Abstract When a beam of Canal-Rays passes into a high vacuum, it is well known that the light-intensity falls exponentially to a constant value which, if the pressure is sufficiently low, is very small indeed. The experiments of Wien, and the theory developed by him, showed that the observations could be equally well accounted for on either of two hypotheses, namely that the light from each atom falls off exponentially at the observed rate, or alternatively, that all the energy from any one atom is emitted in a very short time indeed, and that what is measured is essentially the decay-curve of those atoms that are in the initial state for the radiation in question. As to whether the maximum emission (on either theory) occurred exactly at the point of excitation, however, these experiments provided very little evidence; the excitation was in fact, so far as can be determined, distributed continuously over a region about which all that can be said is that it was bounded more or less definitely on one side by the back of the cathode, and it was possible that the curves for a point-excitation would really resemble, for instance, the activity-curve of an intermediate radioactive substance initially absent from a preparation. In fact plausible grounds for believing that this should be the true form were ready to hand, since the atom might very well be expected to spend some time in radiating infra-red lines before it reached the initial state for the radiation observed; on this point all that could be inferred from Wien’s curves was that the interval must at most be appreciably smaller than the half-period of the exponential fall. This interval is not the same as the “Yerweilzeit” considered by him; the Verweilzeit is the basis of a demonstration that the curves due to point-excitation should be exponential, while the present interval would have the effect that they should not. In order to decide whether there was in fact any delay of this nature at all, it seemed desirable to try whether non-luminous canal-rays, some distance after their entry into the vacuum, could be excited anew to luminescence at approximately one known point only, so that any interval between excitation and emission could be directly measured. The experiments devoted to this object have been somewhat interrupted, and it is further necessary to discontinue them, at least for a time, once more. The present paper is thus an interim account, containing a description of the method that has been worked out and the result that has been obtained; there are, however, a number of further questions which it was hoped to decide, and which presumably can be decided by the same method, that are still unsettled.
Abstract When a beam of Canal-Rays passes into a high vacuum, it is well known that the light-intensity falls exponentially to a constant value which, if the pressure is sufficiently low, is very small indeed. The experiments of Wien, and the theory developed by him, showed that the observations could be equally well accounted for on either of two hypotheses, namely that the light from each atom falls off exponentially at the observed rate, or alternatively, that all the energy from any one atom is emitted in a very short time indeed, and that what is measured is essentially the decay-curve of those atoms that are in the initial state for the radiation in question. As to whether the maximum emission (on either theory) occurred exactly at the point of excitation, however, these experiments provided very little evidence; the excitation was in fact, so far as can be determined, distributed continuously over a region about which all that can be said is that it was bounded more or less definitely on one side by the back of the cathode, and it was possible that the curves for a point-excitation would really resemble, for instance, the activity-curve of an intermediate radioactive substance initially absent from a preparation. In fact plausible grounds for believing that this should be the true form were ready to hand, since the atom might very well be expected to spend some time in radiating infra-red lines before it reached the initial state for the radiation observed; on this point all that could be inferred from Wien’s curves was that the interval must at most be appreciably smaller than the half-period of the exponential fall. This interval is not the same as the “Yerweilzeit” considered by him; the Verweilzeit is the basis of a demonstration that the curves due to point-excitation should be exponential, while the present interval would have the effect that they should not. In order to decide whether there was in fact any delay of this nature at all, it seemed desirable to try whether non-luminous canal-rays, some distance after their entry into the vacuum, could be excited anew to luminescence at approximately one known point only, so that any interval between excitation and emission could be directly measured. The experiments devoted to this object have been somewhat interrupted, and it is further necessary to discontinue them, at least for a time, once more. The present paper is thus an interim account, containing a description of the method that has been worked out and the result that has been obtained; there are, however, a number of further questions which it was hoped to decide, and which presumably can be decided by the same method, that are still unsettled.
A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.
Abstract As I have shown in numerous publications, the vapour of mercury in vacuo, even at temperatures below the solidifying point of the metal, when illuminated by the light of a water-cooled quartz mercury lamp, emits the monochromatic radiation of wave-length 2536. According to Bohr’s theory, this resonance radiation results from the fall of an electron from the 2p2 orbit, to which it has been raised from the IS orbit by the absorption of one quantum of 2536 radiation, back to the IS orbit, the orbit occupied by the electron when the atom is in its normal state. An atom in which the electron has been brought to the 2p2 orbit is in a condition to absorb the radiation of any other line produced by the fall of electrons from outer orbits to orbit 2p2, and Fuchtbauer has shown, in an extremely interesting and important paper (‘Physik. Zeit.,’ 1920), that a large number of the arc lines of mercury are radiated by cold mercury vapour illuminated by the mercury arc. He showed that, if the 2536 radiation was screened off by glass, none of the other lines appeared, and explained their presence as the result of the absorption by excited atoms with electrons at the 2p2 level, of the radiations terminating at this level, by which the electrons are carried out to other-orbits, from which they can fall back in various ways, giving rise to emission lines in the process. Though Fuchtbauer‘s experiment did not prove that this process actually took place, his explanation was the only one possible on modern theory, and the experiments which are to be considered in the present paper show that it was correct. Twenty years ago the most that could have been said would have been that the absorption of the 2536 radiation “shook” the atom in some way, causing it to emit its other frequencies as a sort of fluorescence, since Fuchtbauer’s experiment did not show that excitation by lines other than the 2536 line was necessary for the emission of the arc spectrum. To demonstrate experimentally that the step-by-step transfer of electrons from orbit to orbit actually takes place, it is necessary to illuminate the mercury vapour with various combinations of monochromatic light, and study the resultant emission spectrum.
In this work we shall study the interaction of matter and light. In so doing we shall rely heavily on the description of such processes provided by quantum mechanics. This appears on a number of levels: firstly, a quantum description of matter is indispensable if one wants to understand on the microscopic scale the different kinds of interaction processes that can occur. Secondly, a quantum description of light often turns out to be useful, sometimes necessary, to better understand these processes. We shall study phenomena such as spontaneous emission, which can only be properly treated by a theory taking into account the quantum nature of both light and matter. In the following chapters we shall address, amongst others, the following question: ‘given an atom prepared at a given time in a particular state and subjected from this time onwards to electromagnetic radiation, what is the state of the atom and radiation at any later moment in time?’ In order to be able to answer this question it will be necessary for us to know how to calculate the evolution of a quantum system in a small number of typical situations. These methods we shall demonstrate in the first chapter. The evolution of the coupled atom–light system depends on the temporal dependence of the applied light field, which could, for example, be applied from a given moment and thereafter remain unchanged in intensity, or, perhaps, be appreciable only for a finite period of time (pulsed excitation).
The spontaneous decay of an excited atom by photon emission is one of the most common and elementary physical process present in nature and in laboratories. The decay is random in time with constant probability density, as it can be inferred by the exponential law observed experimentally. Despite the simplicity of the process, in Quantum Mechanics the decay itself is considered a law of nature which is not further analyzed or explained. However it is legitimate to ask for the reason of its randomness and for the dynamics of the atom around the decay. The decay process of an isolated atom is usually assumed to be instantaneous, the so called Quantum jumps. Particular experimental arrangements can widen the duration of the transition from an excited state to the ground state to a finite time. In general this is due to the quantum back action of the detector. The development of Quantum Optics has enormously enriched the possibilities to study in detail the process, and new illuminating insights have been obtained, but the question is still valid and its answer is still elusive. In this paper we analyze the spontaneous decay within the new prospect introduced by a recent model that complete standard Quantum Mechanics in a formalism which is able to describe the dynamics of measurement and of spontaneous decay. Comparison is made with more phenomenological theories, the Quantum Mechanics of open system and the Stochastic wave function model. In the new model the stochastic photon emission by an excited atom is triggered by the vacuum fluctuations. It is suggested how to re-analyze three types of experiments, already realized in many laboratories, in order to reveal the presence and the effect of these fluctuations.
The phenomena which are principally responsible for the interaction of radiation with matter in everyday life are absorption and spontaneous emis sion. Light falling on matter is absorbed, leaving atoms in excited states which spontaneously emit radiation with a spread of frequency inversely proportional to the decay time. The situation is modified when the atomic decay times are sufficiently long and the radiation sufficiently intense, for then radiation may fall on excited atoms, resulting in stimulated emission rather than absorption. We saw in Chap. 2 that this third process was required to obtain the Planck blackbody radiation law in Einstein’s derivation. It is stimulated emission which enables an atomic medium to amplify incident radiation, and therefore is an essential ingredient in light amplifiers and oscillators. In fact, the word LASER is an acronym for Light Amplification by Stimulated Emission of Radiation.
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.