by Gary Zukav
Where the crest of one wave meets the trough of another wave, they cancel each other and the surface of the water along this line of interaction is calm. These calm areas are the nodes which separate the standing waves. In the double-slit experiment, the nodes are the dark bands in the pattern of alternating light and dark areas. The light bands are the crests of the standing waves.
Schrödinger chose the model of a small tub of water with its complex and intricate interference pattern to explain the nature of the atom. This model is, as he put it, an “analogue” of electron waves in an atom-sized basin.
The ingenious but nevertheless somewhat artificial assumptions of [Bohr’s model of the atom]…are replaced by a much more natural assumption in de Broglie’s wave phenomena. The wave phenomenon forms the real “body” of the atom. It replaces the individual punctiform [pointlike] electrons, which in Bohr’s model swarm around the nucleus.4
Standing waves on clotheslines have two dimensions: length and width. Standing waves in mediums like water, or on the head of a conga drum, have three dimensions: length, width, and depth. Schrödinger analyzed the standing wave patterns of the simplest atom, hydrogen, which has only one electron. In hydrogen alone he calculated, using his new wave equation, a multitude of different possible shapes of standing waves. All of the standing waves on a rope are identical. This is not true of the standing waves in an atom. All of them are three-dimensional and all of them are different. Some of them look like concentric circles. Some of them look like butterflies, and others look like mandalas, as in the illustration on the next page.
Shortly before Schrödinger’s discovery, another Austrian physicist, Wolfgang Pauli, discovered that no two electrons in an atom can be exactly alike. The presence of an electron with one particular set of properties (“quantum numbers”) excludes the presence of another electron with exactly the same properties (quantum numbers) within the same atom. For this reason, Pauli’s discovery became known as the Pauli exclusion principle. In terms of Schrödinger’s standing wave theory, Pauli’s exclusion principle means that once a particular wave pattern forms in an atom, it excludes all others of its kind.
Schrödinger’s equation, modified by Pauli’s discovery, shows that there are only two possible wave patterns in the lowest of Bohr’s energy levels, or shells. Therefore, there can be only two electrons in it. There are eight different standing-wave patterns possible in the next energy level, therefore there can be only eight electrons in it, and so on.*, †
From Modern College Physics, Harvey White, N.Y., Van Nostrand, 1972.
These are exactly the numbers of electrons that Bohr’s model assigns to these energy levels. In this respect, the two models are alike. In another important way, however, they are different.
Bohr’s theory was entirely empirical. That is, he built it around the experimentally observed facts to explain them. In contrast, Schrödinger built his theory on de Broglie’s matter-wave hypothesis. Not only does it yield mathematical values which have been verified experimentally, but it also provides a consistent explanation for them.
For example, there are only a certain number of electrons in each energy level, because there are only a certain number of standing wave patterns possible at each energy level. The energy level of an atom jumps only from certain specific values to other certain specific values, because standing-wave patterns of only certain dimensions can form with the atom, and none other.
Although Shrödinger was sure that electrons were standing waves, he was not sure what was waving.* He was convinced, nonetheless, that something was waving, and he called it psi, a Greek letter pronounced “sigh.” (A “wave function” and a “psi function” are the same thing.)
To use the Schrödinger wave equation, we feed it certain characteristics of the atom in question. It then gives us the evolution in time of standing-wave patterns which occur in the atom. If we prepare an atom in an initial state and let it propagate in isolation, that initial state, while propagating in isolation, evolves in time into different standing-wave patterns. The order of these patterns is calculable. The Schrödinger wave equation is the mathematical device which physicists use to calculate the order of these patterns. Said another way, the development of standing wave patterns in an atom is deterministic. Given initial conditions, one pattern always follows another in accordance with the Schrödinger wave equation.†
The Schrödinger wave equation also provides a self-consistent explanation of the size of the hydrogen atom. According to it, the wave pattern of a system with one electron and one proton, which is what we call a hydrogen atom, in its lowest energy state, has an appreciable magnitude only within a sphere which is just the diameter of the smallest Bohr orbit. In other words, such a wave pattern turns out to be the same size as the ground state of a hydrogen atom!
Although Schrödinger’s wave mechanics became a pillar of today’s quantum mechanics, the useful aspects of Bohr’s model of subatomic phenomena still are used when the wave theory does not yield appropriate results. In such cases, physicists simply stop thinking in terms of standing waves and start thinking again in terms of particles. No one can say that they are not adaptable in this matter (wave).
Schrödinger was convinced that his equations described real things, and not mathematical abstractions. He pictured electrons as actually being spread out over their wave patterns in the form of a tenuous cloud. If the picture is limited to the one-electron hydrogen atom, whose standing waves have only three dimensions (length, width, and depth), this is possible to imagine. However, the standing waves in an atom with two electrons exist in six mathematical dimensions; the standing waves in an atom with four electrons exist in twelve dimensions, etc. To visualize this is quite an exercise.
At this point Max Born, a German physicist, put the final touch to the new wave interpretation of subatomic phenomena. According to him, it is not necessary or possible to visualize these waves because they are not real things, they are probability waves.
…the whole course of events is determined by the laws of probability; to a state in space there corresponds a definite probability, which is given by the de Broglie wave associated with the state.5
To obtain the probability of a given state we square (multiply by itself) the amplitude of the matter wave associated with the state.
The question of whether de Broglie’s equations and Schrödinger’s equations represent real things or abstractions was clear to Born. It did not make sense to him to try to think of a real thing that exists in more than three dimensions.
We have two possibilities. Either we use waves in spaces of more than three dimensions…or we remain in three-dimensional space, but give up the simple picture of the wave amplitude as an ordinary physical magnitude, and replace it by a purely abstract mathematical concept…into which we cannot enter.6
This is exactly what he did. “Physics,” he wrote,
is in the nature of the case indeterminate, and therefore the affair of statistics.7
This is the same idea (probability waves) that Bohr, Kramers, and Slater had thought of earlier. This time, however, using the mathematics of de Broglie and Schrödinger, the numbers came out right.
Born’s contribution to Schrödinger’s theory is what enables quantum mechanics to predict probabilities. Since the probability of a state is found by squaring the amplitude of the matter wave associated with it, and, given initial conditions, the Schrödinger equation predicts the evolution of these wave patterns, the two taken together give a determinable evolution of probabilities. Given any initial state, physicists can predict the probability that an observed system will be observed to be in any other given state at any particular time. Whether or not the observed system is observed to be in that state, however, even if that state is the most probable state for that time, is a matter of chance. In other words, the “probability” of quantum mechanics is the probability of observing an observed system in a given state at a given time if it was prepared in a given initial st
ate.*
Thus it was that the wave aspect of quantum mechanics developed. Just as waves have particle-like characteristics (Planck, Einstein), particles also have wave-like characteristics (de Broglie). In fact, particles can be understood in terms of standing waves (Schrödinger). Given initial conditions, a precise evolution of standing-wave patterns can be calculated via the Schrödinger wave equation. Squaring the amplitude of a matter wave (wave function) gives the probability of the state that corresponds to that wave (Born). Therefore, a sequence of probabilities can be calculated from initial conditions by using the Schrödinger wave equation and Born’s simple formula.
We have come a long way from Galileo’s experiments with falling bodies. Each step along the path has taken us to a higher level of abstraction: first to the creation of things that no one has ever seen (like electrons), and then to the abandonment of all attempts even to picture our abstractions.
The problem is, however, that human nature being what it is, we do not stop trying to picture these abstractions. We keep asking “What are these abstractions of?” and then we try to visualize whatever that is.
Earlier we dismissed Bohr’s planetary model of the atom with the promise that we later would see “how physicists currently think of an atom.” Well, the time has come, but the task is a thorny one. We gave up our old picture of the atom so easily because we assumed that it would be replaced by one more meaningful, but equally as lucid. Now it develops that our replacement picture is not a picture at all, but an unvisualizable abstraction. This is uncomfortable because it reminds us that atoms were never “real” things anyway. Atoms are hypothetical entities constructed to make experimental observations intelligible. No one, not one person, has ever seen an atom. Yet we are so used to the idea that an atom is a thing that we forget that it is an idea. Now we are told that not only is an atom an idea, it is an idea that we cannot even picture.
Nonetheless, when physicists refer to mathematical entities in English (or German or Danish), the words that they use are bound to create images for laymen who hear them, but who are not familiar with the mathematics to which they refer. Therefore, given this lengthy explanation of why it cannot be done, we come now to how physicists today picture an atom.
An atom consists of a nucleus and electrons. The nucleus is located at the center of the atom. It occupies only a small part of the atom’s volume, but almost all of its mass. This is the same nucleus as in the planetary model. As in the planetary model, electrons move in the general area of the nucleus. In this model, however, the electrons may be anywhere within an “electron cloud.” The electron cloud is made of various standing waves which surround the nucleus. These standing waves are not material. They are patterns of potential. The shape of the various standing waves which comprise the electron cloud tells physicists the probability of finding the point electron at any given place in the cloud.
In short, physicists still think of an atom as a nucleus around which move electrons, but the picture is not so simple as that of a tiny solar system. The electron cloud is a mathematical concept which physicists have constructed to correlate their experiences. Electron clouds may or may not exist within an atom. No one really knows. However, we do know that the concept of an electron cloud yields the probabilities of finding the electron at various places around the nucleus of an atom, and that these probabilities have been determined empirically to be accurate.
In this sense, electron clouds are like wave functions. A wave function also is a mathematical concept which physicists have constructed to correlate their experiences. Wave functions may or may not “actually exist.” (This type of statement assumes a qualitative difference between thought and matter, which may not be a good assumption). However, the concept of a wave function undeniably yields the probabilities of observing a system to be in a given state at a given time if it was prepared in a given way.
Like wave functions, electron clouds generally cannot be visualized. An electron cloud containing only one electron (like the electron cloud of a hydrogen atom) exists in three dimensions. All other electron clouds, however, contain more than one electron and therefore exist in more than three dimensions. The nucleus of the simple carbon atom, for example, with its six electrons, is surrounded by an electron cloud with eighteen dimensions. Uranium, with ninety-two electrons, has an electron cloud of two hundred and seventy-six dimensions. (Similarly, a wave function contains three dimensions for each particle that it represents). The situation, in terms of mental pictures, is clearly unclear.
This ambiguity results from attempting to depict with limited concepts (language) situations which are not bound by the same limitations. It also masks the fact that we do not know what actually is going on in the invisible subatomic realm. The models that we use are “free creations of the human mind,” to use Einstein’s words, that satisfy our innate need to correlate experience rationally. They are guesses about what “really” goes on inside the unopenable watch. It is extremely misleading to think that they actually describe anything.
In fact, a young German physicist, Werner Heisenberg, decided that we never can know what actually goes on in the invisible subatomic realm, and that, therefore, we should “abandon all attempts to construct perceptual models of atomic processes.”8 All that we legitimately can work with, according to this theory, is what we observe directly. All we know is what we have at the beginning of an experiment, and what we have at the end of it. Any explanation of what actually happens between these two states—which are the observables—is speculation.
Therefore, about the same time (1925), but independently of de Broglie and Schrödinger, the twenty-five-year-old Heisenberg set about developing a means of organizing experimental data into tabular form. He was fortunate in that sixty-six years earlier an Irish mathematician named W. R. Hamilton had developed a method of organizing data into arrays, or mathematical tables, called matrices. At that time, Hamilton’s matrices were considered the fringe of pure mathematics. Who could have guessed that one day they would fit, like a precut piece, into the structure of a revolutionary physics?
To use Heisenberg’s tables, we simply read from them, or calculate from them, what probabilities are associated with what initial conditions. Using this method, which Heisenberg called matrix mechanics, we deal only with physical observables, which means those things that we know at the beginning of an experiment, and those things that we know about it at the end. We make no speculation about what happens in between.
After twenty-five years of struggling for a theory to replace Newtonian physics, physicists suddenly found themselves with two different theories, each one a unique way of approaching the same thing: Schrödinger’s wave mechanics, based on de Broglie’s matter waves, and Heisenberg’s matrix mechanics, based on the unanalyzability of subatomic phenomena.
Within a year after Heisenberg developed his matrix mechanics, Schrödinger discovered that it was mathematically equivalent to his own wave mechanics. Since both of these theories were valuable tools for subatomic research, both of them were incorporated into the new branch of physics which became known as quantum mechanics.
Much later, Heisenberg applied matrix mathematics to the particle collision experiments of high-energy particle physics. Because such collisions always result in a scattering of particles, it was called the Scattering Matrix, which was shortened to the S Matrix. Today, physicists have two ways to calculate the transition probabilities between what they observe at the beginning of a quantum mechanical experiment and what they observe at the end of it.
The first method is the Schrödinger wave equation, and the second method is the S Matrix. The Schrödinger wave equation describes a temporal development of possibilities, one of which suddenly actualizes when we make a measurement in the course of a quantum mechanical experiment. The S Matrix gives directly the transition probabilities between the observables without giving any indication of a development in time, or the lack of it, or anything else. Both of them work.*
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As important as was Heisenberg’s introduction of matrix mathematics into the new physics, his next discovery shook the very foundations of “the exact sciences.” He proved that, at the subatomic level, there is no such thing as “the exact sciences.”
Heisenberg’s remarkable discovery was that there are limits beyond which we cannot measure accurately, at the same time, the processes of nature. These limits are not imposed by the clumsy nature of our measuring devices or the extremely small size of the entities that we attempt to measure, but rather by the very way that nature presents itself to us. In other words, there exists an ambiguity barrier beyond which we never can pass without venturing into the realm of uncertainty. For this reason, Heisenberg’s discovery became known as the “uncertainty principle.”
The uncertainty principle reveals that as we penetrate deeper and deeper into the subatomic realm, we reach a certain point at which one part or another of our picture of nature becomes blurred, and there is no way to reclarify that part without blurring another part of the picture! It is as though we are adjusting a moving picture that is slightly out of focus. As we make the final adjustments, we are astonished to discover that when the right side of the picture clears, the left side of the picture becomes completely unfocused and nothing in it is recognizable. When we try to focus the left side of the picture, the right side starts to blur and soon the situation is reversed. If we try to strike a balance between these two extremes, both sides of the picture return to a recognizable condition, but in no way can we remove the original fuzziness from them.
The right side of the picture, in the original formulation of the uncertainty principle, corresponds to the position in space of a moving particle. The left side of the picture corresponds to its momentum. According to the uncertainty principle, we cannot measure accurately, at the same time, both the position and the momentum of a moving particle. The more precisely we determine one of these properties, the less we know about the other. If we precisely determine the position of the particle, then, strange as it sounds, there is nothing that we can know about its momentum. If we precisely determine the momentum of the particle, there is no way to determine its position.