field and the magnetization field of a material.