In his general theory of relativity, however, Einstein had had to resort to the use of a branch of advanced mathematics called “tensor analysis” in order to give an adequate description of physical phenomena in four dimensional non-Euclidean space. With the complication in the calculations that this entailed, Einstein began to find the need for an assistant who was well trained in mathematics. For this purpose Einstein preferred young people who had a scientific education and ambition, but who because of external circumstances were unable to get a job at a public institution. Thus one of his first assistants in Berlin was a Russian Jew who suffered from a pathological enlargement of his bones (leontiasis) and as a result made such a repulsive impression on people that no one wanted to engage him as an assistant, let alone as a teacher. In time the young man understandably wanted to advance to an independent position. He expected Einstein to get him a position as teacher in a school although it was obvious that with his unfortunate appearance no school would hire him. Nevertheless he blamed Einstein for not trying hard enough and finally quarreled with him.
It was not easy for Einstein to find a suitable assistant. This may appear strange, but there were reasons for it. Students who wanted to study physics could wish for no better opportunity than to watch and help a man like Einstein at his creative work, and to this was added the pleasure of being in contact with a man with a very interesting personality, who was extremely friendly and adept in the art of conversation. But in large measure Einstein’s trouble was due to the fact that he did not carry on any ordinary teaching in Berlin. The students at the university who were working toward the doctorate or to pass examinations as physics teachers were busy enough trying to satisfy all the demands made on them. They studied with the professors at the university who gave the examinations, and received from them the subjects for their doctoral dissertations. Only rarely did one of them come into personal contact with Einstein. As a result Einstein usually had as assistants students from outside Germany. These foreigners did not come to Berlin to pass examinations or to find positions, but to learn from the outstanding scientists there. They immediately turned to men like Planck, Nernst, or Einstein. In this way Einstein had as collaborators first the aforementioned Russian and later the Hungarian Cornelius Lanczos and the Austrian Walter Mayer. The last two were of great help to Einstein, and published valuable contributions to the general theory of relativity. They are now both teaching in American institutions.
2. Structure of the Atom
The world believed that Einstein’s theory of relativity was the oddest and the most radical change in physics that had occurred for a long time. Actually new conceptions of matter even more baffling and far-reaching in their effects were being developed simultaneously.
In 1905, while still at Bern, Einstein had made outstanding contributions to the structure of light, as described in Section 10 of Chapter III. Since then he had turned his attention to his theories of relativity and gravitation, which dealt mainly with large objects such as stars and planets and not with the ultimate particle of nature — the atom. He had considered properties of light rays in gravitational fields, but in these cases it had made no difference whether light was simply a wave phenomenon or consisted of a stream of photons.
Einstein himself had realized in 1905 when he proposed the idea of light quanta (photons) that it was only a provisional hypothesis. Numerous difficulties had remained unsolved. For instance, the theory of the photon had had amazing success in explaining properties of heat radiation and the photoelectric effect, but it could not explain the whole set of phenomena dealing with the interference and diffraction of light. On the other hand, the wave theory, which could cope with these latter properties, was useless for those phenomena for which the photon theory was successful.
In conversation Einstein expressed this dual character of light as follows: “Somewhere in the continuous light waves there are certain ‘peas,’ the light quanta.” The amplitude of the waves determines how many “peas” are present at any spot, but only as a statistical average. One can never know whether such a “pea” will be present at a particular point at a specific instant of time. From the beginning Einstein thought that this could not be the ultimate truth. “I shall never believe,” he once said, “that God plays dice with the world.” Nevertheless, “God’s dice” penetrated into physics at several points. For instance, in the disintegration of radioactive substances a certain percentage of the atoms present disintegrate every second, but there is no way by which we can tell which particular atom will disintegrate in the next second.
But Einstein’s early suggestion of “photons (light quanta) in every light ray” had fallen on fertile soil. The “heuristic point of view” turned out to stimulate actually new discoveries. In 1913 the Danish physicist Niels Bohr attempted to correlate the structure of atoms with the light emitted by them. Rutherford in England had shown in 1911 that the atom consists of a central nucleus with positive charge and a number of negatively charged electrons around it. Also it had been known for a long time that free atoms, unlike glowing solid bodies that emit light with continuous distribution of different colors, emit light of only certain definite frequencies which are characteristic of the atom. In trying to explain this unique character of light emitted by free atoms Bohr found that it was completely impossible if he assumed that the electrons circulate around the nucleus according to Newton’s laws of motion in the same way that the planets revolve around the sun. He was thus led to setting up a separate hypothesis, with which he modified Newton’s laws in much the same way that Planck had done in explaining properties of heat radiation. Bohr assumed that only certain discrete sets of circular orbits (preferred orbits) were allowed for the electrons moving around the nucleus. Electrons in different orbits had different energies, and when an electron jumped from one of higher to one of lower energy, the difference in energy was emitted in the form of a light quantum (photon). This concept of emission of photons may be considered as a sort of inversion of Einstein’s photoelectric law, in which a photon is absorbed and an electron liberated. But here again, as in the case of radioactive atoms, only the average behavior of the atoms, and nothing of individual cases, could be predicted. At first this deficiency did not cause much concern. It was thought that the behavior of the atoms was not unlike the mortality statistics of life-insurance companies, from which the average life expectancy of man can be predicted accurately, but not that of individuals. Nevertheless every single death has its cause. The physicists believed then that similarily causes exist for the behavior of the individual atoms, but that they are as yet unknown.
3. Mechanics of the Atom
The feeling that God did not play dice for the fate of the world began to be shaken about the time Einstein settled down in Berlin after his trips. In 1924 Prince Louis de Broglie, a graduate student in Paris, submitted a doctoral thesis to Professor Langevin in which he proposed even greater changes in Newtonian mechanics than Einstein had done in his theory of relativity. Langevin, who was well known as a radical in politics, was staggered by the boldness of the new proposals. De Broglie’s work seemed fairly absurd to him, but considering that the idea of Bohr’s “preferred” orbits was also very baffling, he thought there might be something in his student’s thesis.
De Broglie had noted that Einstein’s “heuristic point of view” in optics had been helpful by attributing to light properties that are usually ascribed to material particles; namely, energy and momentum of photons. De Broglie took his cue from Einstein and introduced an analogous “heuristic viewpoint” into mechanics. To resolve the difficulties in the description of the motion of subatomic particles (particles within the atom) de Broglie suggested that certain wave properties be attributed to particles. He assumed that, just as the motion of photons in a light ray is determined by the electromagnetic field that constitutes the light wave, so the motion of particles is guided or “steered” by a new type of waves, called “matter waves” by de Broglie and “de Brogl
ie waves” by other physicists. According to this view, the “preferred” orbits of Bohr are orbits along which the de Broglie waves are built up by interference, while along all other orbits the waves are annihilated by interference. This phenomenon is exactly analogous to the interference patterns of light passing a small hole where there are light and dark regions depending on whether light falling on the regions from different directions builds up or interferes. De Broglie waves, however, have wave lengths inversely proportional to the momentum of the particles, and manifest themselves only in the case of very small masses, in particular in the case of subatomic particles. For any ordinary body like a billiard ball, the wave length is so small that it has no observable wave property.
Two years later Erwin Schrödinger, an Austrian, developed on the basis of de Broglie’s idea a new mechanics of the atom according to which the motion of atomic particles could be calculated for any field of force. In Bohr’s theory of the atom Newtonian laws and arbitrary assumptions (preferred orbits) are mixed to give satisfactory results. Schrödinger, however, obtained the same results by means of a coherent theory.
Originally de Broglie and Schrödinger had assumed that the connection between the particles and the “steering” waves by which the motion of these particles was directed was a strictly “causal” connection. But in 1926 the German physicists Max Born and P. Jordan interpreted the intensity of de Broglie waves as the average number of particles situated in a unit volume of space. The relation between the intensity of matter waves and the number of particles is thus exactly the same as that between the intensity of light and the number of Einstein’s photons.
Five winners of the Nobel Prize in physics. This picture, taken in Berlin-Zehlendorf, shows Walter Nernst, Einstein, Planck, Millikan, and Von Laue (Illustration Credit 9.1)
A recent portrait of Einstein (Illustration Credit 9.2)
This theory, developed by de Broglie, Schrödinger, and Born, by means of which not the position itself but only the average position of atomic particles could be calculated, began to be known as wave mechanics, so named appropriately since it laid stress on the wave property of material particles. By this theory future observable events cannot be predicted precisely, but only statistically. For example, we cannot predict the exact point where a particle or photon will hit a screen, but only what percentage of incoming protons or particles will hit within any given region of the screen. If science could not advance beyond this stage, “God would,” as Einstein said, “play dice indeed.”
The idea that there are waves associated with material particles received a striking experimental verification. In 1927 two Americans, Clinton J. Davisson and L. H. Germer, proved that a beam of electrons is diffracted by a metal crystal in exactly the same way that light is diffracted by a grating, and X-rays by crystals. This confirmation is all the more amazing since diffraction is a phenomenon that is purely characteristic of waves, and nobody had ever even dreamed that it could be caused by material particles such as the electron until de Broglie suggested and Davisson and Germer actually observed it. Moreover, the wave length associated with the electrons, which could be calculated from the size of the diffraction pattern, agreed exactly with the value predicted by de Broglie.
At about the same time, W. Heisenberg, a young German, approached the interaction between subatomic particles and radiation from another direction. He broke away completely from the fundamental notion in Newtonian mechanics that a particle changes its location continuously and can thus be pursued.
Einstein in his general theory of relativity started from “Mach’s requirement” that a physical theory should lead eventually to relations between quantities that can actually be measured. Accordingly, “absolute motion” was replaced by “motion relative to material bodies.” Heisenberg started similarly. He abandoned the computation of the exact motions of electrons in an atom. For the laws of nature are such that it is impossible to determine the path of electrons by any measurement. The only properties of an atom that are accessible to actual measurement are the intensity and frequency of the emitted radiation. Therefore Heisenberg suggested to formulate the basic laws governing subatomic phenomena in terms of intensity and frequency of radiation. This suggestion implies a radical break with mechanistic physics, which uses “position and velocity of particles” as the basic concepts occurring in the fundamental laws of nature.
If we accept Heisenberg’s suggestion, subatomic particles (like electrons or photons) are no longer “full-fledged particles” in the Newtonian sense, as their behavior cannot be described in the Newtonian way. But they are physical objects possessing some of the properties of particles.
This aproach to the theory of atoms has come to be known as quantum mechanics. It received a logically more satisfying form when Heisenberg went to Copenhagen and collaborated with Niels Bohr.
4. Bohr’s Complementarity Principle
According to Bohr, it was not advisable to throw overboard completely the motion of particles as the basis of a description of subatomic phenomena. The presentation in terms of the observable intensity and frequency of radiation, as originally suggested by Heisenberg, should be replaced by a restricted or qualified use of the “moving particle” as the principal means of description. Heisenberg had certainly proved his point that the motion of atomic particles cannot be described in the Newtonian sense. According to Newton, once the forces that act on a particle and its initial position and momentum are given, its subsequent position and momentum at any instant can be calculated with any desired precision. Heisenberg discovered that this is not true of subatomic particles. There are no laws which connect the position and the momentum of such a particle in one instant of time with the values of these quantities at a future instant. The laws have in this domain a different character. If the initial position and momentum of a particle of very small mass (a subatomic particle) are known within a certain margin, the position at a future instant can be computed within a certain margin. By making the initial margin sufficiently narrow, however, we cannot achieve, as in Newton’s mechanics, a final margin as narrow as we desire. In other words, if we want to hit a definite point of a target, we cannot be sure to achieve the desired result even if we aim very accurately. If we want to hit our target point at least within a reasonable margin, we have to consider that according to Heisenberg there is a definite relation between the initial margins of position and momentum: the product of these two margins has to equal a definite quantity, which is, roughly speaking, Planck’s constant h. This relation has become famous under the name of “Heisenberg’s relation of indeterminacy.”
Soon afterward Bohr gave a more satisfactory interpretation of this strange behavior of atomic particles. He pointed out that “position” and “momentum” are two different aspects of a small mass (e.g., an electron) in much the same way that the particle properties and wave properties are two aspects of the photon. To say that a particle is located in a certain limited region of space is exactly analogous to the statement that light-energy is concentrated in a photon, and to define the momentum of a particle is analogous to the emphasis on the wave aspect of light. Both material particles and light have the dual characteristics of particles and waves, but their behavior is neither contradictory nor haphazard. Bohr emphasized again “Mach’s requirement” that we should make only such statements as can be tested by definite physical experiments. According to him, it depends solely on the specific arrangement of apparatus used whether the emission of light and of electrons has to be described as a wave or as a beam of moving particles. According to this view, the two types of properties exhibited are “complementary” features of the same physical object. What we observe depends on what observable reaction of our subatomic phenomena we bring to a test. This conception has been called Bohr’s theory of complementarity.
Bohr’s point of view is therefore even more different from Newtonian mechanics than Einstein’s theory of relativity. In Bohr’s conception, we cannot describe what
“actually” occurs in space while, say, light is emitted by the sun before it hits the earth. We can describe only what we observe when a measuring apparatus is hit by light. We can, for example, describe whether or not the light from the sun hits a certain spot on a screen. Or, to express it more precisely: We cannot describe “physical reality” by describing the path that a particle traverses in space, but we can and must describe only the observations made on various physical instruments arranged at different points in space and time. Physical laws link together these observations, but not the positions or paths of the particles or photons. This viewpoint was interpreted as being in agreement with positivistic philosophy which asserts that science cannot discover what actually happens in the world, but can only describe and combine the results of different observations.
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