On the Heisenberg Uncertainty Relations and Their Theoretical-Epistemological Significance*

M. Laue, Berlin

Two years ago I pointed out** that the conclusion drawn from the formal relations of quantum mechanics, in particular from the uncertainty relations, concerning the necessity of abandoning the principle of causality is, from my point of view, not obligatory, since this conclusion rests on concepts borrowed from Newtonian mechanics, and the latter has an empirical origin. These conclusions, in my understanding, refute only the applicability of the concepts mentioned to atomic phenomena. However, in discussing this there arises—at that time not analyzed by me—a thought which at first glance seems very tempting; I wish to deal with it in this article. The thought is the following: “A truly causal understanding of atomic phenomena is impossible because one cannot use, for measuring these phenomena, objects smaller than the atom; the action of the means of measurement on the atom (or combination of atoms) under investigation substantially disturbs the state of the latter. If in other branches of physics one may assume that the test body used in measurement can be chosen sufficiently small to exclude its action on the phenomenon being measured, this possibility disappears as soon as atoms and their states are investigated.”

No essential difference arises if, in this reasoning, the smallest quantum of action $h$ is taken as the smallest body.

The posing of the question concerning the action of the means of measurement on the phenomenon under investigation, and the significance of this action for the investigation of atomic states, appears to me to be a great merit of Bohr and Heisenberg. Nevertheless, I do not believe that the considerations cited indicate a boundary to knowledge which it is altogether impossible to cross. Indeed, at the basis of this conclusion lies a tacitly made premise: “For the discovery of new measurement possibilities it is necessary to resort to new experimental means.” Only in the event that this

* Naturwiss., 26, 439, 1934, translated by L. Chernov.
** Naturwiss., 20, 915, 1932.

premise has been accepted, one may continue further: “Since we have arrived at the most delicate auxiliary means—the atoms themselves—then we shall never be able to go any further.” But is this assumption correct?

Let us consider, for example, Hertz’s discovery, which undoubtedly expanded our knowledge. With what did Hertz experiment? With an inductor and the current source feeding it, with several conductors, metal plates and spheres—in short, with well-known things, which others had already used a thousand times before Hertz. What did he observe? Electric sparks, truly long since known.

Thus this conquest of science, apparently, was not at all due to new experimental means, but to the experimenter’s ingenious logic.

In the history of physics one can perhaps find many more examples of this kind; from the most recent period Nernst’s theorem may serve as an example. But this appears more clearly, perhaps, than in any historical case, in a fairy tale that I therefore wish to tell. Incidentally, despite all its correctness in a physical respect, in certain other respects it avails itself of the rights of a fairy tale.

“There once lived a young man who wished to devote himself to the study of physics, but, because of poverty, was compelled to earn his bread by his own labor. He applied to an electrotechnical firm, which, in view of his poor school education in the field of physics, took him into its service more out of compassion than for other reasons. He was given, however, quite a simple task, namely: to test the electromotive force of galvanic cells with the aid of voltmeters, in the manner customary in technology. He took several such instruments of different designs, so as to test each galvanic cell with several voltmeters, because he could rely on the correctness of a measurement only when all of them indicated one and the same number of volts. The young man did this, and at first everything went well.

But one day there came the turn of a cell for which all his voltmeters showed different voltages. The fault did not lie in errors of the experiment; despite all his diligence, the student could not obtain identical results. Having thought the situation over, our young man finally came to the conclusion: it is self-evident that the use of a voltmeter entails an effect of it upon the cell. In other cases it had been imperceptible; here, however, evidently, the effect is so strong that it substantially affects the results of the measurement.

Thus, a measurement that does not affect the magnitude of the electromotive force is impossible by my means.

He communicated this consideration to an old friend. The latter replied: “As for the effect of the voltmeter on the results of measure—

ON HEISENBERG’S UNCERTAINTY RELATIONS

...then in this respect you are quite right; however, it can be calculated, since the internal resistance of the voltmeter and of the cell is known.” And the friend gave our student a short lesson on Ohm’s law.

After this our young man connected all his voltmeters in succession into one and the same circuit and directly determined (in arbitrary units), from their deflections, the internal resistances of these voltmeters. When he then connected to the galvanic cell—which had created difficulties for him—first one voltmeter and then another, he was able, by means of elementary calculations, to determine, first, the internal resistance of the cell (in the very same units) and, second, the desired effect on the electromotive force. For this he did not need any new experimental auxiliary means; he now had no more conductors than before, but he possessed new knowledge.

It is in general risky to draw too far-reaching theoretical-cognitive conclusions from the present state of physical knowledge. Apart from the principal objection, which consists in the fact that one cannot reject the principle of the accessibility of nature to investigation merely because we have not yet fully understood how to apply it, one must always proceed from foundations that are logically sound and contain no internal contradictions.

This, unfortunately, cannot be asserted with respect to contemporary physics. I shall point here to one profound internal contradiction in this science.

The old conception of the filling of space by matter does not fit with the assumption of the existence of minute particles, and yet both conceptions are made to fit together in contemporary physics. If the smallest particles of space are understood in the usual sense of the word, then, however small that space may be, it—and along with it the corpuscle itself—can be divided further. The further question necessarily arises: what happens if parts of the corpuscle are displaced with respect to one another?

Whether this question is ripe for solution at the present time, or whether it should be left to subsequent generations of physicists, we shall not discuss here. In doing so, however, we are already speaking of particles smaller than the “smallest.” If, despite this, we believe in the existence of these smallest particles, then this, in my opinion, is justified only by the fact that, both according to the experimental results concerning the diffraction of electrons and atoms and according to the wave-mechanical interpretation of spectra and other things of the same kind, the old conception of the filling of space by corpuscles is incorrect, for it cannot be reconciled with these experiments. And nevertheless, in the literature there is still always talk of the radius of the electron in the sense of spatial-

of a substantial extent—is an obvious sign of an unnoticed* contradiction.

It would be very unpleasant for me if these considerations regarding the misunderstanding were taken as directed against contemporary quantum and atomic theory. There exists, it seems to me, a completely objective measure of the success that this theory signifies; this measure lies in the so often censured “lack of visualizability” of it. What is regarded as visualizable depends on the time. A theory that compels the contemporaries of its emergence, especially the older generation, to alter their habitual notions of the external world is always and inevitably regarded as non-visualizable. This was already the case with the theories of Copernicus and of Faraday–Maxwell. With these lines I should like only to warn physicists against the notion that, in view of the brilliant mathematical formalism of contemporary atomic theory, the latter consciously refuses to answer certain questions, to draw conclusions about the existence of what is in principle unknowable. Let us grant that these questions may be such, at least some of them, that it is impossible to answer them, i.e. they may have no physical meaning. We object only to the conclusion that even with a modified formulation of the question we shall never be able to arrive at a complete understanding of physical phenomena. The uncertainty relations set a limit—this is my opinion—to corpuscular mechanics, but not to physical knowledge.

When, strictly speaking, will causality be able to be considered “empirically proven”? When will the last problems of natural science be exhaustively solved? But such a state, in all probability, will never come. There was, indeed, a time several decades ago when it seemed that, at least in physics, we were already close to this. At that time people thought that it was easy to answer all the open questions of physics, and therefore they considered physics, in its essential features, to be complete. Physics was cured of this naive optimism in its subsequent development. But now people fall into an equally uncritical pessimism: the task of physics in general is insoluble. This pessimism seems to me, despite all the supposed grounds cited in its favor, only an echo of a general deep pessimism in the sphere of culture. To yield to it is not the business of the natural scientist. His science stands above human moods.

* That the quantity \(\frac{e^2}{mc^2}\) sometimes has this meaning for the electron is, of course, entirely possible.