Einstein 1938: The Waves of Matter

Reference: Evolution of Physics

This paper presents Chapter IV section 5 from the book THE EVOLUTION OF PHYSICS by A. EINSTEIN and L. INFELD. The contents are from the original publication of this book by Simon and Schuster, New York (1942).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.


The Waves of Matter

How can we understand the fact that only certain characteristic wave-lengths appear in the spectra of the elements?

It has often happened in physics that an essential advance was achieved by carrying out a consistent analogy between apparently unrelated phenomena. In these pages we have often seen how ideas created and developed in one branch of science were afterwards successfully applied to another. The development of the mechanical and field views gives many examples of this kind. The association of solved problems with those unsolved may throw new light on our difficulties by suggesting new ideas. It is easy to find a superficial analogy which really expresses nothing. But to discover some essential common features, hidden beneath a surface of external differences, to form, on this basis, a new successful theory, is important creative work. The development of the so-called wave mechanics, begun by de Broglie and Schrodinger, less than fifteen years ago, is a typical example of the achievement of a successful theory by means of a deep and fortunate analogy.

Our starting-point is a classical example having nothing to do with modern physics. We take in our hand the end of a very long flexible rubber tube, or a very long spring, and try to move it rhythmically up and down, so that the end oscillates. Then, as we have seen in many other examples, a wave is created by the oscillation which spreads through the tube with a certain velocity. If we imagine an infinitely long tube, then the portions of waves, once started, will pursue their endless journey without interference.

Now another case. The two ends of the same tube are fastened. If preferred, a violin string may be used. What happens now if a wave is created at one end of the rubber tube or cord? The wave begins its journey as in the previous example, but it is soon reflected by the other end of the tube. We now have two waves: one creation by oscillation, the other by reflection; they travel in opposite directions and interfere with each other. It would not be difficult to trace the interference of the two waves and discover the one wave resulting from their superposition; it is called the standing wave. The two words “standing” and “wave” seem to contradict each other; their combination is, nevertheless, justified by the result of the superposition of the two waves.

The simplest example of a standing wave is the motion of a cord with the two ends fixed, an up-and down motion, as shown in our drawing. This motion is the result of one wave lying on the other when the two are travelling in opposite directions. The characteristic feature of this motion is: only the two end-points are at rest. They are called nodes. The wave stands, so to speak, between the two nodes, all points of the cord reaching simultaneously the maxima and minima of their deviation.

But this is only the simplest kind of a standing wave. There are others. For example, a standing wave can have three nodes, one at each end and one in the centre. In this case three points are always at rest. A glance at the drawings shows that here the wave-length is half as great as the one with two nodes. Similarly, standing waves can have four, five, and more nodes. The wavelength in each case will depend on the number of nodes. This number can only be an integer and can change only by jumps. The sentence “the number of nodes in a standing wave is 3.576″—is pure nonsense. Thus the wave-length can only change discontinuously. Here, in this most classical problem, we recognize the familiar features of the quantum theory. The standing wave produced by a violin player is, in fact, still more complicated, being a mixture of very many waves with two, three, four, five, and more nodes and, therefore, a mixture of several wave-lengths. Physics can analyse such a mixture into the simple standing waves from which it is composed. Or, using our previous terminology, we could say that the oscillating string has its spectrum, just as an element emitting radiation. And, in the same way as for the spectrum of an element, only certain wave-lengths are allowed, all others being prohibited.

We have thus discovered some similarity between the oscillating cord and the atom emitting radiation. Strange as this analogy may seem, let us draw further conclusions from it and try to proceed with the comparison, once having chosen it. The atoms of every element are composed of elementary particles, the heavier constituting the nucleus, and the lighter the electrons. Such a system of particles behaves like a small acoustical instrument in which standing waves are produced.

Yet the standing wave is the result of interference between two or, generally, even more moving waves. If there is some truth in our analogy, a still simpler arrangement than that of the atom should correspond to a spreading wave. What is the simplest arrangement? In our material world, nothing can be simpler than an electron, an elementary particle, on which no forces are acting, that is, an electron at rest or in uniform motion. We could guess a further link in the chain of our analogy: electron moving uniformly → waves of a definite length. This was de Broglie’s new and courageous idea.

It was previously shown that there are phenomena in which light reveals its wave-like character and others in which light reveals its corpuscular character. After becoming used to the idea that light is a wave, we found, to our astonishment, that in some cases, for instance in the photoelectric effect, it behaves like a shower of photons. Now we have just the opposite state of affairs for electrons. We accustomed ourselves to the idea that electrons are particles, elementary quanta of electricity and matter. Their charge and mass were investigated. If there is any truth in de Broglie’s idea, then there must be some phenomena in which matter reveals its wave-like character. At first, this conclusion, reached by following the acoustical analogy, seems strange and incomprehensible. How can a moving corpuscle have anything to do with a wave? But this is not the first time we have faced a difficulty of this kind in physics. We met the same problem in the domain of light phenomena.

Fundamental ideas play the most essential role in forming a physical theory. Books on physics are full of complicated mathematical formulae. But thought and ideas, not formulae, are the beginning of every physical theory. The ideas must later take the mathematical form of a quantitative theory, to make possible the comparison with experiment. This can be explained by the example of the problem with which we are now dealing. The principal guess is that the uniformly moving electron will behave, in some phenomena, like a wave. Assume that an electron or a shower of electrons, provided they all have the same velocity, is moving uniformly. The mass, charge, and velocity of each individual electron are known. If we wish to associate in some way a wave concept with a uniformly moving electron or electrons, our next question must be: what is the wave-length? This is a quantitative question and a more or less quantitative theory must be built up to answer it. This is indeed a simple matter. The mathematical simplicity of de Broglie’s work, providing an answer to this question, is most astonishing. At the time his work was done, the mathematical technique of other physical theories was very subtle and complicated, comparatively speaking. The mathematics dealing with the problem of waves of matter is extremely simple and elementary but the fundamental ideas are deep and far-reaching.

Previously, in the case of light waves and photons, it was shown that every statement formulated in the wave language can be translated into the language of photons or light corpuscles. The same is true for electronic waves. For uniformly moving electrons, the corpuscular language is already known. But every statement expressed in the corpuscular language can be translated into the wave language, just as in the case of photons. Two clues laid down the rules of translation. The analogy between light waves and electronic waves or photons and electrons is one clue. We try to use the same method of translation for matter as for light. The special relativity theory furnished the other clue. The laws of nature must be invariant with respect to the Lorentz and not to the classical transformation. These two clues together determine the wave-length corresponding to a moving electron. It follows from the theory that an electron moving with a velocity of, say, 10,000 miles per second, has a wave-length which can be easily calculated, and which turns out to lie in the same region as the X-ray wave-lengths. Thus we conclude further that if the wave character of matter can be detected, it should be done experimentally in an analogous way to that of X-rays.

Imagine an electron beam moving uniformly with a given velocity, or, to use the wave terminology, a homogeneous electronic wave, and assume that it falls on a very thin crystal, playing the part of a diffraction grating. The distances between the diffracting obstacles in the crystal are so small that diffraction for X-rays can be produced. One might expect a similar effect for electronic waves with the same order of wave-length. A photographic plate would register this diffraction of electronic waves passing through the thin layer of crystal. Indeed, the experiment produces what is undoubtedly one of the great achievements of the theory: the phenomenon of diffraction for electronic waves. The similarity between the diffraction of an electronic wave and that of an X-ray is particularly marked as seen from a comparison of the patterns in Plate III. We know that such pictures enable us to determine the wave-lengths of X-rays. The same holds good for electronic waves. The diffraction pattern gives the length of a wave of matter and the perfect quantitative agreement between theory and experiment confirms the chain of our argument splendidly.

Our previous difficulties are broadened and deepened by this result. This can be made clear by an example similar to the one given for a light wave. An electron shot at a very small hole will bend like a light wave. Light and dark rings appear on the photographic plate. There may be some hope of explaining this phenomenon by the interaction between the electron and the rim, though such an explanation does not seem to be very promising. But what about the two pinholes? Stripes appear instead of rings. How is it possible that the presence of the other hole completely changes the effect? The electron is indivisible and can, it would seem, pass through only one of the two holes. How could an electron passing through a hole possibly know that another hole has been made some distance away?

We asked before: what is light? Is it a shower of corpuscles or a wave? We now ask: what is matter, what is an electron? Is it a particle or a wave? The electron behaves like a particle when moving in an external electric or magnetic field. It behaves like a wave when diffracted by a crystal. With the elementary quanta of matter we came across the same difficulty that we met with in the light quanta. One of the most fundamental questions raised by recent advance in science is how to reconcile the two contradictory views of matter and wave. It is one of those fundamental difficulties which, once formulated, must lead, in the long run, to scientific progress. Physics has tried to solve this problem. The future must decide whether the solution suggested by modern physics is enduring or temporary.



As described in Chapter II-8, light has both wave and particle like properties. It is only when very small obstacles and apertures are used that light reveals its wave-like nature. Later, Einstein showed more directly the particle-like nature of light through the photoelectric phenomenon.

The radiation emitted by atoms has discrete and orderly wavelengths, much like those of a standing wave, where it depends on the number of nodes. Thus, atom as a system of particles behaves like a small acoustical instrument in which standing waves are produced. This observation provided de Broglie with the new and courageous idea that electrons moving uniformly may also represent waves of definite wavelength.

If there is any truth in de Broglie’s idea, then there must be some phenomena, in which electron reveals its wave-like character. It was calculated mathematically that electron may have a wavelength in the same region as the X-ray wave-lengths. This was experimentally confirmed when an electronic beam was sent through a very thin crystal, and the resulting diffraction was registered on a photographic plate.

The only way we can reconcile the two contradictory views of matter and wave is that a moving particle is not like a “moving ball” but more like a vibrating string.


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