Based on a lecture to the Scandinavian Meeting of Natural Scientists and published in Danish in Fysisk Tidsskrift in 1929. First published in English in 1934 by Cambridge University Press.

Natural phenomena, as experienced through the medium of our senses, often appear to be extremely variable and unstable. To explain this, it has been assumed, since early times, that the phenomena arise from the combined action and interplay of a large number of minute particles, the so-called atoms, which are themselves unchangeable and stable, but which, owing to their smallness, escape an immediate perception. Quite apart from the fundamental question of whether we are justified in demanding visualizable pictures in fields which lie outside the reach of our senses, the atomic theory originally was of necessity of a hypothetical character; and, since it was believed that a direct insight into the world of atoms would, from the very nature of the matter, never be possible, one had to assume that the atomic theory would always retain this character. However, what has happened in so many other fields has happened also here; because of the development of observational technique, the limit of possible observations has continually been shifted. We need only think of the insight into the structure of the universe which we have gained by the aid of the telescope and the spectroscope, or of the knowledge of the finer structure of organisms which we owe to the microscope. Similarly, the extraordinary development in the methods of experimental physics has made known to us a large number of phenomena which in a direct way inform us of the motions of atoms and of their number. We are aware even of phenomena which with certainty may be assumed to arise from the action of a single atom, or even of a part of an atom. However, at the same time as every doubt regarding the reality of atoms has been removed and as we have gained a detailed knowledge even of the inner structure of atoms, we have been reminded in an instructive manner of the natural limitation of our forms of perception. It is this peculiar situation which I shall attempt to portray here.

Time does not permit of my describing in detail the extraordinary extension of our experience, here dealt with, which is characterized by the discoveries of cathode rays, Röntgen rays, and the radioactive substances. I shall merely remind you of the main features of the picture of the atom which we have gained through these discoveries. Negatively charged particles, the so-called electrons, which are held within the atom by the attraction of a much heavier positively charged atomic nucleus, enter as common building stones in all atoms. The mass of the nucleus determines the atomic weight of the element but has otherwise only a slight influence on the properties of the substance, these depending primarily on the electric charge of the nucleus which, apart from the sign, is always an integral multiple of the charge of an electron. Now, this whole number, which determines how many electrons are present in the neutral atom, has turned out to be just the atomic number that gives the place of the element in the so-called natural system, in which the peculiar relationships of the elements as regards their physical and chemical properties are so appropriately expressed. This interpretation of the atomic number may be said to signify an important step towards the solution of a problem which for a long time has been one of the boldest dreams of natural science, namely, to build up an understanding of the regularities of nature upon the consideration of pure numbers.

The development mentioned above has, to be sure, produced a certain change in the fundamental concepts of the atomic theory. Instead of assuming that the atoms are unchangeable, it is now assumed that it is the parts of the atoms which are constant. In particular, the great stability of the elements depends upon the fact that the atomic nucleus is not affected by the ordinary physical and chemical influences which produce changes only in the binding of the electrons within the atom. While all our experience strengthens the assumption of the permanence of electrons, we know, however, that the stability of atomic nuclei is of a more limited character. Indeed, the peculiar radiations from the radioactive elements provide us with direct evidence of a disruption of atomic nuclei, whereby electrons or positively charged nuclear particles are ejected with great energy. These disintegrations, so far as we are able to judge from all evidence, take place without any external cause. If we have a given number of radium atoms, we can merely say that there is a definite probability that a certain fraction of them will break down during the next second. We shall return later to this peculiar failure of the causal mode of description which we come upon here and which is closely connected with fundamental features of our description of atomic phenomena. Here, I shall call to mind only the important discovery of Rutherford that a disruption of atomic nuclei may, under certain circumstances, be brought about by external influence. As we all know, he succeeded in showing that the nuclei of certain, otherwise stable, elements may be split up when hit by the particles expelled from the radioactive nuclei. This first example of a transmutation of an element, regulated by man, may be said to mark an epoch in the history of natural science and to open up an entirely new field of physics, namely, the exploration of the interior of atomic nuclei. However, I shall, not dwell upon the prospects opened up by this new field, but shall confine myself to discussing the general information that we have gained through our endeavours to account for the ordinary physical and chemical properties of the elements on the basis of the conceptions of atomic structure mentioned above.

At first glance, it might appear that the solution of the problem considered would be quite simple. The picture of the atom with which we are dealing is that of a small mechanical system which even, resembles in certain main features our own solar system, in the, description of which mechanics has won such great triumphs and has given us a principal example of the fulfilment of the claim of causality in ordinary physics. Indeed, from a knowledge of the instantaneous positions and motions of the planets, we can calculate, with apparently unlimited accuracy, their positions and motions at any later time. However, the fact that in such a mechanical description an arbitrary initial state may be chosen presents great difficulties when the problem of atomic structure is considered. In fact, if we must reckon with an infinite number of continuously varying states of motion of the atoms, then we come into obvious contradiction with our experimental knowledge that the elements possess definite properties. One might believe perhaps that the properties of the elements do not inform us directly of the behaviour of individual atoms but, rather, that we are always concerned only with statistical regularities holding for the average conditions of a large number of atoms. In the mechanical theory of heat, which not only permits of our accounting for the fundamental laws of thermodynamics, but also gives us an understanding of many of the general properties of matter, we have a well-known example of the fruitfulness of statistical mechanical considerations in the atomic theory. The elements have, however, other properties which permit of more direct conclusions being drawn with respect to the states of motion of the atomic constituents. Above all, we must assume that the quality of the light which the elements in certain circumstances emit and which is characteristic of each element is essentially determined by what occurs in a single atom. Just as the wireless waves tell us about the nature of the electrical oscillations in the apparatus of the broadcasting station, so should we expect, on the basis of the electromagnetic theory of light, that the frequencies of the individual lines in the characteristic spectra of the elements should give us information as to the motions of the electrons within the atom. However, mechanics does not offer us a sufficient basis for interpreting this information ; indeed, owing to the possibility of a continuous variation of the mechanical states of motion mentioned above, it is not possible even to understand the occurrence of sharp spectral lines.

The missing element in our description of nature, evidently required to account for the behaviour of the atoms' has been supplied, however, by Planck's discovery of the so-called quantum of action. This discovery had its origin in his investigation of black body radiation, which, because of its independence of the special properties of the substances, offered a decisive test of the range of validity of the mechanical theory of heat and of the electromagnetic theory of radiation. It was the very inability of these theories to account for the law of black body radiation which led Planck to recognize a general feature of the laws of nature that had hitherto remained unnoticed. This feature, to be sure, is not obvious in the description of ordinary physical phenomena, but it has, nevertheless, caused a complete revolution in our account of such effects which depend on individual atoms. Thus, in contrast with the demand of continuity which characterizes the customary description of nature, the indivisibility of the quantum of action requires an essential element of discontinuity in the description of atomic phenomena. The difficulty of combining the new knowledge with our ordinary scheme of physical ideas became especially apparent through the discussion of the nature of light, which was renewed by Einstein in connection with his explanation of the photo-electric effect, although the question, judging from all earlier experimental results, had found a perfectly satisfactory solution within the frame of the electromagnetic theory. The situation which we meet here is characterized by the fact that we are apparently forced to choose between two mutually contradictory conceptions of the propagation of light: one, the idea of light waves, the other, the corpuscular view of the theory of light quanta, each conception expressing fundamental aspects of our experience. As we shall see in the following, this apparent dilemma marks a peculiar limitation of our forms of perception which is bound up with the quantum of action. This limitation is brought to light by a closer analysis of the applicability of the basic physical concepts in describing atomic phenomena.

Indeed, only by a conscious resignation of our usual demands for visualization and causality was it possible to make Planck's discovery fruitful in explaining the properties of the elements on the basis of our knowledge of the building stones of atoms. Taking the indivisibility of the quantum of action as a starting-point, the author suggested that every change in the state of an atom should be regarded as an individual process, incapable of more detailed description, by which the atom goes over from one so-called stationary state into another. According to this view, the spectra of the elements do not give us immediate information about the motions of the atomic parts, but each spectral line is associated with a transition process between two stationary states, the product of the frequency and the quantum of action giving the energy change of the atom in the process. In this way, it proved possible to obtain a simple interpretation of the general spectroscopic laws which Balmer, Rydberg and Ritz had succeeded in deriving from the experimental data. This view of the origin of spectra was directly supported also by the well-known experiments of Franck and Hertz on collisions between atoms and free electrons. The amounts of energy which can be exchanged in such collisions were found to agree exactly with the energy differences, computed from the spectra, between the stationary state in which the atom was before the collision and one of the stationary states in which it can exist after the collision. On the whole, this point of view offers a consistent way of ordering the experimental data, but the consistency is admittedly only achieved by the renunciation of all attempts to obtain a detailed description of the individual transition processes. We are here so far removed from a causal description that an atom in a stationary state may in general even be said to possess a free choice between various possible transitions to other stationary states. From the very nature of the matter, we can only employ probability considerations to predict the occurrence of the individual processes, which fact, as Einstein has emphasized, exhibits a close similarity to the conditions holding for the spontaneous radioactive transformations.

A peculiar feature of this attack on the problem of atomic structure is the extensive use of whole numbers which also play an important role in the empirical spectroscopic laws. Thus, the classification of stationary states, besides depending upon the atomic number, also depends on the so-called quantum numbers, to the systematics of which Sommerfeld has contributed so much. On the whole, the views considered have permitted us to account, to a considerable extent, for the properties and relationships of the elements on the basis of our general conceptions of atomic structure. Considering the great departure from our customary physical ideas, one might wonder that such an account has been possible, since, after all, our entire knowledge of the building stones of the atoms rests upon these ideas. Indeed, any use of concepts like mass and electric charge is obviously equivalent to the invocation of mechanical and electrodynamical laws. A method of making such concepts useful in other fields than that in which the classical theories are valid has been found, however, in the demand of a direct concurrence of the quantum-mechanical description with the customary description in the border region where the quantum of action may be neglected. The endeavours to utilize in the quantum theory every classical concept in a reinterpretation which fulfils this demand without being at variance with the postulate of the indivisibility of the quantum of action, found their expression in the so-called correspondence principle. However, there were many difficulties to overcome before a complete description based on the correspondence principle was actually accomplished, and only in recent years has it been possible to formulate a coherent quantum mechanics which may be regarded as a natural generalization of the classical mechanics, and in which the continuous, causal description is replaced by a fundamentally statistical mode of description.

A decisive step towards the attainment of this goal was made by the young German physicist, Werner Heisenberg, who showed how the ordinary ideas of motion may be replaced in a consistent way by a formal application of the classical laws of motion, the quantum of action appearing only in certain rules of calculation holding for the symbols which replace the mechanical quantities. This ingenious attack upon the problem of the quantum theory makes, however, great demands on our power of abstraction, and the discovery of new artifices which, in spite of their formal character, more closely meet our demands for visualization has, therefore, been of profound significance in the development and clarification of the quantum mechanics. I am referring to the ideas of matter waves, introduced by Louis de Broglie, which have proved so fruitful in the hands of Schrodinger, especially in connection with the conception of stationary states, the quantum numbers of which are interpreted as the number of nodes of the standing waves symbolizing these states. De Broglie's starting-point was the similarity, which had already been so important in the development of classical mechanics, existing between the laws governing the propagation of light and those holding for the motion of material bodies. In fact, the wave mechanics forms a natural counterpart to the above-mentioned light quantum theory of Einstein. As in this theory, so also in the wave mechanics, we are not dealing with a self-contained conceptual scheme but, rather, as especially emphasized by Born, with an expedient to formulate the statistical laws which govern atomic phenomena. It is true that the confirmation of the idea of matter waves, provided by the experiments on the reflection of electrons by metal crystals, is, in its way, just as decisive as the experimental evidence for the wave conception of the propagation of light. However, we must bear in mind that the application of matter waves is limited to those phenomena, in the description of which it is essential that the quantum of action be taken into account and which, therefore, lie outside the domain where it is possible to carry out a causal description corresponding to our customary forms of perception and where we can ascribe to words like "the nature of matter" and "the nature of light" meanings in the ordinary sense.

With the help of quantum mechanics, we master an extensive range of experience. Especially are we able to account for a large number of details concerning the physical and chemical properties of the elements. Recently, it has been possible even to obtain an interpretation of the radioactive transformations, in which the empirical probability laws holding for these processes appear as an immediate consequence of the peculiar statistical mode of description that characterizes the quantum theory. This interpretation provides an excellent example of the fruitfulness as well as of the formal nature of the wave conceptions. On one hand, we have here a direct connection with the customary ideas of motion, since, owing to the great energy of the fragments expelled by the atomic nuclei, the paths of these particles may be directly observed. On the other hand, the ordinary mechanical conceptions completely fail to provide us with a description of the course of the disintegration process, since the field of force surrounding the atomic nucleus would, according to these ideas, prevent the particles from escaping from the nucleus. On the quantum mechanics, however, the state of affairs is quite different. Though the field of force is still a hindrance which, for the most part, holds the matter waves back, yet it permits a small fraction of them to leak through. The part of the waves which escapes in this way in a certain time gives us a measure of the probability that the disruption of the atomic nucleus takes place during this time. The difficulty of speaking of "the nature of matter" without the above-mentioned proviso could scarcely be more strikingly brought to light.

In the case of the idea of light quanta, there exists a similar relationship between our conceptual pictures and the calculation of the probability of occurrence of the observable light effects. In accordance with the classical electromagnetic conceptions, we cannot, however, ascribe any proper material nature to light, since observation of light phenomena always depends on a transfer of energy and momentum to material particles. The tangible content of the idea of light quanta is limited, rather, to the account which it enables us to make of the conservation of energy and momentum. It is, after all, one of the most peculiar features of quantum mechanics that, in spite of the limitation of the classical mechanical and electromagnetic conceptions, it is possible to maintain the conservation laws of energy and momentum. In certain respects, these laws form a perfect counterpart to the assumption, basic for the atomic theory, of the permanence of the material particles, which is strictly upheld in the quantum theory even though the conceptions of motion are renounced.

As with classical mechanics, so quantum mechanics, too, claims to give an exhaustive account of all phenomena which come within its scope. Indeed, the inevitability of using, for atomic phenomena, a mode of description which is fundamentally statistical arises from a closer investigation of the information which we are able to obtain by direct measurement of these phenomena and of the meaning which we may ascribe, in this connection, to the application of the fundamental physical concepts. On one hand, we must bear in mind that the meaning of these concepts is wholly tied up with customary physical ideas. Thus, any reference to space-time relationships presupposes the permanence of the elementary particles, just as the laws of the conservation of energy and momentum form the basis of any application of the concepts of energy and momentum. On the other hand, the postulate of the indivisibility of the quantum of action represents an element which is completely foreign to the classical conceptions ; an element which, in the case of measurements, demands not only a finite interaction between the object and the measuring instrument but even a definite latitude in our account of this mutual action. Because of this state of affairs, any measurement which aims at an ordering of the elementary particles in time and space requires us to forego a strict account of the exchange of energy and momentum between the particles and the measuring rods and clocks used as a reference system. Similarly, any determination of the energy and the momentum of the particles demands that we renounce their exact co-ordination in time and space. In both cases, the invocation of classical ideas, necessitated by the very nature of measurement, is, beforehand, tantamount to a renunciation of a strictly causal description. Such considerations lead immediately to the reciprocal uncertainty relations set up by Heisenberg and applied by him as the basis of a thorough investigation of the logical consistency of quantum mechanics. The fundamental indeterminacy which we meet here may, as the writer has shown, be considered as a direct expression of the absolute limitation of the applicability of visualizable conceptions in the description of atomic phenomena, a limitation that appears in the apparent dilemma which presents itself in the question of the nature of light and of matter.

The resignation as regards visualization and causality, to which we are thus forced in our description of atomic phenomena, might well be regarded as a frustration of the hopes which formed the starting-point of the atomic conceptions. Nevertheless, from the present standpoint of the atomic theory, we must consider this very renunciation as an essential advance in our understanding. Indeed, there is no question of a failure of the general fundamental principles of science within the domain where we could justly expect them to apply. The discovery of the quantum of action shows us, in fact, not only the natural limitation of classical physics, but, by throwing a new light upon the old philosophical problem of the objective existence of phenomena independently of our observations, confronts us with a situation hitherto unknown in natural science. As we have seen, any observation necessitates an interference with the course of the phenomena, which is of such a nature that it deprives us of the foundation underlying the causal mode of description. The limit, which nature herself has thus imposed upon us, of the possibility of speaking about phenomena as existing objectively finds its expression, as far as we can judge, just in the formulation of quantum mechanics. However, this should not be regarded as a hindrance to further advance; we must only be prepared for the necessity of an ever extending abstraction from our customary demands for a directly visualizable description of nature. Above all, we may expect new surprises in the domain where the quantum theory meets with the theory of relativity and where unsolved difficulties still stand as a hindrance to a complete fusion of the extension of our knowledge and of the expedients to account for natural phenomena which these theories have given us.

Even though it be at the end of the lecture, yet I am glad to have the opportunity of emphasizing the great significance of Einstein's theory of relativity in the recent development of physics with respect to our emancipation from the demand for visualization. We have learned from the theory of relativity that the expediency of the sharp separation of space and time, required by our senses, depends merely upon the fact that the velocities commonly occurring are small compared with the velocity of light. Similarly, we may say that Planck's discovery has led us to recognize that the adequacy of our whole customary attitude, which is characterized by the demand for causality, depends solely upon the smallness of the quantum of action in comparison with the actions with which we are concerned in ordinary phenomena. While the theory of relativity reminds us of the subjective character of all physical phenomena, a character which depends essentially upon the state of motion of the observer, so does the linkage of the atomic phenomena and their observation, elucidated by the quantum theory, compel us to exercise a caution in the use of our means of expression similar to that necessary in psychological problems where we continually come upon the difficulty of demarcating the objective content.

Before I conclude, it would be natural, at such a joint meeting of natural scientists, to touch upon the question as to what light can be thrown upon the problems regarding living organisms by the latest development of our knowledge of atomic phenomena which I have here described. Even though it may not yet be possible to give an exhaustive answer to this question, we can perhaps already catch a glimpse of a certain connection between these problems and the ideas of the quantum theory. A first hint in this direction we find in the circumstance that the mutual action between the organisms and the external world, upon which the sense impressions depend, may, at any rate in certain circumstances, be so small that it approaches the quantum of action. As it has often been remarked, a few light quanta are sufficient to produce a visual impression. We see, therefore, that the needs of the organism for independence and sensibility are here satisfied to the utmost limit permitted by the laws of nature, and we must be prepared to come upon similar conditions also at other points of decisive significance for the formulation of biological problems. If, however, the physiological phenomena exhibit a refinement which is developed to the above-mentioned limit, then, indeed, this means that we at the same time approach the limit for an unambiguous description of them with the help of our ordinary visualizable conceptions. This in no way contradicts the fact that the living organisms to a wide extent present problems to us which lie within the range of our visualizable forms of perception and have formed a fruitful field for the application of physical and chemical points of view. Neither do we see any immediate limit for the applicability of these view-points. Just as we do not need to distinguish, in principle, between the current in a water pipe and the flow of blood in the vessels, no more should we expect, beforehand, any profound fundamental difference between the propagation of sense impressions in the nerves and the conduction of electricity in a metal wire. It is true, for all such phenomena, that a detailed account carries us into the domain of atomic physics; indeed, so far as the conduction of electricity is concerned, we have just learned, in quite recent years, that only that limitation of our visualizable conceptions of motion, which is characteristic of the quantum theory, enables us to understand how the electrons can make their way between the metal atoms of the wire. However, in the case of these phenomena, such, a refinement in the mode of description is not necessary to account for those effects which first call for our consideration. With regard to the more profound biological problems, however, in which we are concerned with the freedom and power of adaptation of the organism in its reaction to external stimuli, we must expect to find that the recognition of relationships of wider scope will require that the same conditions be taken into consideration which determine the limitation of the causal mode of description in the case of atomic phenomena. Besides, the fact that consciousness, as we know it, is inseparably connected with life ought to prepare us for finding that the very problem of the distinction between the living and the dead escapes comprehension in the ordinary sense of the word. That a physicist touches upon such questions may perhaps be excused on the ground that the new situation in physics has so forcibly reminded us of the old truth that we are both onlookers and actors in the great drama of existence.

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