The idea of energy quanta was first introduced by Planck at the beginning of this century in order to explain some effects much more complicated than the photoelectric effect. But the photo-effect shows most clearly and simply the necessity for changing our old concepts.
It is at once evident that this quantum theory of light explains the photoelectric effect. A shower of photons is falling on a metal plate. The action between radiation and matter consists here of very many single processes in which a photon impinges on the atom and tears out an electron. These single processes are all alike and the extracted electron will have the same energy in every case. We also understand that increasing the intensity of the light means, in our new language, increasing the number of falling photons. In this case, a different number of electrons would be thrown out of the metal plate, but the energy of any single one would not change. Thus we see that this theory is in perfect agreement with observation.
What will happen if a beam of homogeneous light of a different color, say, red instead of violet, falls on the metal surface? Let us leave experiment to answer this question. The energy of the extracted electrons must be measured and compared with the energy of electrons thrown out by violet light. The energy of the electron extracted by red light turns out to be smaller than the energy of the electron extracted by violet light. This means that the energy of the light quanta is different for different colors. The photons belonging to the color red have half the energy of those belonging to the color violet. Or, more rigorously: the energy of a light quantum belonging to a homogeneous color decreases proportionally as the wave-length increases. There is an essential difference between quanta of energy and quanta of electricity. Light quanta differ for every wave-length, whereas quanta of electricity are always the same. If we were to use one of our previous analogies, we should compare light quanta to the smallest monetary quanta, differing in each country.
Let us continue to discard the wave theory of light and assume that the structure of light is granular and is formed by light quanta, that is, photons speeding through space with the velocity of light. Thus, in our new picture, light is a shower of photons, and the photon is the elementary quantum of light energy. If, however, the wave theory is discarded, the concept of a wave-length disappears. What new concept takes its place? The energy of the light quanta! Statements expressed in the terminology of the wave theory can be translated into statements of the quantum theory of radiation. For example:
TERMINOLOGY OF THE WAVE THEORY TERMINOLOGY OF THE QUANTUM THEORY
Homogeneous light has a definite wave-length. The wave-length of the red end of the spectrum is twice that of the violet end. Homogeneous light contains photons of a definite energy. The energy of the photon for the red end of the spectrum is half that of the violet end.
The state of affairs can be summarized in the following way: there are phenomena which can be explained by the quantum theory but not by the wave theory. Photo-effect furnishes an example, though other phenomena of this kind are known. There are phenomena which can be explained by the wave theory but not by the quantum theory. The bending of light around obstacles is a typical example. Finally, there are phenomena, such as the rectilinear propagation of light, which can be equally well explained by the quantum and the wave theory of light.
But what is light really? Is it a wave or a shower of photons? Once before we put a similar question when we asked: is light a wave or a shower of light corpuscles? At that time there was every reason for discarding the corpuscular theory of light and accepting the wave theory, which covered all phenomena. Now, however, the problem is much more complicated. There seems no likelihood of forming a consistent description of the phenomena of light by a choice of only one of the two possible languages. It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do!
How is it possible to combine these two pictures? How can we understand these two utterly different aspects of light? It is not easy to account for this new difficulty. Again we are faced with a fundamental problem.
For the moment let us accept the photon theory of light and try, by its help, to understand the facts so far explained by the wave theory. In this way we shall stress the difficulties which make the two theories appear, at first sight, irreconcilable.
We remember: a beam of homogeneous light passing through a pinhole gives light and dark rings. How is it possible to understand this phenomena by the help of the quantum theory of light, disregarding the wave theory? A photon passes through the hole. We could expect the screen to appear light if the photon passes through and dark if it does not. Instead, we find fight and dark rings. We could try to account for it as follows: perhaps there is some interaction between the rim of the hole and the photon which is responsible for the appearance of the diffraction rings. This sentence can, of course, hardly be regarded as an explanation. At best, it outlines a program for an explanation holding out at least some hope of a future understanding of diffraction by interaction between matter and photons.
But even this feeble hope is dashed by our previous discussion of another experimental arrangement. Let us take two pinholes. Homogeneous light passing through the two holes gives light and dark stripes on the screen. How is this effect to be understood from the point of view of the quantum theory of light? We could argue: a photon passes through either one of the two pinholes. If a photon of homogeneous light represents an elementary light particle, we can hardly imagine its division and its passage through the two holes. But then the effect should be exactly as in the first case, light and dark rings and not light and dark stripes. How is it possible then that the presence of another pinhole completely changes the effect? Apparently the hole through which the photon does not pass, even though it may be at a fair distance, changes the rings into stripes! If the photon behaves like a corpuscle in classical physics it must pass through one of the two holes. But in this case, the phenomena of diffraction seem quite incomprehensible.
Science forces us to create new ideas, new theories. Their aim is to break down the wall of contradictions which frequently blocks the way of scientific progress. All the essential ideas in science were born in a dramatic conflict between reality and our attempts at understanding. Here again is a problem for the solution of which new principles are needed. Before we try to account for the attempts of modern physics to explain the contradiction between the quantum and the wave aspects of light, we shall show that exactly the same difficulty appears when dealing with quanta of matter instead of quanta of light.
LIGHT SPECTRA
We already know that all matter is built of only a few kinds of particles. Electrons were the first elementary particles of matter to be discovered. But electrons are also the elementary quanta of negative electricity. We learned furthermore that some phenomena force us to assume that light is composed of elementary light quanta, differing for different wave-lengths. Before proceeding we must discuss some physical phenomena in which matter as well as radiation plays an essential role.
The sun emits radiation which can be split into its components by a prism. The continuous spectrum of the sun can thus be obtained. Every wave-length between the two ends of the visible spectrum is represented. Let us take another example. It was previously mentioned that sodium when incandescent emits homogeneous light, light of one color or one wave-length. If incandescent sodium is placed before the prism we see only one yellow line. In general, if a radiating body is placed before the prism, then the light it emits is split up into its components, revealing the spectrum characteristic of the emitting body.
The discharge of electricity in a tube containing gas produces a source of light such as seen in the neon tubes used for luminous advertisements. Suppose such a tube is placed before a spectroscope. The spectroscope is an instrument which acts like
a prism, but with much greater accuracy and sensitiveness; it splits light into its components, that is, it analyzes it. Light from the sun, seen through a spectroscope, gives a continuous spectrum; all wave-lengths are represented in it. If, however, the source of light is a gas through which a current of electricity passes, the spectrum is of a different character. Instead of the continuous, multi-colored design of the sun’s spectrum, bright, separated stripes appear on a continuous dark background. Every stripe, if it is very narrow, corresponds to a definite color or, in the language of the wave theory, to a definite wave-length. For example, if twenty lines are visible in the spectrum, each of them will be designated by one of twenty numbers expressing the corresponding wave-length. The vapors of the various elements possess different systems of lines, and thus different combinations of numbers designating the wave-lengths composing the emitted light spectrum. No two elements have identical systems of stripes in their characteristic spectra, just as no two persons have exactly identical fingerprints. As a catalogue of these lines was worked out by physicists, the existence of laws gradually became evident, and it was possible to replace some of the columns of seemingly disconnected numbers expressing the length of the various waves by one simple mathematical formula.
All that has just been said can now be translated into the photon language. The stripes correspond to certain definite wave-lengths or, in other words, to photons with a definite energy. Luminous gases do not, therefore, emit photons with all possible energies, but only those characteristic of the substance. Reality again limits the wealth of possibilities.
Atoms of a particular element, say, hydrogen, can emit only photons with definite energies. Only the emission of definite energy quanta is permissible, all others being prohibited. Imagine, for the sake of simplicity, that some element emits only one line, that is, photons of a quite definite energy. The atom is richer in energy before the emission and poorer afterwards. From the energy principle it must follow that the energy level of an atom is higher before emission and lower afterwards, and that the difference between the two levels must be equal to the energy of the emitted photon. Thus the fact that an atom of a certain element emits radiation of one wave-length only, that is photons of a definite energy only, could be expressed differently: only two energy levels are permissible in an atom of this element and the emission of a photon corresponds to the transition of the atom from the higher to the lower energy level.
But more than one line appears in the spectra of the elements, as a rule. The photons emitted correspond to many energies and not to one only. Or, in other words, we must assume that many energy levels are allowed in an atom and that the emission of a photon corresponds to the transition of the atom from a higher energy level to a lower one. But it is essential that not every energy level should be permitted, since not every wave-length, not every photon-energy, appears in the spectra of an element. Instead of saying that some definite lines, some definite wavelengths, belong to the spectrum of every atom, we can say that every atom has some definite energy levels, and that the emission of light quanta is associated with the transition of the atom from one energy level to another. The energy levels are, as a rule, not continuous but discontinuous. Again we see that the possibilities are restricted by reality.
It was Bohr who showed for the first time why just these and no other lines appear in the spectra. His theory, formulated twenty-five years ago, draws a picture of the atom from which, at any rate in simple cases, the spectra of the elements can be calculated and the apparently dull and unrelated numbers are suddenly made coherent in the light of the theory.
Bohr’s theory forms an intermediate step toward a deeper and more general theory, called the wave or quantum mechanics. It is our aim in these last pages to characterize the principal ideas of this theory. Before doing so, we must mention one more theoretical and experimental result of a more special character.
Our visible spectrum begins with a certain wave-length for the violet color and ends with a certain wave-length for the red color. Or, in other words, the energies of the photons in the visible spectrum are always enclosed within the limits formed by the photon energies of the violet and red lights. This limitation is, of course, only a property of the human eye. If the difference in energy of some of the energy levels is sufficiently great, then an ultraviolet photon will be sent out, giving a line beyond the visible spectrum. Its presence cannot be detected by the naked eye; a photographic plate must be used.
X rays are also composed of photons of a much greater energy than those of visible light, or in other words, their wave-lengths are much smaller, thousands of times smaller in fact, than those of visible light.
But is it possible to determine such small wavelengths experimentally? It was difficult enough to do so for ordinary light. We had to have small obstacles or small apertures. Two pinholes very near to each other, showing diffraction for ordinary light, would have to be many thousands of times smaller and closer together to show diffraction for X rays.
How then can we measure the wave-lengths of these rays? Nature herself comes to our aid.
FIG. 12.
A crystal is a conglomeration of atoms arranged at very short distances from each other on a perfectly regular plan. Our drawing shows a simple model of the structure of a crystal. Instead of minute apertures, there are extremely small obstacles formed by the atoms of the element, arranged very close to each other in absolutely regular order. The distances between the atoms, as found from the theory of the crystal structure, are so small that they might be expected to show the effect of diffraction for X rays. Experiment proved that it is, in fact, possible to diffract the X-ray wave by means of these closely packed obstacles disposed in the regular three-dimensional arrangement occurring in a crystal.
Suppose that a beam of X rays falls upon a crystal and, after passing through it, is recorded on a photographic plate. The plate then shows the diffraction pattern. Various methods have been used to study the X-ray spectra, to deduce data concerning the wave-length from the diffraction pattern. What has been said here in a few words would fill volumes if all theoretical and experimental details were set forth. In Plate III (page 317) we give only one diffraction pattern obtained by one of the various methods. We again see the dark and light rings so characteristic of the wave theory. In the center the non-diffracted ray is visible. If the crystal were not brought between the X rays and the photographic plate, only the light spot in the center would be seen. From photographs of this kind the wave-lengths of the X-ray spectra can be calculated and, on the other hand, if the wave-length is known, conclusions can be drawn about the structure of the crystal.
THE WAVES OF MATTER
How can we understand the fact that only certain characteristic wavelengths 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 Schrödinger, less than fifteen years ago, is a typical example of the achievement of a successful theory by means of a deep and fortunate analogy.
FIG. 13.
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 th
e 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.
FIG. 14.
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 created 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.
FIG. 15.
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 (figure 15). This motion is the result of one wave lying on the other when the two are traveling 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 center. In this case three points are always at rest. A glance at figures 16 and 17 shows that here the wave-length is half as
A Stubbornly Persistent Illusion Page 29