The Current State of Atomic Physics1

A. Sommerfeld, Munich.

One now often has occasion to hear or read pessimistic judgments concerning the state of atomic physics and concerning the logical system of all physics in general. The coherent system of the theory of the continuum, of field physics, which by the end of the last century seemed to have formed itself into a unified electromagnetic picture of the world, has been destroyed by the peculiar demands of quantum theory. And although at first it did not prove possible to construct a uniform theory of the discontinuum, an all-embracing quantum theory, the facts from the domain of atomic physics unquestionably point to the existence of such a physics of the discontinuum; spectra point to it at first glance, as does all of chemistry. Therefore Bohr’s atomic physics could be created only because, from the very beginning, he connected it with Planck’s discontinuous quantum of action.

But in constructing his theory Bohr had to make a “loan” from classical theory in the form of his principle of correspondence, which was necessary in order to fill the essential gaps of quantum methods. I myself have always been especially willing to emphasize the discontinuous, integral character of the quantum rules; I did so both in my book and, especially, in my academic address of 1925, where I compared the integral point of view of quantum theory with the old views of the Pythagoreans concerning the role of whole numbers in the laws of nature—with views that flowed from the integral relations of the strings of the lyre, reflected in the harmony of the spheres. And indeed, so long as the quantum conditions, which to a certain degree served to determine atoms, were put forward without proof as initial propositions, we were dealing here with a special axiom of integrality which seemed to determine the essence of physical things.

soon switched to Hertz’s point of view, and the entire further development of physics in general pushed the ether models aside. Will not the same thing happen with the present models of the atom; will they not be replaced, for example, by Schrödinger’s wave equation or by Heisenberg’s corresponding matrix algebra? Of course not. For Bohr’s model of the atom enters into Schrödinger’s equation just as it does into the condition imposed on Heisenberg’s Hamiltonian function, as an integral component. Indeed, the number of electrons in the model of the atom determines the number of independent variables in the aforementioned equation, and the potential energy between the electrons and the nuclei enters explicitly in Schrödinger and Heisenberg. Consequently, without a model of the atom there exists no wave equation and no matrix calculus. The modern development of the theory cannot and does not wish to displace the model of the atom; it provides the long-sought micromechanics adequate to atomic physics. By contrast, the old ether models were connected with Maxwell’s equations only in an entirely external way, for in these equations there are no traces whatever of the rollers and flywheels of the ether models. These mechanisms, including even the most solidly materialized ether, can be removed without harm to Maxwell’s equations. If, however, one removes the model of the atom, for example, from Schrödinger’s wave equation, then absolutely nothing remains.

But what, then, is the situation with the intuitive reality of the model of the atom? Since the new mechanics correctly conveys real facts, the model of the atom inseparably connected with it is real. But is this the ordinary reality in space and time? According to our present knowledge, we must answer this question in the negative. Models of the atom for atoms built from several electrons are real not in three-dimensional, but in many-dimensional space; moreover, here each electron may be imagined as a genuine point corpuscle. But in three-dimensional space the electron cannot be localized. Heisenberg emphasizes this in his Düsseldorf lecture, and Schrödinger illustrates it by “smearing” the electron’s charge into a continuous spatial mass. Personally I do not greatly believe in this smeared-out, spreading electron, if only because outside the atom corpuscularly concentrated electrons of high velocity can undoubtedly be established experimentally. On the other hand, it is an indisputable fact that Schrödinger’s continuous densities, in calculating the physical and chemical effects of the atom, render invaluable assistance and in this sense are real to a greater degree than the point-localized electron of the old theory. It is quite possible that the continuous charge density and the continuous charge current connected with it in Schrödinger’s theory must be understood statistically in the sense of several important works by Bohr,

namely, as the probability of the possible localization of those electrons which lie at the basis of three-dimensional phenomena, but possess an exact real existence in another, many-dimensional phase space.

The situation with time is the same as with space. The elementary conception of the temporal circulation of the model of the atom cannot be preserved in the new theory. The wondrous visual clarity of a microscopic planetary system, according to the present state of quantum theory, has apparently perished irretrievably.

But visual clarity in physics is a special matter. When Maxwell’s theory appeared, the older physicists in particular said: this is not a physical theory, it is a mathematical theory; Maxwell’s equations cannot be given a visual interpretation. At present, after thirty years of practice, Maxwell’s equations seem to us visual to the highest degree; we would be happy if we could subject atomic physics to Maxwell’s scheme. The same happened with the theory of relativity. It now seems to many of us, owing to the emphasis on the empirical origin of our space-time perceptions, more physically visual than the old Newtonian doctrine of absolute space and absolute time. Therefore we shall not approach quantum mechanics with the yardstick of premature and ill-defined visual clarity, but shall wait until this visual clarity comes of itself, when its time arrives.

Far more important than visual clarity is mathematical simplicity. A judgment about the visual clarity of a theory is subjectively conditioned; a judgment about mathematical simplicity is objective and is characterized by the problems that can be solved by the theory. In this respect there can be no doubt that the new theory, especially in the form given to it by Schrödinger, possesses, in comparison with the old one, the advantage of simplicity to a high degree.

Instead of the visual \(n_k\)-orbits of the old theory, which must be traversed by a point electron with a definite rhythm, there now appear quantum states, called by Schrödinger characteristic numbers or eigenoscillations, which are much more difficult to picture. However, they possess all the features that were formerly essential for the characterization of electronic orbits. Namely, they are determined by the same (integer or half-integer) quantum numbers \(n, k, j \ldots\) as before, and moreover through calculations which, step by step, run parallel to the former calculations. Schrödinger already emphasizes this by using, for example, in considering the problem of the Stark effect, the very same notations that have ordinarily been used up to now in presenting this problem.

I believe that it is permissible even now, as before, to speak of electronic orbits in the atom, for they provide a correct, mathematically unambiguous representation of the abstract quantum states now under consideration—a representation that stands in the same relation to quantum states as, in optics, the description of a process by means of light rays stands to its description by light waves, or, in chemistry, the depiction of a compound by ordinary valence strokes stands to its depiction by electron fields. Indeed, even in the future, when the conditions of the bonds in the atom are precisely known, the simple picture of Kekulé’s benzene ring, as a means of illustration, cannot be forbidden. In exactly the same way, to illustrate the quantum states of the new theory one may use an aggregate of \(n_k\) orbits, with the proviso that these illustrations are not to be understood too literally.

We have already spoken earlier of the, perhaps, merely statistical, i.e. non-causal, interpretation of the state quantities in wave mechanics. Such an interpretation would bring with it a certain indeterminism into our views, without, however, depriving the phenomena accessible to observation of their strict causal determination. The latter, i.e. the exact prediction of what is to be observed under specified conditions, we must demand, since natural science exists at all, and we are attaining this to an ever greater degree.

Whether the form of causality familiar to us will be preserved in this case is doubtful. This form is based on classical mechanics, and it may be characterized, in the spirit of Laplace, as follows: if the magnitudes and the rates of change of all state quantities in the universe are known at some definite initial moment, then from this one can mathematically derive all the state quantities for any future moment of time. This preference for the initial state in the description of natural phenomena is entirely in accord with the structure of classical mechanics, with its differential equations of second order in time. On the contrary, the more subtle data of the quantum world point to an equal role for the initial and the final state. I emphasized this already two years ago in a report to the Innsbruck Society of Natural Scientists, and then substantiated it by considering the intensity of multiplet lines, which, according to the so-called sum rules, are determined symmetrically from the weights of the initial and final states. We can now formulate this more rigorously and generally, saying that the Heisenberg amplitudes \(a_{nm}\) are computed symmetrically from the characteristic functions \(\psi_n\) and \(\psi_m\) of the initial and final states. Connected with this is also Einstein’s equality of transition probabilities \(B_{nm} = B_{mn}\). Let us consider, from another side, the frequencies \(\omega_{nm}\), which are connected with \(a_{nm}\). The equation \(\omega_{nm} = \omega_{mn}\) means spectroscopically that the same line which is preferentially emitted by the atom,

the same atom and is preferentially absorbed—roughly speaking, this is what we call Kirchhoff’s law. By mechanical analogies this law was easy to understand so long as it was possible to assume that the atom, in emission and absorption, is in identical states. Now, however, we know that the initial state in emission differs from the initial state in absorption, and that only the transition between the initial and final states in the two cases is connected by reciprocity. Thus Kirchhoff’s law is difficult to understand from the standpoint of ordinary causality; it requires an expanded form of causality, in which the final result of the process is also taken into account.

Let us consider, on the other hand, an excited atom, i.e. one whose energy level lies above the ground level. We know that a transition will occur from this higher level to a lower one, but we do not know whether this transition will take place directly to the ground level or whether, before that, there will be a transition to an intermediate level. Consequently, what happens in an individual case remains indeterminate. But the statistical mean of all these individual cases is quite determinate. Moreover, in order to find the probability of a transition from the initial state \(n\) to some possible final state \(m\), i.e. essentially the quantity \(a_{nm}\), we must take into account the final state, i.e. its weight and its state function \(\psi_m\), just as we do the weight and the state function for the initial state. This procedure is indicated to us not by a subjective preference, but by the present state of our physical knowledge. While the individual phenomenon remains indeterminate, the mean, taken over many cases—which alone is accessible to observation—is completely predetermined by the calculation of \(a_{nm}\). Or let us also consider the problem of the Compton effect. In what direction the electron will fly is unknown to us in an individual case. But how much is scattered on average in any direction, what the intensity and wavelength will be that the light scattered simultaneously with it will have in any direction—this we can determine if we take into account the initial and final states. Can we imagine that we might also succeed in determining the directions of individual electrons if we made more detailed assumptions about the incident light—for example, if we imagined that the light quantum enters the atom to the right or to the left of the middle? I think that such a conception would be chimerical and non-physical.

In exactly the same way, the most fruitful principle of classical mechanics—the principle of least action—has, as has already been noted more than once, a quasi-teleological form; and yet it is known that it is equivalent to the differential equations of mechanics, having—

a strictly causal character. I repeat once again—the question is not that of casting doubt on the lawlike determinacy of physical processes; I touch only upon the question, now widely discussed, of the extent to which the rigid causality handed down to us by the eighteenth century and by rationalistic mechanics corresponds to the present state of our knowledge. I especially emphasize the indication concerning the mathematical form of the fundamental quantum formulae, their symmetry with respect to the initial and final states.

Here a problem of great importance for philosophy arises. As is known, Kant, in his Critique of Pure Reason, held the view that the law of causality is given to us a priori, in other words, that all phenomena of nature must necessarily obey it. By causality he understood the so-called strict causality of the eighteenth century. But Kant also considered the Euclidean structure of space to be given a priori, whereas the works of Riemann, Helmholtz, and Einstein have shown us that geometry is possible with a much smaller number of assumptions. The question is: what is the least number of epistemological presuppositions required for a quantitative world-picture to be possible? The new Kant will in this case be based not on Newtonian, but on quantum mechanics, and he will see that so-called strict causality requires a certain extension in order for a prediction of the phenomena of nature, corresponding to modern knowledge, to be possible.

After these general questions let us turn to the consideration of certain particulars.

In order to elucidate the relation of the former atomic orbits to the images constructed in accordance with Schrödinger’s wave mechanics, let us consider an interesting phenomenon described by Stark, consisting in a dissymmetry of the intensities between the components with the longer and the shorter wavelength in the electric splitting of the Balmer lines. This phenomenon in Bohr’s theory, as is known, is explained as follows: the components with the longer wavelength in the initial state correspond to orbits which pass more behind the nucleus; the components with the shorter wavelength—to orbits which pass more in front of the nucleus. Here “in front” and “behind” are interpreted with respect to the direction of the electric field. If now we are dealing with hydrogen canal rays, for example in nitrogen, then collisions of nitrogen molecules with the hydrogen atom occur in the negative direction with respect to the field; the orbits passing behind the nucleus are better protected from perturbations, and the corresponding components with the longer wavelength are brighter. If, on the contrary, we have nitrogen canal rays in hydrogen, then the emission of the Balmer lines is produced by resting

hydrogen atoms: the front orbits (from the point of view of the electric field) turn out to be unperturbed and therefore more probable. In this case the dissymmetry of the intensities is reversed. This was discovered by Wierl in work carried out under the direction of my colleague W. Wien: he showed, moreover, that in a high vacuum, together with the cessation of collisions, the dissymmetry also disappears. We thus have a complete confirmation of Bohr’s interpretation of the phenomenon. The question arises: if there are no localized orbits, if, together with Schrödinger, we replace them by a continuous density distribution, can we then understand the symmetry of the phenomenon? I suggested to F. G. Slack of New York that he find this density distribution in accordance with Schrödinger’s theory.

Fig. 1. Density distribution in the initial state 202.

Slack found this density distribution in agreement with Schrödinger’s theory. Fig. 1 shows the result obtained for the component 202 of the initial state \(H_\beta\). The drawing presents curves of equal intensity for the density contained in a circular tube of radius \(r\), described around the direction of the field, i.e. the quantity \(\int \rho r\,dr\). The principal maximum of this density lies in front of the nucleus; behind it (on both sides of the middle line) we have two secondary maxima. If one imagines a stationary hydrogen atom in the natural oscillation of this density distribution, then it will be better shielded against impacts coming from the left than in the opposite one corresponding to the quantum numbers 022 of the distribution, for which the principal maximum would lie on the left, so that it would discharge to the left. Fig. 2 shows the final state 002, which, together with

1. Lecture delivered at the invitation of the Faculty of Natural Sciences of the University of Hamburg. Phys. Zeitschr. 28, 231, 1927 (15 March 1927). ↩