THE QUANTUM POSTULATE AND THE NEW DEVELOPMENT OF ATOMISM1

N. Bohr, Copenhagen.

In connection with the discussion concerning the physical interpretation of the quantum methods that have developed in recent years, I should very much like to set forth the following general remarks on the principles underlying the description of atomic phenomena. These remarks, I hope, will help toward some reconciliation of sharply divergent views in this field.

§ 1. The quantum postulate and causality.

Characteristic of quantum theory is the acknowledgment of a fundamental limitation of classical physical concepts in their application to atomic phenomena. The situation arising as a consequence of this is very peculiar, since our interpretation of the experimental material is in its essential features based precisely on the application of classical concepts. Yet despite the difficulties that arise from this, for the formulation of mod—

...of the content of quantum theory, its meaning can evidently be expressed, as we shall see, by the so-called quantum postulate. According to this postulate, every atomic process contains features of discontinuity or, rather, individuality, expressed by Planck’s quantum of action and entirely foreign to classical theory.

A consequence of the postulate is the renunciation of a causal space-time description, or coordination, of atomic phenomena. Indeed, our usual description of natural phenomena rests ultimately on the assumption that the very observation of a given phenomenon does not essentially affect it. This is clearly seen, for example, in the formulation of the theory of relativity, which proved so fruitful for the clarification of classical theories. Every observation, or measurement, as Einstein has shown, is based in the last analysis on the coincidence of two independent events at one and the same point of space-time. This coincidence must not depend on the difference which may otherwise exist in the space-time description of different observers. According to the quantum postulate, however, every observation of atomic phenomena is connected with such an interaction of the latter with the means of observation that it cannot be neglected, and therefore it is impossible to ascribe an independent physical reality, in the usual sense, either to the phenomenon or to the means of observation. In general, the concept of observation involves a certain arbitrariness, since in essence it depends on what objects are included together with the system subject to observation. Of course, in the final analysis, every observation can be reduced to the sensations of our senses. But in interpreting observations we must always employ theoretical conceptions, as a result of which, in each individual case, it is a question of expediency at what point the concept of observation should be introduced, together with the quantum postulate and the irrationality inherent in the latter.

Such a state of affairs entails important consequences. On the one hand, the definition of the state of a physical...

systems, in the usual sense, requires the exclusion of all external influences; but in that case, according to the quantum postulate, every possibility of observation is also excluded, and above all the concepts of time and space lose their immediate meaning. If, on the other hand, in order to make observations possible, one admits certain interactions with suitable means of observation that do not belong to the system, then, in essence, an unambiguous definition of the state of the system is no longer possible, and there can be no question of causality in the usual sense. In accordance with the essence of quantum theory, we must therefore regard the space-time representation and the demand of causality, the combination of which characterizes classical theories, as complementary features that exclude one another in the description of the content of experience—as features symbolizing the idealization of the possibilities of observation and, correspondingly, of definition. Just as in the theory of relativity the expediency of a sharp separation of space and time, required by our senses, is based on the fact that ordinary relative velocities are small in comparison with the velocity of light, so quantum theory must lead to the recognition that the admissibility of the entire causal space-time view is conditioned only by the smallness of the quantum of action in comparison with the ordinary actions that manifest themselves in sensations. Indeed, in the description of atomic phenomena, the quantum postulate sets before us the task of developing a certain “theory of complementarity” (Komplementaritätstheorie), in which the absence of contradictions can be judged only by weighing the possibilities of definitions and observations.

Such an interpretation becomes necessary already in the much-discussed question of the nature of light and of the particles of matter. The propagation of light is represented, as is known, in a rational manner in the electromagnetic theory of light. In particular, interference phenomena in empty space and the optical properties of material media are governed entirely by the wave principle of superposition. Nevertheless, the conservation of energy and momentum in the interaction of radiation

tion and matter, manifested in the photoelectric and Compton effects, finds a rational expression in the conception of light quanta developed by Einstein. The doubts as to the strict fulfillment of the superposition principle and as to the exact validity of the conservation principles, to which this apparent contradiction led, have, as is known, been refuted by decisive direct experiments. Such a state of affairs clearly shows the impossibility of a causal space-time description of light phenomena. In wishing to trace the space-time propagation of light actions, we are condemned, according to the quantum postulate, to a statistical treatment. On the other hand, in insisting on the requirement of causality in an individual light process characterized by a quantum of action, we are forced to renounce space-time relations. Of course, there can never be any question of a completely independent application of the space-time description and of the concept of causality. Both interpretations of the nature of light are rather two different attempts to adapt the experimental facts to our customary views; in these attempts the limitation of classical concepts finds mutually complementary expression.

The consideration of the properties of material particles leads to an analogous conclusion. The individuality of electric elementary particles would seem to follow from the most general experiments. Nevertheless, in order to explain various facts, in particular the recently discovered selective reflection of electrons from metallic crystals, we are forced to resort to the wave principle of superposition in accordance with the ideas originally expressed by de Broglie. Just as in the question of light, so also with respect to the essence of matter, while retaining the classical concepts we stand before an unavoidable dilemma, which may be regarded as the precise expression of the analysis of the experimental material. In reality, here we have before us not mutually contradictory, but mutually complementary interpretations of phenomena, which only together give a natural generalization of the classical mode of description. In considering these questions one cannot

one must not lose sight of the fact that, in speaking of radiation in empty space or of an isolated material particle, we are dealing, in accordance with the interpretation set forth here, with abstractions, since their properties, according to the quantum postulate, are accessible to observation and determination only in interaction with other systems. Nevertheless these abstractions, as we shall see, are a necessary means for expressing the content of experiments on the basis of our customary way of viewing things.

The difficulties that stand in the way of a causal space-time description in quantum theory, and that have long been the subject of discussion, have recently been brought to the fore thanks to the development of new symbolic methods. An important step forward in the question of the irreproachable application of these methods is the new work of Heisenberg (ZS. f. Phys. 43, 172, 1927). He pointed out a peculiar mutual indeterminacy inherent in every measurement of atomic quantities. Before proceeding to a more detailed consideration of his arguments, it will be useful to show how the mutually complementary features of description, manifested in this indeterminacy, are already unavoidable in the analysis of the very simplest concepts that underlie the interpretation of experiments.

§ 2. The quantum of action and kinematics.

The fundamental contradiction between the quantum of action and the classical concepts becomes immediately clear from the simple formulae that constitute the common basis of the theory of light quanta and of the wave theory of material particles. Denoting Planck’s constant by \(h\), we have, as is known:

\[ E\tau = J\lambda = h, \tag{1} \]

where \(E\) is energy, \(J\) is momentum, and \(\tau\) and \(\lambda\) are the corresponding period of oscillation and wavelength. In these formulae the above-mentioned two interpretations of light stand in sharp opposition to one another. Energy and momentum correspond to the concept of a particle

and from the classical point of view can be characterized by the coordinates of space-time; on the other hand, the period and wavelength stand, in the space-time relation, in connection with an unbounded, plane harmonic alternation of waves. Only with the aid of the superposition principle can one here establish a connection with the ordinary method of description. Indeed, the limitation of the space-time extent of wave “fields” can always be interpreted as a consequence of interference within a group of harmonic elementary waves. De Broglie proved that the translational velocity of individuals associated with waves can be represented by means of the so-called group velocity of waves. Let us denote a plane elementary wave as follows:

\[ A \cos 2\pi (t\nu - x\sigma_x - y\sigma_y - z\sigma_z + \delta), \]

where \(A\) and \(\delta\) are constants determining respectively the amplitude and phase, the quantity \(\nu = \frac{1}{\tau}\) is the frequency of oscillations, \(\sigma_x, \sigma_y, \sigma_z\) are the wave numbers in the directions of the coordinates (they may be regarded as components of the wave number \(\sigma = \frac{1}{\lambda}\) in the direction of propagation); \(\frac{\nu}{\sigma}\) is the wave or phase velocity, while the group velocity is defined through \(\frac{d\nu}{d\sigma}\). According to the theory of relativity, for a particle with velocity \(v\):

\[ J = \frac{v}{c^2} E \quad \text{and} \quad v\,dJ = dE, \]

where \(c\) is the speed of light. By formula (1) the phase velocity is therefore equal to \(\frac{c^2}{v}\), and the group velocity is equal to \(v\). On the one hand, the circumstance that, generally speaking, \(\frac{c^2}{v} > c\) immediately indicates the symbolic character of these considerations. On the other hand, the possibility of identifying

of the particle’s velocity with the group velocity is an indication of the domain of applicability of space-time images in quantum theory. Here, at the same time, the additional character of the description is expressed, since the use of wave groups is necessarily connected with the indeterminacy of the determination of the period and wavelength, and consequently, according to (1), of the conjugate quantities of energy and momentum.

A bounded wave field can, strictly speaking, be represented only by the superposition of a multitude of elementary waves corresponding to all possible values of \(\nu, \sigma_x, \sigma_y, \sigma_z\). As regards the order of magnitude, the mean difference of these values for two elementary waves of the group is given in the most favorable case by the condition:

\[ \Delta t \cdot \Delta \nu = \Delta x \Delta \sigma_x = \Delta y \Delta \sigma_y = \Delta z \Delta \sigma_z = 1, \]

where \(\Delta t, \Delta x, \Delta y, \Delta z\) indicate the extension of the wave field in time and in the directions of the spatial coordinate axes. These relations, well known from the theory of optical instruments, in particular on the basis of Rayleigh’s considerations on the resolving power of spectral instruments, express the condition that wave segments can be extinguished as a result of interference at the space-time boundary surfaces of the wave field. This same condition can also be interpreted as the absence of phase in the group as a whole, in the sense in which the latter is attributed to individual elementary waves. From formula (1) it then follows that:

\[ \Delta t \Delta E = \Delta x \Delta J_x = \Delta y \Delta J_y = \Delta z \Delta J_z = h, \tag{2} \]

which expresses the greatest possible accuracy in determining the energy and momentum of an individual conjugated with the wave field. The conditions for the conjugation of certain values of energy and momentum with the wave field according to formula (1) will in general be still less favorable. If the properties of the wave group initially satisfy relations (2), then in the course of time the extension of the group will undergo such changes that it must become less and less suitable for ...

suitable for the representation of an individual. It is precisely in this circumstance that the paradoxical nature of the question of the nature of light and of material particles lies. The limitation of classical concepts, expressed by relation (2), is closely connected with the restricted applicability of classical mechanics, which corresponds in the wave theory of matter to geometrical optics, in which the propagation of waves is represented by “rays.” Only in the limiting case of such “rays” can one unambiguously determine the energy and momentum on the basis of the space-time picture. For the general definition of these concepts we must resort directly to the conservation principles, whose rational formulation constitutes the fundamental problem of the symbolic methods to which we shall return later.

In the language of the theory of relativity, the content of relations (2) may be formulated as follows: according to the quantum theory there exists a general reciprocal connection between the maximum sharpness of the definition of the space-time vector and, correspondingly, of the energy-momentum vectors. Such a connection may be regarded as a simple symbolic expression of the mutually complementary nature of the space-time description and of the requirement of causality. At the same time, the general character of this connection makes it possible, to a certain extent, to unite the conservation principles with the space-time representation of observations; instead of the coincidence at one point of space-time of certain precisely determined events, one may speak of the meeting of imprecisely determined individuals within a finite space-time region.

This circumstance makes it possible to avoid the familiar paradoxes that characterize the description of the scattering of radiation by free electric particles and the collision of two particles. According to classical concepts, the description of scattering requires a finite extension of the radiation in space and time, whereas in the change in the motion of the electron required by the quantum postulate it seems as though one were dealing with an instantaneous action taking place at a spatial point. But for both the radiation and the electron one cannot

determine energy and momentum, without resorting to a finite space-time region. Further, the application of conservation laws to the given process presupposes that the accuracy of the determination of the momentum-energy vector is one and the same for both the radiation and the electron. According to (2), one and the same space-time region may therefore be associated with both individuals in the interaction.

Exactly the same is true for a collision between two material particles; it is true that, until the inevitability of the wave representation had been established, attention was not paid to the significance of the quantum postulate for this phenomenon. Here this postulate replaces the assumption of the individuality of particles, which meets the requirement of causality but goes beyond the limits of a space-time description. Whereas the concrete content of the idea of light quanta is given only by the laws of conservation of energy and momentum, in the case of electrically elementary particles one must also take into account, in this respect, the further invariability of electric charge. It scarcely needs to be recalled that, for a detailed description of the interactions of individuals, one cannot be satisfied only with the facts that find their expression in formulas (1) and (2); one must also introduce auxiliary means that make it possible to take account of the mutual connection characterizing the interactions, in which the significance of electric charge is also manifested. As we shall see below, these auxiliary means require, however, an even deeper renunciation of visualizability in the usual sense.

§ 3. Measurements in the theory of quanta.

In the already mentioned investigation of the absence of contradictions in quantum methods, Heisenberg established relation (2) as an expression of the greatest possible accuracy with which the space-time coordinates and the values of momentum-energy for a certain particle can be measured simultaneously. In doing so he bases himself on

on the following consideration. On the one hand, the position of a particle can be established with any desired degree of accuracy, for example, by means of some optical instrument, if only, in obtaining the image, radiation of sufficiently short wavelength is used. But according to the quantum theory, the scattering of radiation by an object is always connected with a finite change of momentum, which is the greater the shorter the waves. On the other hand, the momentum of a particle can be measured with any accuracy, for example by the Doppler effect of the radiation under consideration, if only the wavelength employed is so large that the recoil in emission can be neglected; but, correspondingly, the determination of position will become inaccurate.

The core of this argument lies in the inevitability of the quantum postulate in judging the possibilities of measurement. A more precise investigation of the possibilities of determination is required, however, in order to clarify in every respect the mutually complementary character of the description. The discontinuous change of energy and momentum by itself could not serve as an obstacle to our assigning perfectly exact values to the space-time coordinates and to the momentum-energy quantities before and after the process. The mutually connected uncertainty, always inherent in the indication of these values, is due, as is clear from what has been set forth above, above all to the limited accuracy with which changes of energy and momentum can be determined when the wave fields used for observation must be sufficiently small that the space-time coordinates of the particles can be established.

In determining position with the aid of an optical instrument, one must remember that the image is always obtained only with a convergent beam. If the wavelength is denoted by $\lambda$, then the resolving power of the microscope will be determined by the well-known expression $\frac{\lambda}{2\varepsilon}$, where $\varepsilon$ is the so-called numerical aperture, i.e. the sine of half the corresponding angle. Even in the case of illumination of the object by parallel—

by light, when the momentum of the incident quantum \(h/\lambda\) is known also in direction, the finite aperture will nevertheless hinder us in the exact determination of the recoil in scattering.

Even if the momentum of the particle before the scattering process is known exactly, the component of the momentum in the plane of the object will be observed by us with an inaccuracy lying evidently within the limits \(2\varepsilon h/\lambda\). The product of the accuracies with which the coordinates of position and the component of momentum in a definite direction can be specified is therefore expressed again by formula (2). One might think that, for judging the accuracy of determining position, not only the convergence of the rays is important, but also the length of the wave train, since during the finite time of illumination the particle may change its position. But since the wavelength of light is not essential for our calculation, it is easy to see that, for each aperture, the wave train may be chosen so short that the change in the position of the particle during the time of observation can be neglected in comparison with the limits of accuracy in determining position that depend on the resolving power.

When measuring momentum by means of the Doppler effect (taking into account the Compton effect), it will be necessary to use a parallel wave train. For the accuracy with which the change in the wavelength of the scattered light can be measured, what is essential, however, is the extension of the wave train in the direction of propagation. Let the directions of the incident and scattered radiation be the same, or opposite, as in the case of the components of position and momentum subject to measurement; then the expression \(c/2l\) may be regarded as a measure of the accuracy of measuring the velocity (\(l\) is the length of the wave train), the velocity of light here being taken, for simplicity, as large in comparison with the velocity of light. If \(m\) is the mass of the particle, then the uncertainty connected with the determination of the momentum from observations is equal to \(cm/2l\). In this

in this case the magnitude of the recoil

\[ \frac{2h}{\lambda} \]

is sufficiently well determined and cannot lead to any appreciable uncertainty in finding the momentum of the particle by observation. The general theory of the Compton effect makes it possible to determine the components of the velocity in the direction of the radiation before and after the change of momentum from the difference of the wavelengths of the incident and scattered radiation. If, in addition, the initial coordinates of the particle’s position are precisely known, there still remains an uncertainty in the determination of the position after the observation. Owing to the impossibility of assigning to the recoil a definite moment of time, we are able to determine the mean velocity in the direction of observation during the scattering process only with an accuracy

\[ \frac{2h}{m\lambda}. \]

Hence the uncertainty in specifying the position by observation reaches

\[ \frac{2hl}{mc\lambda}. \]

Here too the product of the accuracies of the measurement of position and momentum is expressed by the general formula (2).

Just as in the determination of position, the duration of the process of observation in measurements of momentum may be made arbitrarily short, provided only that sufficiently short waves are used. The increase of the recoil, or back-impact, in this case, as we have seen, does not affect the accuracy of the measurement. It may also be noted that, in speaking repeatedly of the velocity of a particle, we had in mind only a connection with the usual space-time description that is appropriate in the given case. As follows from the considerations set forth above by de Broglie, the concept of velocity must be applied with caution. An unambiguous definition of this concept is also excluded by the quantum postulate, which must especially be borne in mind when comparing the results of many successive observations. The position of a certain individual at two given moments of time can be determined with any desired accuracy, but if we wish from this to calculate, in the customary way, the velocity during the intervening time, we shall obtain a certain ideal result from which no unambiguous conclusions can be drawn about the previous or future behavior of the individual.

In view of what was said above concerning the possibilities of determining the properties of individuals, the consideration of the accuracy with which the position and momentum of a particle are measured cannot, evidently, be at all different if, instead of the scattering of radiation, we turn to collisions of material particles. In both cases we find that the uncertainty under consideration is equally inherent both in the description of the means of measurement and in the description of the object. Indeed, this uncertainty is inevitable in describing the behavior of individuals with respect to the ordinary coordinate system with rigid bodies and unperturbed clocks. The conditions of the experiment—the opening and closing of diaphragms, etc.—allow conclusions to be drawn only about the space-time extension of the associated wave fields.

Reducing observations to our sensations, we again encounter the quantum postulate in the perception of the means of observation, whether it be a direct action on the eye or on a suitable auxiliary means in the form of a photographic plate, cloud tracks in a Wilson chamber, etc. But it is easy to see that the statistical element thereby introduced cannot affect the degree of uncertainty in the description of the object. One might even suppose that arbitrariness in the choice of what is to be regarded as the object and what as the means of observation opens the possibility of circumventing this uncertainty. One may, for example, ask whether, in measuring the position of a particle by means of an optical instrument, it is not possible, with the aid of the conservation law, to determine the momentum imparted in the scattering, by measuring the change of momentum experienced during observation by the microscope (together with the light source and the photographic plate). Closer examination shows, however, that such a measurement is impossible if at the same time it is desired to know with sufficient accuracy the position of the microscope. In fact, from experiments finding their expression in the wave theory of matter, it follows that the position of the center of gravity of any body and its total momentum can be determined only within the limits of accuracy indicated by formula (2).

Strictly speaking, the concept of observation is inherent precisely in the causal space-time mode of description. But, owing to the general character of relation (2), this concept can be applied without contradiction also in quantum theory, provided only that one takes into account the uncertainty expressed by the said relation. As Heisenberg points out, we obtain an instructive illustration of the quantum description of atomic (microscopic) phenomena if we compare this uncertainty with the uncertainty inherent in the ordinary description of natural phenomena in each observation because of the imperfection of measurements. He notes, in this connection, that even with respect to macroscopic processes one may in a certain sense say that they arise as a result of repeated observations.

One must not forget, however, that according to classical theories each subsequent observation makes it possible to predict the further course of phenomena with ever greater certainty, since with each new observation an increasingly accurate knowledge of the initial state of the system is acquired.

According to quantum theory, however, with each new observation there enters an entirely new element, not subject to control, as a result of the interaction with the measuring apparatus in each observation; moreover, this interaction cannot be neglected. As is clear from the preceding, the measurement of the coordinates of a particle’s position is connected not only with a finite change of the dynamical variables: the establishment of the particle’s position signifies a complete rupture with the causal description of the particle’s dynamical behavior; in exactly the same way, knowledge of its momentum is acquired only at the cost of an unfillable gap in tracing its space-time propagation. This circumstance clearly shows the mutually complementary character of the quantum description of atomic phenomena, which may be regarded as a direct consequence of the contradiction between the quantum postulate and the separation of the object and the measuring apparatus, characteristic of the concept of observation.

§ 4. The correspondence principle and matrix theory.

Until now we have considered only certain general features of the quantum problem. In essence, however, the center of gravity lies in the formulation of the laws of interaction of objects symbolized by the abstract images of the isolated particle and radiation. Points of support for such a formulation were found first of all in the question of the structure of the atom. Here, as is well known, it is possible to illuminate many experimental facts simply by applying classical concepts in direct connection with the quantum postulate. The possibility of this is based on the fact that, for these questions, one may abstract from a detailed description of the space-time course of processes. For example, experiments on the excitation of spectra by electron impacts or by radiation find a rational interpretation on the assumption of discrete stationary states and individual transition processes.

Here the difference from the usual method of description appears especially sharply: spectral lines, which in the classical interpretation are ascribed to one and the same state of the atom, correspond, according to the quantum postulate, to different transition processes, which are offered to the atom as a choice after excitation. Despite this contradiction, it is possible here to establish a formal connection with classical conceptions in limiting cases, when the relative difference in the properties of neighboring states asymptotically disappears and, in statistical applications, discontinuities may be neglected. A connection of this kind made it possible, over wide ranges, to interpret the regularities in spectra on the basis of our conceptions of the structure of the atom.

The tendency to regard the quantum theory as a rational generalization of classical theories led to the establishment of the so-called correspondence principle. The use of this principle for explaining spectral results was based on the symbolic application of classical electrodynamics: with each transition process there was associated one of the harmonic components of the motion of the atom-

of the particle, which could be expected in the given case on the basis of ordinary mechanics. With the exception of the aforementioned limiting cases, where one may neglect the relative difference of successive stationary states, such partial application of the classical theories made it possible only in isolated cases to obtain a strictly quantitative description of the phenomena. In this respect one should especially recall the connection, established by Ladenburg and Kramers, between the classical interpretation of dispersion and the statistical laws developed by Einstein for transition processes associated with radiation phenomena. Kramers’s theory of dispersion led to conclusions of great importance for the rational elaboration of the idea of correspondence. However, the aims implicit in the correspondence principle became fully attainable only with the aid of quantum methods created in recent years.

The beginning of a new stage of development was laid, as is well known, by Heisenberg’s fundamental work, in which he succeeded in freeing himself entirely from the classical concept of motion. Heisenberg from the very outset replaced the customary kinematical and mechanical quantities by symbols relating directly to the individual processes required by the quantum postulate. This was achieved by replacing the Fourier expansion of classical-mechanical quantities with a certain scheme—a matrix—whose elements symbolize purely harmonic oscillations and are associated with possible transitions between stationary states. On the basis of the requirement that the frequencies associated with the elements of the matrix always satisfy the combination principle of spectral lines, there are obtained, as Heisenberg showed, simple rules of calculation for the symbols, making it possible directly to translate the fundamental equations of classical mechanics into the language of quantum theory. This bold and rational attack on the dynamical problem of atomic theory proved from the very beginning to be an extremely powerful and fruitful means for the quantitative interpretation of experimental results. In the collaboration of Born, Jordan, and also Dirac, the theory received a formulation

which, in respect of completeness and generality, can vie with classical mechanics. It is especially noteworthy that the element characteristic of quantum theory, namely Planck’s constant, enters in explicit form only in the rules for calculations with symbols. For matrices corresponding to canonically conjugate variables in the sense of Hamilton’s equations, the rule of commutativity under multiplication is not satisfied; instead, for two such quantities \(q\) and \(p\) the following rule for permuting the factors holds:

\[ pq - qp = \sqrt{-1}\,\frac{h}{2\pi}; \tag{3} \]

this relation sharply expresses the symbolic character of the theory. Matrix theory is often called a calculus with directly observable quantities. It should be remembered, however, that the procedure described is limited only to those problems in which it is possible, to a broad extent, to dispense with a space-time description when applying the quantum postulate; and therefore the question of observation in the true sense recedes into the background.

For the further development of the correspondence between quantum laws and classical mechanics, the deepening of the statistical character of the quantum description, conditioned by the quantum postulate, acquired fundamental importance. A great step forward in this respect was achieved through the generalization of symbolic methods by Dirac and Jordan. They succeeded in operating with matrices arranged not according to stationary states, but precisely with matrices in which the admissible values of any variables enter as the indices of the matrix elements. In the original form of the theory, the “diagonal elements” of a matrix, referring only to a single stationary state, could be interpreted as the time averages of the quantities represented. In exactly the same way, the general theory of matrix transformations makes it possible to represent the mean values of a certain mechanical quantity. In calculating these mean values, a certain number of variables characterizing the “state” have given values, while other variables, canonically conjugate to them, remain indeterminate; and thereby the precisely defined values of the latter quantities are replaced by a kind of statistical average.

tions, while the canonically conjugate quantities run through all possible values. On the basis of the method developed by the authors mentioned, and also on the ideas of Bohr and Pauli, Heisenberg attempted, in the work already indicated above, to give a more detailed analysis of the physical content of quantum theory and especially of the displacement rule (3), paradoxical at first sight. In this connection he established the relation:

\[ \Delta q \cdot \Delta p \sim h, \tag{4} \]

which, in the most general form, is to indicate the maximum possible accuracy of simultaneous observation of two canonically conjugate variables. In this way Heisenberg succeeded, in a very interesting manner, in clarifying many paradoxes encountered in the application of the quantum postulate, and also in proving, within broad limits, the absence of contradictions in the symbolic method.

In connection with the mutually complementary character of the quantum description emphasized here, when judging the absence of contradictions in it one must, as has already repeatedly been pointed out, take simultaneous account of the possibilities of observation and definition. It is precisely in considering this question, as we shall see, that great assistance was provided by the method of wave mechanics developed by Schrödinger. This method gives a general application of the superposition principle also to particles that are in interaction, and thus makes it possible to establish a direct connection with the question of radiation and free particles. Later we shall return to the connection of wave mechanics with the general formulation of the quantum laws by means of the theory of matrix transformations.

§ 5. Wave mechanics and the quantum postulate.

In his considerations on the wave description of material particles, de Broglie from the very beginning pointed out the possibility of representing the stationary states of the atom as an interference phenomenon of phase waves conjugate with

with bound electrons. This point of view initially yielded, in quantitative terms, no more than the older methods based on the application of classical-mechanical concepts, in the development of which Sommerfeld in particular did a great deal. But Schrödinger succeeded in developing the wave method, which opened new horizons and has been of decisive importance for the great successes of atomic theory in recent times. As is known, the natural oscillations of Schrödinger’s wave equation give a rational representation of the stationary states in the atom. At the same time, the energy of each state is connected with the corresponding period of oscillation by the general quantum relation (1). Counting the nodes of the natural oscillations gives a simple interpretation of the concept of a quantum number, already known from the old methods but at first disappearing in the matrix formulation. Furthermore, Schrödinger was able to connect with the solutions of the wave equation the continuous densities of electricity and current which, when applied to a certain natural oscillation, express the electrostatic and magnetic properties of the atom in the corresponding stationary state. In a similar way, the superposition of two natural solutions corresponds to a continuous distribution of electrical oscillations. Their radiation, calculated according to classical electrodynamics, serves as an instructive illustration of the consequences of the quantum postulate and the correspondence requirements with respect to transition processes formulated in matrix theory. Born pointed out the application, important for the further development of Schrödinger’s method, of this method to the investigation of collisions of atoms and free electrical particles. In connection with this he gave a statistical interpretation of the wave function, which makes it possible to calculate the probabilities of individual processes of transition between stationary states required by the quantum postulate. This corresponds to the wave formulation of Ehrenfest’s adiabatic principle, the fruitfulness of which is especially evident from Hund’s very promising investigations of the problem of molecule formation.

On the basis of these results, Schrödinger expressed the hope that the consistent development of wave theory,

perhaps will make it possible to dispense entirely with the irrationality contained in the quantum postulate, and gradually to approach a description of atomic phenomena corresponding to the fundamental features of the classical theories. In support of such a view, in a recently published paper (Ann. d. Phys. 83, 956, 1927) Schrödinger emphasized that in wave theory we are dealing with a simple resonance problem in the case where, according to the quantum postulate, there is a discontinuous exchange of energy between atoms. In particular, the representation of individual stationary states would in that case turn out to be a fiction, and its applicability would serve only as an illustration of the resonance just mentioned. It should, however, be taken into account that in the indicated resonance problem the question concerns a closed system which, according to the interpretation that we have made fundamental, is not amenable to any observation. In general, from this point of view, wave mechanics (and matrix theory) must be regarded as a symbolic translation of the corresponding problem of motion in classical mechanics—a translation adapted to the requirements of quantum theory and capable of interpretation only when the quantum postulate is explicitly invoked. In general, both formulations of the problem of interaction (with respect to their starting points, the wave and corpuscular interpretations of free individuals) should be regarded as mutually complementary. Connected with this is the apparent contradiction in the applications of the concept of energy in the two theories.

The fundamental difficulties standing in the way of a space-time description of an interacting system of particles by means of classical concepts are immediately evident from the inevitability of the principle of superposition in describing the behavior of individual particles. As we have seen, knowledge of the momentum and energy of a free particle already excludes a precise indication of the coordinates of space-time. It follows from this that the direct use of the concept of energy in connection with the classical conception of the potential energy of a system is impossible here. In Schrödinger’s wave equation these difficulties are avoided by the fact that the classical

the expression of the Hamiltonian function is replaced by a certain differential operator by means of the relation:

\[ p=\sqrt{-1}\,\frac{h}{2\pi}\frac{\partial}{\partial q}, \tag{5} \]

where \(p\) is the generalized component of momentum and \(q\) the canonically conjugate variable. In this, the negative value of the energy is regarded as conjugate to time. Thus both time and space, and energy and momentum, are at first applied in a purely formal manner.

The symbolic character of Schrödinger’s method is clear not only because its simplicity, like that of the matrix method, is based on the essential use of imaginary arithmetical quantities. Here there can be no question of a direct connection with our customary mode of representation, above all because the “geometrical” problem represented by the wave equation is assigned to the so-called coordinate space, the number of dimensions of which is equal to the number of degrees of freedom of the system and therefore, generally speaking, different from the three dimensions of ordinary space. Moreover, the formulation of the problem of interaction in the wave equation, just as the matrix formulation of quantum theory, is limited by the fact that in the classical-mechanical problem taken as its basis, the finite velocity of propagation of forces required by the theory of relativity is not taken into account.

The desire for visualizability with respect to space-time pictures in the problem of interaction is also illegitimate. All our information about the properties of atoms, if it does not concern their motion as a whole, is based on reactions of radiation or of collisions. In the final analysis the interpretation of observation is made with the help of radiation in empty space and of free material particles; on these abstractions rests all our space-time interpretation of phenomena, as well as the definition of the concepts of momentum and energy. Speaking of the application of these auxiliary means—we can judge only of the absence of internal contra-

... speech; moreover, the possibilities of definition and observation must be taken into account.

The eigenoscillations of Schrödinger’s wave equation give a convenient representation of stationary states precisely because, in connection with the general quantum relation (1), they make it possible to determine the energy of the system unambiguously.

At the same time, however, in the interpretation of observations there is inevitably a broad renunciation of the space-time description. As we shall see, the consistent application of the concept of stationary states excludes any possibility of a more detailed understanding of the behavior of individual particles in the atom. In problems where the description of this behavior is essential for the interpretation of observations, we can turn to the investigation of the general solution of the wave equation, obtained from the superposition of eigen-solutions. Here we are faced with a mutual supplementing of the possibilities of definition of the same type as in the question considered earlier concerning the properties of light and of a free material particle. While the definition of the energy and momentum of individual entities is connected with the concept of a harmonic elementary wave, every space-time feature of the description of phenomena is based, as we have seen, on the consideration of interferences occurring in a group of such waves. And in the case now under consideration, one can directly prove the agreement of the possibilities of observation with the possibilities of definition.

According to the quantum postulate, every observation of the behavior of an electron is always accompanied by a change in the state of the atom. As Heisenberg noted, in observing atoms in stationary states with small quantum numbers, this change consists in the general case even in the ejection of the corresponding electron from the atom. A description of the electron’s “orbit” by means of successive observations in such cases is therefore excluded. This is connected with the fact that from eigenoscillations with few nodes one cannot construct a wave group that could even approximately represent the “motion” of the particle.

The supplementary nature of the description is expressed, however, above all by the fact that an unambiguous use of observations of the behavior of particles in the atom is always based on the possibility of neglecting the interaction in the process of observation and of regarding the particles as free. But for this it is required that the duration of the process of observation be small in comparison with the natural period of the atom, which inevitably entails an inaccuracy in the determination of the energy changing in the process—an inaccuracy greater than the difference of the energies of neighboring stationary states.

In judging the possibilities of observation in general, one should remember that the solutions of wave mechanics can receive a visual interpretation only insofar as they are describable by means of the concept of free particles. Here the difference between classical mechanics and the quantum treatment in the problem of interaction is revealed with particular vividness. In the first case the indicated reservation is unnecessary, because an immediate “reality” is ascribed to the particle, independently of whether it is free or bound. This should especially be borne in mind in the unobjectionable application of the Schrödinger density of electricity as a measure of the probability of finding electrons within a definite spatial region in the atom. With the reservation mentioned, such an interpretation reduces directly to the assumption that the probability of the presence of a free electron is determined by the electric density associated with the wave field, just as the probability of the presence of a light quantum is determined by the density of radiation calculated according to the wave theory.

As has already been indicated, the method for the general unobjectionable use of classical concepts in quantum theory is given in the Dirac–Heisenberg transformation theory; with the aid of this theory Heisenberg formulated his general uncertainty relation (4). It is precisely in this theory that Schrödinger’s wave equation has found an instructive application. The proper solutions of this equation serve as auxiliary functions for the transformation of matrices in which the values of the system’s energy serve as indices into matrices where the indices are the spatial co-

coordinates of particles. In this connection it should be mentioned that recently Jordan and Klein (ZS. f. Phys. 45, 751, 1927) obtained a formulation of the interaction problem contained in Schrödinger’s wave equation; relying on the wave representation of an individual particle, they made use of a symbolic device connected with the broad interpretation of the problem of radiation developed by Dirac on the basis of matrix theory. We shall return to this below.

§ 6. The Reality of Stationary States.

In the concept of stationary states, as has been said, we have before us a characteristic application of the quantum postulate. In its essence this concept requires a complete renunciation of description in time. From our point of view, precisely this renunciation is the condition for an unambiguous determination of the energy of the atom. Strictly speaking, the concept of a stationary state requires the elimination of every external interaction with individuals not belonging to the system. In ascribing to such a closed system a definite value of the energy, we directly express the requirement of causality, hidden in the principle of conservation of energy. In this we see the justification of the assumption underlying the application of the quantum postulate to questions of atomic structure, namely the supermechanical stability of stationary states; according to this assumption, the atom, both before and after external actions, is in a precisely determined stationary state.

In judging the well-known paradoxes that arise when describing collision and radiation reactions on the basis of this assumption, it is essential to take into account the limited possibilities of defining the means of reaction, expressed by relation (2). Indeed, a determination of the energy of the reacting individuals so precise that one could speak of conservation of energy in the reaction requires, according to (2), the coupling of such an interval of time with the reaction that is large in comparison with the period connected with the transition process; this period, according to (1), depends on the difference of the energies of the stationary states. This circumst—

…manifests itself in an interesting way when considering processes that take place when rapidly moving particles pass through an atom. According to ordinary kinematics, the effective duration of the collision here is very small in comparison with the natural periods of the atom, and, apparently, serious difficulties arise in reconciling the conservation law with the assumption of the stability of stationary states (cf. ZS. f. Phys. 34, 142, 1925). From the wave point of view, however, the reaction time under consideration is connected directly with the accuracy of determining the energy of the incident particle, and therefore there can never be any question of a contradiction with the conservation law. Concerning paradoxes of this kind, Campbell (Phil. Mag. 1, 1106, 1926) proposed regarding the very concept of time as essentially statistical. From our point of view, in which the basis of the space-time description should be sought in the abstract image of free individuals, it is impossible to reconcile with the requirement of relativity such a fundamental distinction between the concepts of time and space. The special position of time in connection with the concept of stationary states must have its basis in the special character of the corresponding problems.

The application of the concept of stationary states requires that in any observation which makes it possible to distinguish individual stationary states (by means of collision reactions or radiation), one should be able to disregard the previous history of the atom. At first glance, it may seem a difficulty here that the symbolic quantum methods assign to each stationary state a certain phase of oscillation, which seems to establish a connection—contrary to the idea of stationary states—with one or another preceding influence. But if, in general, one has to deal with some problem of time, then there can never be any question of a completely closed system. The use of purely harmonic proper oscillations in the interpretation of observations is in reality only an expedient idealization, which, upon exact consideration, must always be replaced by a group of harmonic oscillations corresponding to a finite interval of frequencies.

As has already been said, the absence, for the group as a whole, of a phase in the sense in which it exists for individual elementary waves, or proper oscillations, is a general consequence of the principle of superposition.

Such unobservability of phase, well known from the theory of optical instruments, appears especially simply when considering the Stern–Gerlach experiment with a molecular beam, so important for the investigation of the properties of individual atoms. Heisenberg explained that, in order to separate atoms with different orientations in a magnetic field, it is necessary that the deflection of the rays be greater than the diffraction from the slit for the de Broglie waves representing the translational motion of the atoms. A simple calculation shows that, according to this condition, the product of the time required for passage through the field and the uncertainty in the determination of the atom’s energy in the field (arising as a consequence of the finite width of the beam of rays) must be at least equal to the quantum of action. Heisenberg sees in this result a confirmation of relation (2) with respect to the reciprocal uncertainty connected with indications of energy and time. In the present case, however, there is no simple measurement of the energy of the atom at a given moment. The periods of the atom’s proper oscillations in the field are connected with its total energy by the general relation (1); therefore we see that the indicated condition for the separation of atoms corresponds precisely to the loss of information about phase. This circumstance makes it possible to avoid the apparent contradictions that arise in certain, often discussed, thought experiments concerning the coherence of resonant radiation, considered also by Heisenberg.

When we spoke above of the atom as a closed system, this meant neglecting radiation, which, even without external actions, limits the duration of stationary states. In many cases radiation may be neglected because the mutual coupling of the atom and the radiation field, which may be expected according to classical electrodynamics, is, generally speaking, very weak in comparison with the coupling of the particles in the atom. Indeed, in describing the state of an atom one may, within wide limits, neglect the reciprocal

by the action of radiation, if one abstracts from the blurring of the energy values, whose connection with the lifetime of stationary states corresponds to formula (2) (cf. ZS. f. Phys. 13, 117, 1923). It is precisely on this that the possibility is based of drawing conclusions about the properties of radiation in connection with classical electrodynamics.

The problem of radiation in the new quantum methods was at first treated precisely on the basis of a quantitative application of these considerations of correspondence. Such was the starting point of Heisenberg’s initial conclusions. A careful analysis of Schrödinger’s treatment of radiation phenomena on the basis of the correspondence principle has recently been given by Klein (ZS. f. Phys. 41, 407, 1927). In a treatment founded more rigorously (Dirac, Proc. Roy. Soc. of London A, 114, 243, 1927), the radiation field is included in the closed system under consideration. Thanks to this, it became possible to take rational account of the individual character of the radiation processes required by quantum theory, and to construct a theory of dispersion in which the finite width of spectral lines is taken into consideration. The abandonment of space-time visualization, which distinguishes this analysis, is a significant indication of the fundamentally supplementary nature of the quantum description. No less do the sharp deviations from the causal mode of description, which we encounter in radiation phenomena and of which we have already spoken in connection with the question of the excitation of spectra, remind us of this.

The mutual exclusion of the concept of stationary states and the description of the behavior of an individual particle in an atom may seem difficult, if one recalls the asymptotic approximation of the properties of atoms to the requirements of classical electrodynamics according to the correspondence principle. This approximation means that mechanical pictures of electronic motions can be rationally used in the region of large quantum numbers, where the relative difference between neighboring stationary states gradually disappears. However, in all this there is no gradual transition to the classical theory, where the quantum postulate

would gradually become superfluous. On the contrary, the conclusions drawn from the correspondence principle by means of classical images are based precisely on the preservation of the concept of stationary states and individual transition processes even in the limit.

In this question it is precisely the new methods that find an instructive application. Schrödinger (Naturwissenschaften 14, 664, 1926) proved the possibility of constructing wave groups in this limiting region by means of the superposition of proper oscillations, the extents of the groups being small in comparison with the “magnitude” of the atom, and the propagation of the groups approaching as closely as desired the classical conception of moving material particles, provided only that the quantum numbers are chosen sufficiently large. For the especially simple case of the harmonic oscillator Schrödinger showed that such wave groups exist even for an unlimited time and oscillate back and forth in accordance with the classical picture of the motion of the oscillator. In this circumstance Schrödinger saw support for the hope of a pure wave theory of matter without the quantum postulate. Heisenberg explained, however, that the simple relations in the oscillator are an exception connected with the purely harmonic nature of the corresponding classical motions. Here there is also no question of a gradual approximation to the problem of free particles. In the general case wave groups gradually spread over the whole atomic region, and the motion of any bound electron can be followed only for such a number of revolutions as will be of the order of magnitude of the quantum numbers associated with the proper oscillations. This question is examined in more detail in a recently published paper by Darwin (Proc. Roy. Soc. of London A. 117, 258, 1927), in which several instructive examples of the behavior of wave groups are given. An analogous problem from the point of view of matrix theory was analyzed by Kennard (ZS. f. Phys. 44, 326, 1927).

We encounter here again the contradiction between the wave principle of superposition and the assumption of the individuality of particles, with which we already had to deal for free particles,

an asymptotic approximation to classical mechanics, which knows no essential distinction between free and bound particles, at the same time gives a particularly simple illustration of the considerations set forth above concerning the freedom from contradictions in the application of the concept of stationary states. As we have seen, the detection of any stationary state by means of impact or radiation reactions is connected with a certain omission in tracing temporal relations, the order of magnitude of which is at least equal to the period conjugate with the process of transition between neighboring stationary states. In the limiting case of large quantum numbers this period may be regarded as equal to the period of revolution. We thus see that there is no possibility of establishing a causal connection between observations that permit the stationary state to be determined and earlier observations concerning the behavior of individual particles in the atom.

Summing up, one may say that the concepts of stationary states and of individual transitions, within their domain of application, possess as great—or as little—a degree of reality as do the individual particles themselves. In both the one case and the other, we take into account the requirement of causality, which supplements the space-time mode of description; and the rational application of this requirement is limited only by the possibilities of defining the corresponding concepts.

§ 7. The problem of elementary particles.

Taking into account the complementarity required by the quantum postulate, it apparently is indeed possible, with the aid of symbolic methods, to construct a description of atomic phenomena that is free of contradictions and constitutes a rational generalization of the space-time description. Such a conclusion does not mean, however, that the classical electron theory can be regarded as a simple limiting case with an evanescently small quantum of action. The connection with experience toward which this theory strives is based on pre-

positions which can hardly be separated from the circle of problems of quantum theory. An indication of this is provided by the already known difficulties in attempts to combine the individuality of electric elementary particles with general mechanical electrodynamic principles. The general theory of gravitation, in the form in which it was formulated in the theory of relativity, likewise did not justify the hopes placed upon it in this respect. A satisfactory solution of the questions touched upon here can be expected only from a rational reworking of the general theory of the field, whereby the electric elementary quantum must find its natural place as an expression of the feature of individuality characteristic of quantum theory. Recently Klein (ZS. f. Phys, 46, 188, 1927) pointed out the possibility of connecting this problem with the five-dimensional unified representation of electromagnetism and gravitation proposed by Kaluza. Indeed, in this theory the conservation of electricity is analogous to the principles of conservation of energy and momentum. These latter concepts are supplementary to the space-time description for atomic phenomena. In exactly the same way, according to Klein, the admissibility of the usual four-dimensional description, and also of its symbolic quantum application, is essentially based on the fact that electric charge always figures here as a quite definite elementary quantum; therefore the conjugate fifth dimension does not manifest itself directly in the interpretation of experiments.

Quite independently of these unresolved deep problems, classical electron theory until the most recent time served as a guiding thread in the further development of the description based on correspondence, with the added idea, first expressed by Compton, that an elementary particle, besides mass and charge, also possesses a magnetic moment connected with an angular momentum determined by the quantum of action. This assumption, introduced with decisive success by Goudsmit and Uhlenbeck in considering the anomalous Zeeman effect, has been fully justified in connection with the new methods, as Heisenberg and Jordan especially showed. One may quite defi-

strictly speaking, that the hypothesis of the magnetic electron, together with the resonance problem clearly posed by Heisenberg (ZS. f. Phys. 41, 239, 1927) and arising in the quantum description of the behavior of atoms with many electrons, has to some extent completed the interpretation, on the basis of correspondence, of the regularities in spectra and in the periodic system. The principles underlying this theory have even opened a path to certain conclusions about the properties of atomic nuclei. Dennison (Proc. Roy. Soc. of London A 115, 483, 1927), for example, succeeded in showing, on the basis of the ideas of Heisenberg and Hund, how the difficulties that had hitherto remained in explaining the specific heat of hydrogen could be bypassed, if one ascribes to the proton an angular momentum of the same magnitude as that of the electron. But, owing to its greater mass, the magnetic moment of the proton must be much smaller than that of the electron.

The insufficiency of the methods that have existed up to now in the problem of elementary particles is manifested in the questions just considered by the fact that they do not provide an unambiguous basis for the difference between the behavior of electrically elementary particles, on the one hand, and the “individuals” symbolized in the representation of light quanta, on the other. In Pauli’s exclusion principle, in which this distinction is also expressed, so fruitful for the problem of the structure of atoms as well as for the most recent development of statistical theories, we are dealing with one of many conceivable possibilities, each of which in itself satisfies the requirement of correspondence. In particular, in the question of the magnetic electron we encounter an instructive example of the difficulty of satisfying the requirement of relativity in quantum theory. Up to now it did not seem possible to reconcile the promising propositions of Darwin and Pauli concerning a generalization of quantum methods that is relatively convenient for the present problem with the relativistic-kinematic considerations of Thomas, which have proved so essential for explaining the experimental results. However, very recently Dirac (Proc. Roy. Soc. of London A. 117, 610, 1928) has succeeded in making a successful approach to the problem of the magnetic

electron by means of a new, extraordinarily ingenious extension of symbolic methods; in doing so, agreement with spectral phenomena is obtained, and at the same time the requirement of relativity is satisfied. This theory is based not only on the complexity of the preceding methods, characterized by the use of imaginary quantities; in its very fundamental equations it employs numerical fields (matrices) of an even higher degree of complexity.

In its essence, the very formulation of relativity already presupposes the combination of space-time coordination and the requirement of causality characteristic of classical theories. Therefore, in rationally adapting the requirement of relativity to the quantum postulate, we must be prepared for an even greater renunciation of visualization in the ordinary sense than in the quantum methods considered above. Indeed, here we find ourselves on the path laid out by Einstein, the path of adapting our forms of contemplation, borrowed from sensations, to the gradually deepening knowledge of the laws of nature. The difficulties that we encounter here arise above all from the fact that, so to speak, every word of the language is connected with these forms of contemplation. In quantum theory we meet such a difficulty from the very beginning in the question of the inevitability of the feature of irrationality inherent in the quantum postulate. I hope, however, that the concept of complementarity is capable of characterizing the existing state of affairs, which seems deeply analogous to the general difficulties in the formation of human concepts, based on the separation of subject and object.

1. The article substantially coincides with the contents of a report on the contemporary state of quantum theory delivered during the celebration of the Volta jubilee on September 16, 1927, in Como. A survey of the state of the theory immediately before the development of the new auxiliary means in this field may be found in the author’s report “Atomic Theory and Mechanics” (Naturwissenschaften 14, 1, 1926; Nature, 116, 809, 1925). The extraordinary development of the theory since that time has called forth a very extensive literature. Here we confine ourselves only to a few references in the text to new works especially important for the considerations set forth. [Bohr’s article appeared simultaneously in English (Nature, 121, 580, 1928) and in German (Naturwissenschaften 16, 245, 1928). The Russian translation was made from the German text and checked against the English.] ↩