The Worldview of Modern Physics.1

M. Planck, Berlin.

I.

This winter marks twenty years since I had the honor and good fortune to speak here in Leiden on the unity of the physical worldview. I had then been invited by the natural-historical section of the university’s student corporation. That invitation was vigorously supported by a letter from my colleague Hendrik Antoon Lorentz, who received me in his hospitable home with friendly kindness and for the first time made me feel the charm of his personality. It is precisely for this reason that my visit to Leiden at that time became one of the great events of my life and awakened in me a feeling of gratitude that I preserve as a treasure. And if today, thanks to the special kindness of my colleagues, I must once again speak before you on the same topic, then I cannot but first of all express the feeling of deep sadness that comes over me when I remember that time. Our deeply revered teacher H. A. Lorentz is no longer among us; Kamerlingh Onnes is no longer among us; nor are many others who were present then. But science does not rest upon individual persons; even the most energetic investigator, in the final analysis, must

to hand over to younger men the work he had begun, and the duty of each of them is, to the extent of the powers allotted to him, to take part in the common work.

Today I am making an attempt to characterize the development of the physical picture of the world since that time, although I clearly realize that my exposition can lay even less claim to completeness and finality than it could then, twenty years ago. But I can console myself to some extent with the fact that since then the task has become incomparably more difficult. For in the intervening period there have arisen problems which have penetrated our physical thinking more deeply than could ever have been expected. Therefore it seems expedient to me, in the interests of clarity, to begin somewhat from afar, even at the risk of dwelling on things long since known. In return, later on I shall have to refrain from setting forth particular, perhaps very interesting details, since otherwise I would have to occupy your attention for too long a time.

In any case I shall be very grateful for criticism of my reasoning. That even the sharpest criticism may essentially be combined with benevolence—Lorentz himself gave a particularly vivid example of this.

II.

The construction of physical science is carried out on the basis of measurements. And since every measurement is connected with sensory perception, all the concepts of physics are borrowed from the sensory world. Therefore every physical law, in the final analysis, refers to events of the sensory world. Taking this circumstance into account, some natural scientists and philosophers incline to the view that physics, in the final analysis, has to do with the sensory world—and, moreover, of course, with the sensory world of man; that the so-called “object,” in the physical sense, is only a complex of various associated sensations. It should be emphasized that such a view can in no case be refuted by logical means. For logic

one is not in a position to extract anything whatever from his own sensory world; she cannot compel him to acknowledge the independent existence of a “fellow human being” (Mitmensch).

But in physics, as in every other science, there reigns not only logical understanding (Verstand), but also sound sense (Vernunft). Not everything that turns out to be free of logical contradictions is correct from the standpoint of reason (vernünftig). And reason tells us that when we turn our backs on the so-called “object” and move away from it, something of this object nevertheless remains. It tells us further that an individual person, that we, people, all together, with our sensory world, together with our entire planet, are only a nothing in the vastness of nature, whose laws are not determined by what takes place in the small human brain, but existed even before life appeared on earth and will exist after the last physicist has disappeared from its face.

By such generalizations, grounded in “everyday experience,” and not by logical inferences, we are compelled to recognize, beyond the sensory world, a second, real world, which leads an independent existence independent of man—a world that we cannot comprehend directly, but comprehend through the mediation of the sensory world, through the mediation of certain signs that it communicates to us, exactly as if we could examine the object that interests us only through spectacles whose optical properties are completely unknown to us.

To anyone who cannot follow this line of thought and sees in the introduction of an essentially unknowable real world an insurmountable difficulty, one may point out that there is a great difference between completed physical theories, the content of which can be precisely analyzed and thereby established that, for their formulation, the concepts of the sensory world are entirely sufficient, and the task of constructing a physical theory from a certain number of measurements that are still isolated. The history of physics shows us,

that this latter, immeasurably more difficult task has always been solved only on the basis of accepting a real world independent of human sensations, and there can be no doubt that in the future it will remain so.

To these two worlds—the sensory world and the real world—there is added yet a third world, which perhaps should be distinguished from them: the world of physical science, or the physical picture of the world. This world, in contrast to each of the two preceding ones, is a conscious creation of the human mind serving a definite purpose and, as such, is changeable and subject to a certain evolution. The task of the physical picture of the world can be formulated in a twofold way, depending on what this picture of the world is brought into connection with—the real world or the sensory world. In the first case the task consists in knowing the real world as fully as possible; in the second, in describing the sensory world as simply as possible. It would be useless to try to discover which of these two formulations is the more “correct”: each of them, taken by itself, is one-sided and unsatisfactory, for, on the one hand, direct knowledge of the real world is altogether impossible, while, on the other hand, one cannot answer the question of which description of several connected sensory perceptions is the simplest. It has happened more than once in the history of the development of physics that, of two different descriptions, that which for a certain interval of time was considered the more complicated subsequently proved to be the simpler. The main point is that the two named formulations of the task practically do not contradict one another but, on the contrary, supplement one another. The first gives tangible assistance to the researcher’s forward-striving imagination, creating ideas quite necessary for his work and capable of bearing fruit; the second keeps him on the firm ground of facts. Corresponding to this circumstance is also the fact that individual physicists, depending on whether their cast of mind is more metaphysical or, conversely, more positivistic, direct their work on the physical picture of the world in one direction or the other.

But besides the metaphysicians and the positivists there exists yet a third group of workers on the physical picture of the world. It is characterized by the fact that its principal interest is directed not toward the connection with the real or sensory world, but toward the inner closure and logical construction of the physical picture of the world. These are the axiomatists. Their activity, too, is useful and necessary. But here as well there is a serious danger of one-sidedness, which consists in the fact that the physical picture of the world loses its significance and degenerates into a contentless formalism. For when the connection with reality is broken, the physical law is no longer represented as a relation between quantities which are all measured independently of one another, but as a definition by means of which one of these quantities is reduced to the others. Such an interpretation is especially tempting because, after all, a physical quantity can be defined by an equation much more exactly than by measurement; but in the end it signifies a renunciation of the true meaning of the quantity, and an aggravating circumstance is the further fact that, since the very name of the quantity is preserved, this easily gives rise to ambiguities and misunderstandings.

Thus we see how, simultaneously from different sides and from different points of view, work is being done on the physical picture of the world, striving toward one goal—to connect lawfully the processes of the sensory world with one another and with the processes of the real world. It is understandable that in different epochs of historical development now one, now another tendency comes to the fore. At times when the physical picture of the world displays a stable character, as was the case in the second half of the last century, the metaphysical tendency acquires great significance—the investigators feel that they are already close to knowledge of the real world; but in times of variability and instability, such as the one we are experiencing, the positivists come to the fore—the investigators are inclined even to reduce everything to the single firm point of departure: to the processes of the sensory world.

If we look back upon the various forms of the physical picture of the world, changing with the passage of time and displacing one another, and try to discover the characteristic form of change, then we must above all keep in mind two facts. First, it should be established that in all modifications of the picture of the world, taken as a whole, what takes place is not a rhythmic oscillation to one side and the other, but a continuous progressive development in a definite direction, characterized by the fact that the content of our sensory world is ever more enriched, our knowledge of it ever deepened, and our mastery over it ever strengthened. Practical applications of physical science testify to this most strikingly. That we can now see and hear at considerably greater distances, that we command far greater forces and speeds than previous generations—this cannot be denied even by the most irreconcilable skeptics; and it is all the less permissible to doubt that this progress signifies a lasting enrichment of our knowledge, which in the future will never be recognized as an error.

Second, it is remarkable in the highest degree that, although the impulse toward every improvement and simplification of the physical picture of the world is constantly provided by new observations, i.e. by processes of the sensory world, nevertheless the physical picture of the world, in its structure, moves ever farther away from the sensory world. It increasingly loses its visual, originally anthropomorphic character; sensory perceptions are excluded from it to an ever greater degree—one need only recall physical optics, in which for a long time there has no longer been any question of the human eye—and at the same time, in its very essence, it shifts more and more into the realm of the abstract, with purely formal mathematical operations playing an ever more significant role and qualitative differences increasingly being reduced to quantitative differences.

If we compare this second circumstance with the first named earlier, i.e. with the perfection of the physical picture of the world in respect to its role for the sensory world, then for this striking and, at first glance, paradoxical-

...of the phenomenon, in my opinion, only one explanation can be given. Namely—the fact that continuous improvement is at the same time associated with the continuous removal of the physical picture of the world from the world of the senses means nothing other than an approximation to the real world. There can, of course, be no question of a logical justification of this assertion, since the existence of the real world cannot be deduced by logical means. But to the same extent it is impossible to refute it by logical means. One’s attitude toward it is rather a matter of practical worldview, and the old truth consists in this: the best worldview is the one that bears the richest fruit. Physics would be an exception among the other sciences if the law did not remain valid for it according to which the most fruitful and significant results of research are always obtained on the way toward the fundamentally unattainable goal of knowing real reality.

III.

How has the physical picture of the world changed over the last twenty years? Each of us knows that the shift that has taken place during this time belongs among the most profound ever to have occurred in the history of science, and that the process of transformation has not yet been fully completed. Yet, evidently, even now, in the stream of development, certain characteristic structural forms of the new picture of the world are crystallizing; and it will not be useless to attempt to outline these characteristic forms, if only in order to encourage the improvement of this attempt.

If we compare the old and the new picture of the world, then first of all we discover a further significant step forward in the direction of reducing all qualitative distinctions to quantitative ones. Thus, for example, the variegated diversity of chemical phenomena has apparently been reduced without remainder to numerical and spatial relations. According to modern views, in general there exist only two primary substances: positive electricity and negative electricity.

Both consist of entirely identical tiny particles with opposite and equal charges. A particle of positive electricity is called a proton; a particle of negative electricity, an electron. Every electrically neutral chemical atom consists of a certain number of protons, firmly bound to one another, and the same number of electrons, of which some are firmly bound to the protons and together with them form the nucleus of the atom, while the remaining electrons revolve around the nucleus.

Thus the smallest atom, the hydrogen atom, consists of one single proton, which is its nucleus, and one electron revolving around the nucleus. The largest atom, the uranium atom, consists of 238 protons and the same number of electrons, of which, however, only 92 move around the nucleus, while the rest sit in the nucleus. Between these two extremes lie the atoms of the remaining elements in all possible combinations. The chemical nature of the elements is determined not by the full number of its protons or electrons, but by the number of its mobile electrons, which we also call the atomic number of the element.

Besides this significant success, which, however, is after all only the fortunate realization of an old idea several centuries old, in the modern picture of the world two new ideas are striking, by which it differs from the former one: the principle of relativity and the quantum principle. Both these ideas, in essence, give the new picture its characteristic distinction in comparison with the old. The fact that they arose in science almost simultaneously should be regarded, in a certain sense, as accidental. For both in their content and in their effect upon the physical picture of the world they are entirely different from one another.

The theory of relativity, which at first seemed to introduce a certain confusion into the views on space and time that it created, in fact proved to be the completion of the knowledge of classical physics. To characterize in one word the positive content of the special theory of relativity, it may perhaps be called a fusion

space and time into one single concept. This does not mean that space and time have become entirely equivalent, but they are connected with one another in exactly the same way as a real number and an imaginary number are connected in the single concept of a complex number; from this point of view Einstein did for physics the same thing that Gauss did for mathematics in the last century. And if we continue the comparison somewhat, we may say that the transition from the special to the general theory of relativity in physics signifies something similar to the transition from linear functions to the general theory of functions in mathematics.

If this comparison, like any other, is not entirely satisfactory, it nevertheless gives a correct idea of the fact that the introduction of the theory of relativity into the physical picture of the world signifies one of the most important steps toward unification and completion. This is reflected in the consequences that it entailed—above all in the fusion of momentum and energy, in the reduction of the concept of mass to the concept of energy, in the identification of inertial and gravitational mass, and in the reduction of the law of gravitation to Riemannian geometry.

However brief these epithets are, their content is just as indisputable. Their significance extends to all processes of nature, beginning with radioactive atoms emitting waves and corpuscles, up to the motion of celestial bodies millions of light-years distant.

The theory of relativity has not yet said its final word. It is possible that surprises still await us here, if only we recall that the problem of merging electrodynamics and mechanics still awaits its final solution. Likewise, the cosmological consequences of the theory of relativity apparently have not yet been fully clarified, if only because here everything depends on the still-open question of whether matter, as it exists in world space, possesses a finite density or not. However these questions may be resolved, in any case the fact remains unchanged that the theory of relativity has raised the classical theory to a higher stage of completion and that its

the physical picture of the world has, in formal respects as well, acquired a quite satisfactory completeness.

This circumstance, as well as the indication of the numerous expositions of the theory of relativity intended for readers of the most varied preparation, it seems to me, may serve as sufficient justification for my not dwelling further on its consideration.

IV.

Into the outlined harmonious picture of the world, which, it would seem, fulfills its task almost ideally, the hypothesis of quanta introduced entirely unexpected and vivid features. If we were to try here, in a single word, to characterize the central idea of this hypothesis, we could seek this fundamental idea in the introduction of a new universal constant: the elementary quantum of action. This constant is that mysterious messenger from the real world which appeared again and again on the scene in the most diverse measurements, which at the same time ever more insistently demanded a place for itself in the physical picture of the world, but nevertheless fits so little into this picture that in the end it broke the framework of this picture, which proved too narrow.

There was a time when it seemed not impossible even that classical physics might be completely overthrown. However, it gradually became clear—although for anyone who believes in the uninterrupted progress of science this was evident from the very beginning—that here, too, in the end, the question is not one of destruction, but of a very profound transformation leading to generalization. For if we suppose that the quantum of action is infinitely small, then quantum physics passes over into classical physics. But even in the general case the fundamental pillars of the edifice of classical physics proved not only not to have been shaken, but, thanks to the introduction of new ideas, they even gained in strength and solidity. Therefore it will be useful first to consider these fundamental pillars.

First of all, one must name them. The universal constants—the gravitational constant, the speed of light, the mass and charge of electrons and protons—as the most tangible heralds of the real world, have invariably retained their significance in the new picture of the world as well. Next come the great principles of conservation of energy and momentum. Although for a certain time their validity was subject to doubt, in the end they triumphantly established themselves in all their details. At the same time, contrary to the opinion of many axiomaticians, it again became entirely clear that these principles cannot in any case be regarded as simple definitions. Next come the principles of thermodynamics, especially the second principle, which, thanks to the introduction of the absolute meaning of entropy, received an even more rigorous formulation than in classical physics. Finally—the principle of relativity, which in the new domain of quantum physics proved to be a reliable and informed guide.

Now there naturally arises the question: if all these foundations of classical physics have remained intact, then what, properly speaking, has changed in the new physics? We shall obtain the answer to this question very easily if we consider somewhat more closely what the elementary quantum of action means. It means, in essence, an equivalence between energy and frequency: \(E=h\nu\). From the point of view of classical theory this equivalence is absolutely incomprehensible. It is incomprehensible above all because energy and the number of oscillations have different dimensions: energy is a dynamical quantity, the number of oscillations a kinematical one. However, this is not yet the most important thing. For if the quantum postulate directly connects kinematics and dynamics with each other, reducing the unit of energy, and together with it that of mass, to units of length and time, then this still does not signify a contradiction but, on the contrary, represents a replenishment and enrichment of the content of classical theory. What is absolutely contradictory and therefore wholly incompatible with classical theory is revealed by the following argument. The number of oscillations is a local quantity: it possesses a definite meaning for a certain given place, whatever oscillations may be in question—mecha-

nical, electrical, or magnetic; it is only necessary to observe this place for a sufficiently long time. Energy, however, is an additive quantity. To speak of energy at a definite place, according to classical theory, has no meaning whatsoever; one must first of all indicate the physical image whose energy is meant—just as, in order to be able to speak in a definite sense of velocity, one must indicate the coordinate system. And since a physical image can in general be chosen quite arbitrarily—it may be larger or smaller—there is always a certain arbitrariness in the value of the energy. And yet this energy, arbitrary to some degree, must be equal to a local quantity—the number of oscillations! We see that a glaring discrepancy is revealed between these two concepts. To cover up this discrepancy, it is necessary to take an important step—a step that truly signifies a break with views which, for classical physics, are self-evident.

Until now, among the premises of every causal physical mode of thought there belonged the proposition according to which all processes in the physical world—by the physical world I, as always, mean the physical picture of the world, and not the real world—can be represented as consisting of local processes in various separate infinitely small elements of space, and that each of these separate elementary processes, in its regular course, apart from its connection with all the others, is unambiguously determined by the processes occurring immediately next to it in space and in time. Let us dwell on a concrete, sufficiently general case. Suppose that the physical image under consideration is a system of material points which move in a conservative force field with constant total energy. Then, according to classical physics, each separate point at each moment of time is in a definite state, i.e. it possesses a definite position and a definite velocity, and its motion can be fully computed from its initial state and the local properties of the force field at those points of space which

it passes through during its motion. If, however, the latter is known, then we need not know the remaining properties of the system.

In the new mechanics the matter stands quite differently. According to the new mechanics, purely local relations are just as insufficient for the formulation of the laws of motion as the microscopic investigation of its individual parts is insufficient for understanding the meaning of some picture. Quite the contrary: a usable formulation of law-governed regularity is obtained only when the physical image is considered as a whole. In accordance with this, according to the new mechanics, each individual material point of the system at any moment is, in a certain sense, simultaneously present in all places of the space occupied by the system, and moreover not at all by means of a force field which it spreads around itself—no, it is present with its own mass and with its own charge.

We see that what is at issue is nothing other than the material point—the most elementary concept of classical mechanics. One has to sacrifice the central significance of this concept; it can be preserved only in special limiting cases. At the same time, from the further course of the discussion we shall see what must be put in the place of the material point in the general case.

If the quantum postulate concerning the equivalence of energy and number of oscillations is to have an unambiguous, i.e. system-of-reference independent, meaning, then, according to the theory of relativity, the momentum vector must be equivalent to the wave-number vector, i.e. the absolute value of the momentum must be equivalent to the reciprocal wavelength, whose normal coincides with the direction of the momentum. In this case the wave should be conceived not in ordinary three-dimensional space, but in the so-called “configuration space,” the number of dimensions of which is equal to the number of degrees of freedom of the system, and whose metric is given by twice the kinetic energy or—which is the same thing—by the square of the total momentum. At the same time the wavelength turns out to be reduced to the kinetic energy, i.e. to the difference of the constan...

of the total energy and of the potential energy, which must be regarded as a given function of position.

The number of oscillations, multiplied by the wavelength, is equal to the velocity of propagation, or the phase velocity, of a certain wave in “configuration space”—the so-called wave of matter. Substitution of the corresponding values into the wave equation known from classical mechanics leads to the linear homogeneous partial differential equation found by Schrödinger, which is a clear foundation of modern quantum mechanics and, apparently, will ultimately play the same role as Newton’s or Lagrange’s or Hamilton’s equations in classical mechanics. At the same time, however, Schrödinger’s equation differs sharply from the latter in that in it the coordinates of the “configuration point” are not functions of time, but independent variables. Accordingly, for a given system—in contrast to the more or less significant number, equal to the number of degrees of freedom, of classical equations of motion—there exists only one quantum equation. Whereas the configuration point of the classical theory describes, with the passage of time, a perfectly definite curve, the configuration point of the wave of matter at each given moment fills all infinite space, even those parts of space where the potential energy is greater than the total energy, so that the kinetic energy there is negative and the momentum imaginary. This is entirely analogous to the case of so-called total reflection, in which, only according to geometrical optics, is light really completely reflected, since the angle of refraction becomes imaginary, whereas according to wave optics light penetrates also into the second medium, although not in the form of plane waves.

Be that as it may, the circumstance that there exist regions in configuration space where the potential energy exceeds the total energy—this circumstance has special significance also for quantum mechanics. For calculation shows that in every such case, by no means every ...

to the value of the energy constant there corresponds a final wave, but only certain, quite definite so-called characteristic numbers, which have to be calculated from the wave equation and which—depending on the properties of the given potential energy—turn out to be different.

From the discrete values of the energy corresponding to the characteristic numbers, by the quantum postulate, discrete values of the period of oscillation are obtained—in exactly the same way as for a stretched string fixed at its ends, except that in the latter case the quantization is determined by an external circumstance—the length of the string—whereas in the former case it is determined by the quantum of action, which already enters into the differential equation itself.

To each proper oscillation there corresponds a special wave function $\psi$—a solution of the wave equation; and all these various functions—the fundamental functions—constitute the elements of the description of the process of motion in wave mechanics.

The result obtained is the following: while classical physics carries out a spatial division of the physical image under consideration into its smallest parts and in this way reduces the motion of any material body to the motions of its separate, presumed immutable material points, quantum physics decomposes every process of motion into separate periodic waves of matter. The latter correspond to the proper oscillations and fundamental functions of the given image, and consequently lead to wave mechanics. Therefore, in classical mechanics the simplest motion is the motion of an individual material point; in quantum mechanics it is the motion of a simple periodic wave. And just as, according to the former, the most general motion of a body is regarded as a totality of the motions of its separate points, in quantum mechanics it is regarded as the interaction of all possible kinds of periodic waves of matter. This difference in the method of consideration can be made vivid by the example of a stretched string. Indeed, on the one hand, as an elementary process one may consider the motion

of individual points of the string. Each material particle of the string moves independently of all the others under the influence of the force acting upon it, determined by the curvature of the string at the given place. But, on the other hand, one may regard as elements of the motion the fundamental tone and the overtones of the string—each of them pertains to the whole string, and their interaction again represents the most general type of motion.

Wave mechanics also makes it possible directly to understand one circumstance which has hitherto remained mysterious. According to the unusually fruitful theory of Niels Bohr, electrons move around the nucleus according to laws entirely analogous to the laws of the motion of planets around the sun. Only instead of the force of gravitation there acts the attraction of oppositely charged nuclei and electrons. The peculiar difference consists, however, in the fact that the electrons move along quite definite discrete orbits, whereas in the case of planets no orbit has any advantage over another.

This initially incomprehensible circumstance finds a very clear explanation for itself in the wave theory of electrons. Indeed, if the electronic orbit is closed in itself, then it is clear that an integer number of wavelengths must always fit into it, quite like the way in which the length of a chain closed into a ring and consisting of identical links must always be equal to an integer number of links. In accordance with this, the revolution of an electron resembles not the motion of a planet around the sun, but the rotation of a perfectly symmetrical ring, so that the ring all the time occupies one and the same position in space, and there is no physical sense in speaking of the instantaneous position of the electron.

But now one may pose the following question: if the elements of motion are not material points but waves of matter, then what does wave mechanics do when it has to describe the motion of a separate material point which at a definite moment occupies a definite position? In order to have the possibility of taking up the consideration of this question, the solution of which is shown with particular clarity ...

...lest we regard the entire irreconcilable opposition of the two theories, let us turn first of all to clarifying the physical meaning of the wave function \(\psi\) of a simple periodic wave of matter. This meaning can be established on the basis of the fact that the energy of a matter wave has a twofold significance, for because it determines the period of oscillation of the wave, its initial meaning, following from the principle of conservation of energy, does not disappear. But if the principle of conservation of energy remains in wave mechanics as well, then the energy of a matter wave must be represented not only by means of the number of oscillations, but also with the aid of an integral taken over the whole space of the wave configuration.

In fact, multiplying the wave equation by \(\bar{\psi}\)1 and then integrating over the whole space of the configuration, we obtain a definite expression for the energy, which at first sight can be interpreted in the following way.

Let us imagine the system of material points under consideration in a very large number of copies, each copy in a different configuration, so that we obtain a very large number of points in configuration space. To each configuration point lying in an infinitely small element of space we ascribe a definite energy, which is additively composed of the previously specified value of the local potential energy and a second term proportional to the square of the local gradient of \(\psi\); this second term we may interpret as kinetic energy. If we then set the spatial density of the distribution of configuration points at some place equal to the square of the absolute value of \(\psi\), which we may take as arbitrarily large, since \(\psi\) contains a constant factor of arbitrary magnitude, then the mean energy of the configuration points will represent the energy of the matter wave. In accordance with this, the absolute value of the wave amplitude has in general no physical significance. If we imagine that \(\psi\) is normalized in such a way that the square

1 That is, by the complex quantity conjugate to \(\psi\). Ed.

2 Advances in Physical Sciences, Vol. IX, Issue 4.

of the absolute value of \(\psi\), integrated over configuration space, gives the value 1, then we may briefly denote this square as the probability that the system of material points is located in a certain place of configuration space, and thereby obtain a vivid expression for a certain physical meaning of \(\psi\).

In all these arguments we proceed from a definite fundamental function \(\psi\) and its corresponding simple periodic wave. But we can state the same propositions also for the general case of a superposition of waves with different periods. Then the wave function \(\psi\) is equal to an algebraic sum of periodic fundamental functions multiplied by certain amplitude factors, and the square of the absolute value of \(\psi\) again denotes the probability corresponding to the position of the configuration point.

In the general case, of course, it is no longer possible to speak of one definite period of oscillation of the matter wave; but, naturally, as before, one may speak of a definite energy, so that here the quantum equation \(E=h\nu\) loses its original meaning and refers only to a certain mean number of oscillations \(\nu\). It is worth mentioning in this connection that, in the superposition of an arbitrarily large number of different simple periodic waves with almost identical numbers of oscillations, the energy of the wave function by no means increases with the number of terms in the sum—although this wave function itself is equal to the sum of the individual wave functions—but preserves its original mean value. Just as the energy of a family of simple periodic waves determines the mean number of oscillations, so the momentum of the family determines the mean wavelength.

The amplitudes and phases of the individual simple periodic waves are initially arbitrary. But this also exhausts the variety of mechanical processes accessible to representation by wave mechanics. This circumstance assumes special importance when we turn to the question raised above concerning the description of the motion of an individual definite material point on the basis of wave mechanics. Indeed, it is immediately found that such a description in exact-

in that sense is generally impossible. For already in order to determine the position of a material point or, speaking more generally, in order to determine the position of a known point in configuration space, wave mechanics provides only one means: one must superpose a family of simple periodic waves of a physical type in such a way that their wave functions everywhere in configuration space, by interference with one another, cancel one another out, and only at the prescribed point reinforce one another. In fact, then the probability of all the other points of the configuration would be equal to zero, and only for the chosen point would it be equal to unity. But in order to isolate this point with perfect sharpness, infinitely small wavelengths, and consequently infinitely large momenta, would be necessary. Thus, in order to obtain an at least approximately suitable result, one must put at the basis, instead of a definite point of configuration, a finite, although small, region of configuration space—the so-called wave packet. By this very fact it has already been said that the determination of the position of a point of configuration according to wave theory is always connected with a certain indeterminacy.

Further, if one must assign to the system of material points under consideration, besides a definite configuration, also a definite value of the momentum, then according to the quantum postulate one must make use, strictly speaking, of only a single wave, with a perfectly definite wavelength, and the description is again impossible. But if one likewise introduces into the value of the momentum a certain small indeterminacy, then the desired aim may be achieved, at least with a certain approximation, by applying waves lying in a narrow interval of frequencies.

Thus, according to wave mechanics, both the position and the momentum of a system of material points can be determined only with a certain inaccuracy, and moreover there exists a definite relation between the two inaccuracies. This relation follows from the simple consideration that the waves applied, if by interference they are to extinguish one another outside the limits of a small region of configura-

tions, at opposite edges of the region, despite their small difference in frequencies, must nevertheless exhibit an appreciable difference of course. If, in accordance with the quantum postulate, the difference of course is replaced by the difference of momenta, then one obtains the law formulated by Heisenberg: the product of the uncertainty in the determination of position and the uncertainty in the determination of momentum is at least of the order of magnitude of the quantum of action. The more accurately the position of a configuration point is determined, the less accurately the value of the momentum is known. Thus the two uncertainties reveal, in a certain sense, a complementarity; this, however, is limited by the fact that in wave mechanics, under certain circumstances, the momenta can be determined absolutely precisely, while the position of the configuration point always remains indeterminate within the limits of a finite region.

This “uncertainty relation” of Heisenberg is something altogether unheard of in classical mechanics. Of course, it has always been known that every measurement is associated with an uncertainty; but it was always assumed that, by suitably refining the methods of measurement, the accuracy could be increased without limit. And now it turns out that the accuracy of measurement is subject to a principled limitation, and the most remarkable thing about this is that this limitation applies not to one quantity—position or velocity—but to their combination. Each quantity, in principle, can be measured as accurately as desired, but always at the expense of the accuracy of the other quantity.

However strange such a statement may sound, it is nevertheless manifestly confirmed by various facts. Let us give just one example. The most immediate and most delicate determination of the position of a point is carried out optically—either by direct observation with the naked or aided eye, or by photography. But for this the point must be illuminated. In that case the image will be the sharper, and consequently the measurement the more precise, the shorter the wavelength employed. In accordance with this, the accuracy can be increased as much as desired. But this increase also has its reverse side: by measuring

velocities. For large masses one may neglect the effect of light on the object being illuminated. The situation is otherwise if the object is a very small mass—for example, an individual electron. For every light ray that falls upon the electron and is reflected from it imparts to it a noticeable push, and one that is the stronger the shorter the wavelength. Therefore, although the accuracy of determining position increases as the wavelength is shortened, the inaccuracy of determining velocity increases in the corresponding proportion. And the same holds in analogous cases.

In light of this view, classical mechanics, which proceeds from unchanging corpuscles that can be measured exactly and that move with definite velocities, represents only an ideal limiting case. It is realized only when the image under consideration possesses a comparatively large energy. In this case the discrete values of the energies lie close to one another; a comparatively small region of energy already contains numerous high wave frequencies—or, what is the same thing, short wavelengths—and their superposition makes it possible to delimit in configuration space, with comparative sharpness, a small wave packet with a definite momentum. Then wave mechanics passes over into corpuscular mechanics; Schrödinger’s differential equation passes into Hamilton–Jacobi’s classical differential equation, and the wave packet moves in configuration space according to the same laws that govern the motion of a system of material points in classical mechanics. But this lasts, generally speaking, only for a certain interval of time. For, since the individual waves of matter do not always interfere in the same way, the wave packet spreads out more or less rapidly, the position of the corresponding points of configuration becomes less and less sharp, and in the end only the wave function \(\psi\) remains exactly determined.

Do all these consequences agree with experiment? The investigation of this question, because of the smallness of the quantum of action, can be undertaken only within the framework of atomic physics and therefore always requires, in the highest degree, delicate auxiliary—

of means. Preliminarily, one can only say that so far not a single fact is known that would give occasion for any well-founded doubt as to the physical significance of all these consequences.

Since the establishment of the wave equation, the development and elaboration of the theory has proceeded at an almost impetuous pace. Within the limits of this report it is impossible to set forth all the extensions and applications that the theory has undergone in recent years. Among the former I shall mention only the introduction of the so-called intrinsic rotation of electrons and protons; further, the relativistic formulation of quantum mechanics; among the latter—the application to the problem of the molecule and the consideration of the so-called many-body problem, i.e. the application to systems consisting of several or many completely identical material points. In this last application there arise in particular questions of a statistical character, which pertain to the number of possible distinct states in an isolated system with a given energy and which are also of significance in calculating the entropy of the system.

Finally, I am also compelled to refrain from a special consideration of the physics of light quanta, which has undergone a development in a certain sense opposite to that of the physics of the material point. For in this field Maxwell’s theory of electromagnetic waves originally prevailed in classical physics, and only later did it become clear that the acceptance of discrete light particles is inevitable, i.e. that electromagnetic waves too, like matter waves, may be interpreted as waves of probability.

There is scarcely more vivid proof that a pure wave theory can satisfy the requirements of the new physics just as little as a pure corpuscular theory. Both theories represent limiting extreme cases. Whereas the corpuscular theory characteristic of classical mechanics correctly conveys the position of a system, but proves unsuitable for determining the “proper values” of its energy and momentum, the wave theory characteristic of classical electrodynamics, although it gives the ener-

not momentum, but is alien to the concept of localization of light particles. The general case is an intermediate region in which both theories play practically equivalent roles and which one can approach from either side, but for the time being only at a small distance. Here very many obscure questions still await their resolution, and one must wait to see which of the methods proposed for their solution—the matrix calculus originally invented by Heisenberg, Born, and Jordan, the wave theory established by de Broglie and Schrödinger, or the mathematics of \(q\)-numbers introduced by Dirac—will best lead to the goal.

V.

If we try to sum up the preceding exposition and at the same time obtain a general outline of the characteristic features of the new picture of the world, then the first impression will undoubtedly be wholly unsatisfactory. Above all, it must be unpleasantly surprising that wave mechanics, which after all represents a sharp opposition to classical mechanics, simply makes use of such concepts borrowed from the latter as the concepts of coordinates and momentum of a material point, or the concept of the kinetic and potential energy of a system of points. At the same time, it asserts that it is quite impossible simultaneously to determine exactly the position and the momentum of a point. Nevertheless these concepts are absolutely necessary for wave mechanics, for without them it is impossible to construct configuration space and its measuring apparatus.

Another difficulty in understanding wave theory apparently lies in the fact that matter waves do not possess that degree of visualizability which, for example, acoustic or electromagnetic waves do, since matter waves propagate not in ordinary space, but in configuration space, and their period of oscillation depends on the choice of the physical image to which they refer. The more

the longer the image selected, the greater its energy and, with it, the frequency of the oscillations.

It is not easy to cope with such objections. They would, however, be overcome if the content of the new theory, first, revealed no internal contradictions and, second, in its applications yielded unambiguous results important for experiment. Yet even as to the extent to which quantum mechanics satisfies these requirements, opinions at present still diverge somewhat. It will therefore be permitted me to dwell on this point.

It is often emphasized with particular stress that quantum mechanics deals only with quantities that are observable in principle and only with problems that have physical meaning. This, of course, is so; yet it cannot be counted as a special advantage of quantum theory over other theories. For the question whether a given quantity is observable in principle, or whether a certain problem has physical meaning, can never be decided a priori, but only from the standpoint of a definite theory. The difference between theories lies precisely in the fact that according to one theory a certain quantity is observable in principle and a certain problem is physically meaningful, whereas according to another theory this is not so. Thus, the absolute velocity of the Earth, according to the theory of the stationary luminiferous ether of Fresnel–Lorentz, is observable in principle; according to the theory of relativity, it is not. Or the absolute acceleration of a body, according to Newtonian mechanics, is observable in principle; according to relativistic mechanics, it is not. Likewise, the problem of constructing a perpetuum mobile for the introduction of the principle of conservation of energy had physical meaning; after the establishment of the principle of conservation of energy it lost that meaning. The choice between these contradictory assertions lies not in the nature of the theories themselves—it is made by experience. Therefore, in order to characterize the superiority of quantum mechanics over classical mechanics it is not enough to say that the former deals only with quantities observable in principle—this is, in the corresponding sense, true also as applied to

to classical mechanics—but one must specify precisely those quantities which, according to the new theory, are in principle observable—or else not observable—and then show that experiment confirms this.

In fact, this proof, for example for Heisenberg’s uncertainty relation considered above, has been carried out to the extent that it has been possible up to the present time, and it may be regarded as a substantiation of the superiority of wave mechanics.

Despite these obvious successes, the uncertainty relation characteristic of quantum theory has aroused objections in wide circles—evidently because, in accordance with this relation, the determination of quantities with which one constantly has to deal in calculations becomes in principle inexact. At the same time, the unfavorable attitude is considerably strengthened by the fact that, as we saw above, the concept of probability is introduced into the interpretation of the equations of quantum mechanics. For thereby, apparently, the requirement of strict causality is abolished, and in its place a certain indeterminism is admitted. Indeed, at the present time there are very eminent physicists who consider it necessary, by force of circumstances, to sacrifice strict causality in the physical picture of the world.

If such a step were really necessary, then the aim of physical research would thereby suffer a very serious loss, and we would have to reckon with an enormous defect. For, if a choice can be made at all, in my opinion, under all circumstances determinism should be preferred to indeterminism, if only because a definite answer to a question always has greater value than an indefinite one.

However, as I understand it, nothing at all compels us to perform this act of renunciation. For the impossibility of giving a definite answer to a question sometimes depends not on the properties of the theory, but on the properties of the question posed. To a physically insufficiently formulated question even the most perfect physical theory cannot give a definite answer. This is already generally known and well-established within the framework of classical statistics.

illuminated truth. If, for example, for two elastic spheres colliding on a plane all details are known, both as to the velocities of the spheres before impact and as to the laws of impact, we nevertheless cannot indicate their velocities after impact. Indeed, for the calculation of the four unknown components of the velocity of both spheres after impact, we have at our disposal only three equations: the equation of conservation of energy and the two components of momentum. Yet we do not say that in an impact there is no causality; rather, we say that the available data are insufficient for complete determinacy.

In order to be able to apply this reasoning to the problems of quantum physics, we must now, finally, return to those thoughts which we considered in the introduction.

If it is indeed true that the structure of the physical picture of the world, in its continuous evolution, is ever farther removed from the world of the senses and, to a corresponding degree, comes ever closer to the real, fundamentally unknowable world, then it follows of itself that the picture of the world must free itself more and more from all anthropomorphic elements. Thus it is utterly impossible to introduce into the physical picture of the world concepts which are in any way connected with the art of human techniques of measurement. This is not done in any way in Heisenberg’s uncertainty relation. For the latter follows directly from the consideration that the elements of the new picture of the world are not material corpuscles, but simple periodic waves of matter corresponding to the physical image under consideration, and are a consequence of the mathematical law according to which it is impossible, by the superposition of simple periodic waves of finite length, to determine a given point with a given momentum. This law has nothing whatever to do with measurements; and the waves of matter, for their part, are uniquely determined by the mathematical boundary problem corresponding to the case under consideration. There can be no question of indeterminism here.

Yet another question—that of the relation of the wave of matter to the sensible world, which alone communicates to us information about phy-

The World Picture of Modern Physics

physical processes. For of an image completely closed in upon itself we would, in general, never learn anything.

At first glance it seems that this question has little to do with physics, since it partly intrudes into the domain of physiology and even psychology. Nevertheless, this objection gives rise to no fundamental difficulties. For one can always imagine that the human sense organ is replaced by a suitably constructed measuring instrument, a self-registering apparatus, such as, for example, a photographic plate, which records the influences acting on it from outside and in this way gives us information about the processes taking place in the surroundings. If we include such measuring instruments in the physical system under consideration and remove all other influences, then in this case we obtain a physical image closed off from the external world, about which we can learn something by means of measurements, taking into account, of course, the structure of the instrument and its possible effect on the processes being measured.

If we possessed such a measuring instrument, one capable of reacting to individual waves of matter in the same way as, for example, an acoustic resonator reacts to a sound wave, then in that case we could measure the waves of matter separately and thus analyze the entire wave process. This, of course, is not the case; on the contrary, the indications of a measuring instrument—for example, of a photographic plate—do not make it possible to draw an unambiguous conclusion about all the details of the process under investigation.

A direct basis for the acceptance of indeterminism could be sought in the circumstance that, according to wave mechanics, the processes in a closed system of material points, isolated from the external world, are in no way determined by the initial state of the system, i.e. by the initial configuration and the initial momentum, and are not even determined approximately; for in fact the wave packet corresponding to the initial state spreads out with the passage of time and decomposes into separate waves of probability.

However, closer consideration shows that here indeterminism is due only to the formulation of the question. The latter is borrowed from corpuscular mechanics, in which indeed the initial state uniquely determines the process for future times; but this formulation of the question does not correspond to wave mechanics precisely because, in accordance with the uncertainty relation, a fundamental inaccuracy of a finite quantity figures in it.

But already since the time of Leibniz another formulation of the question has been known in classical mechanics, one which in classical mechanics likewise leads to a definite answer. The process will be fully determined and, moreover, for all times, when, in addition to the configuration at a known moment, there is given not the momentum, but the configuration of that same system at another moment. The variational principle, the principle of least action, then serves for the calculation of the process. Thus, in the example given earlier of the plane elastic collision of two balls, with the initial and final positions of the balls given and with the time interval given, the three unknowns—namely the two coordinates of the point and the instant of collision—are completely determined by the three conservation equations.

This modified formulation of the problem, in contrast to the preceding one, can also be transferred to wave mechanics. Of course, a definite configuration, as we have seen, can never be fixed by the wave theory with complete exactness; however, the inaccuracy can in principle be made arbitrarily small, and therefore the process can be determined with any desired degree of accuracy. As for the spreading of the wave packet, it is by no means evidence of indeterminism. For the wave packet can also gather together again. The sign of time in wave theory plays no role, just as in corpuscular theory. Any process of motion can also proceed in the reverse direction.

Of course, with the indicated formulation of the problem, a definite wave packet, generally speaking, exists only at the two chosen moments of time. In the interval, and likewise before and after, the individual elementary waves lead a separate existence. But whatever we call them—waves

matter or probability waves—they are in any case completely determined. Thus there is explained the seemingly paradoxical assertion that, if a physical image, from some definite configuration, passes in some definite interval of time into another known configuration, then the question of the configuration at intermediate moments of time has no physical meaning; according to this view, it is just as meaningless to ask about the path of a light quantum which is emitted by a point source of light and is absorbed at some point of a screen serving for observation.

It should be emphasized, however, that with this way of looking at things the meaning of determinism is different from that accepted in classical physics. For there it was the configuration that was determined; here—in quantum physics—it is the matter wave that is determined. The difference is especially important because the configuration is connected with the sensible world much more directly than the matter wave. In this sense, in the new physics the connection of the physical picture of the world with the sensible world has become considerably less close.

This is, of course, a shortcoming, but one must reconcile oneself to it in order to save determinism in the picture of the world. Moreover, this step apparently is taken in the direction which, as has already been pointed out several times, is characteristic of the true development of science. For the structure of the physical picture of the world, as it is perfected, moves ever farther away from the sensible world and assumes ever more abstract forms. And from the standpoint of the principle of relativity such a conception is even, as it were, inevitable. Indeed, since according to this principle time has no advantage whatever over space, it follows necessarily that, if for the causal description of a physical process the consideration of a finite region of space is necessary, then a finite interval of time must also be brought in for this purpose.

But it may be that the formulation of the question proposed here is still too one-sided, too anthropomorphic, and therefore unsuitable for a satisfactory construction of a new phys-

...of the physical picture of the world, and perhaps one should seek a different formulation. In any case, many difficult problems still have to be solved here, and many obscure points clarified.

This peculiar predicament in which theoretical physics now finds itself naturally gives rise to doubt as to whether a theory with such radical innovations is on the right path. The resolution of this fatal question depends solely on the extent to which, in the continuously progressing work on the physical picture of the world, the necessary contact of the latter with the world of the senses is preserved. Without this contact, even the most perfect picture of the world as regards form would be only a soap bubble, which would burst at the first breath of wind.

Fortunately, in this respect, at least at the present moment, we may be entirely at ease. We may even assert without exaggeration that in the history of physics there has been no epoch when theory and experiment have gone so amicably hand in hand as at the present time. For it was experimental facts that shattered and overturned classical theory. Every new idea, every new step, was prepared for and even compelled by the results of experiment. Just as the theory of relativity was preceded by the optical-interference experiment of Michelson, so quantum theory was preceded by the measurements of Lummer and Pringsheim, Rubens and Kurlbaum on the distribution of energy in the spectrum, Lenard’s experiments on the photoelectric effect, and Franck and Hertz’s experiments on electron collisions. I would stray too far were I to recall here the numerous, sometimes quite astonishing, experimental results which have driven theory ever farther from the classical point of view and guided it along quite definite paths. One can only wish and hope that in this joint work, in which all the countries of the world take part in peaceful competition, no dissension will ever arise. For in the constant interaction of experimental and theoretical research—an interaction which serves at once as stimulus and as check—lies the sole guarantee of the uninterrupted progress of physical science.

1. An expanded address delivered at the Physical Institute of Leiden University. Published as a separate booklet. Published by J. A. Barth, Leipzig, 1929. —Ed. ↩