Chapter VIII
Pictures vs. Models
I
At this point it may be well to say that condemning the nuclear theory of the atom on the grounds that it does not present a true picture of the atomic structure is, in a way, somewhat unfair to the originators of the theory, in that we are condemning their product for not doing something that it was never designed to do in the first place. Rutherford did not discover the structure of the atom; what he did was to construct a model atom which would be in accord with the results of his experiments.
In the earlier days of the nuclear theory this was well understood. A particularly good discussion of this point, which could profitably be consulted by present-day students of the subject, appears in Karl Darrow’s “Introduction to Contemporary Physics,”70 published in 1926, only fifteen years after the original formulation of the theory. Darrow points out that every atom-model is designed to fit only “a very small fraction of the available knowledge about properties of matter,” and that “…if we were to demand that an atom-model be compatible with all known phenomena not one of those now in the field (including Rutherford’s) would survive.” The use of models is a very convenient and helpful device in the development of physical theory, but it should be remembered that a model is not a picture; as Schrödinger tells us, it is “only a mental help, a tool of thought.”71 In adopting such a model as a conceptual tool we are not in any way committing ourselves to the belief that it is a true representation of the physical facts. Schrödinger himself accepted (with some reservations) the currently popular nuclear atom as a model, a useful conceptual device, but it is evident from the statement which was quoted at the end of Chapter 3, that he did not regard it as a picture. He was willing to accept the orbiting electron as a “mental help,” but he makes it clear that if he could look inside an atom he would not expect to find any electrons in orbit.
Of course, it is the hope of everyone who invents a model to represent some physical entity that sufficient factual evidence may ultimately be discovered to advance this model, either in its original form or after suitable modification, to the standing of a picture: a true representation of the facts. Occasionally this happens. The atom itself was originally nothing but a model—a model of the structure of matter—but we now feel certain that atoms actually exist, since we have accumulated a vast amount of information pointing to the existence of such units and no significant evidence to the contrary. Bridgman says, “It is one of the most fascinating things in physics to trace the accumulation of independent new physical information all pointing to the atom, until now we are as convinced of its physical reality as of our hands and feet.”72 But the number of models proposed is greatly in excess of the number of items for which models are needed, and most models must therefore expect to fall by the wayside sooner or later when the progress of experimental knowledge advances to a new stage.
It does not necessarily follow that these discarded models are entirely without merit. As pointed out in Chapter 4, the mere fact that a hypothesis is advanced at all is an indication that it contains a certain amount of truth (a model is, of course a hypothesis), and even though the remaining aspects may be totally in error, this content of truth may be sufficient to make the hypothesis very definitely useful. Bohr’s original theory of the atom, for example, was based on a completely erroneous assumption, as has been pointed out in the preceding pages, and has now been repudiated by the originator and his associates (on other grounds), but it cannot be denied that it has served a very useful purpose. It is true that the extent of the contribution made by this theory is generally overestimated. When we find the achievements of the Bohr atom in connection with the interpretation of the hydrogen spectrum hailed with such praises as “This is success beyond expectation,”73 it would be well to remember that equally competent observers look at the situation in this manner, “The fact that it gave the correct energy levels for hydrogen is an accident; it failed badly even for helium.”50 It is also evident from the facts brought out in this present work that the advance of physical knowledge has been seriously handicapped in recent years by the retention of Bohr’s basic ideas long after their usefulness had come to an end. But even so we can legitimately give this model considerable credit as an interim expedient pending the accumulation of sufficient experimental knowledge on which to build a better model.
Darrow points out that the same is true of the “billiard ball” model of the atom. It served very satisfactorily as a basis for the kinetic theory and associated developments, and these achievements will stand to its credit even though it had to be superseded when a wider range of atomic properties was discovered.
When the time comes that a model is no longer adequate to meet the demands upon it, good scientific practice requires that it should be ruthlessly cast aside, no matter how faithfully it may have served in the past. Unfortunately, this is easier said than done. A model, or an idea of any kind, which has achieved general acceptance and has had a degree of success establishes itself firmly in the structure of contemporary thought and is extremely difficult to dislodge. As Alfred Lande puts it, the revolutionary hypotheses of yesterday are “today hardened into axioms.”74
So it has been with the nuclear atom-model. It was quite clear even in 1926 when Darrow’s book was published, that this model was far from satisfactory and the possibility of utilizing some different line of approach to devise a new model was already being considered, but as Darrow says, “The desire to retain the nuclear atom-model, and the simple and beautiful explanation of alpha particle deflections which it provides, discouraged and continues to discourage any such attempt.”
This is certainly a striking example of what Louis de Broglie characterized as the “tyrannical influence of certain conceptions that finally came to be considered as dogma.”24 Here is a model, a purely hypothetical atom, that provides a “simple and beautiful” explanation of one of the many properties of the atom, and this one success is then allowed to override all other considerations. Ironically, we now find, on more careful examination of the significance of Rutherford’s findings, that we can keep this “simple and beautiful” explanation without the nuclear hypothesis—without any new hypothesis at all—but for more than fifty years the scientific world has been so obsessed with the idea that Rutherford “discovered” an atomic nucleus that any conflicting fact has simply been pushed out of the way, no matter how drastic a measure has been necessary to accomplish this purpose. Where the theory stands in direct contradiction to the observations or established physical principles, the contradictions are removed by postulating that they do not exist; where there are discrepancies in numerical values, a Principle of Uncertainty makes such discrepancies legal; where causal connections cannot be found, causality is outlawed. Now we have the weird spectacle of the experimental physicists, confident that the atom is constructed of protons, neutrons, electrons, etc., in specific numbers and specific arrangements, spending countless hours trying to determine the exact details of these arrangements, while at the same time the theorists who are responsible for dreaming up the nuclear theory in the first place tell us that this is all a mistake, that the atom has no “immediate and direct properties.” In the words of Jordan, “The atom, as we know it today… can only be characterized by a system of mathematical formulae.”75 Heisenberg expresses exactly the same thing even more forcibly in the statement quoted in Chapter 5.
Surely it is not unreasonable to expect the physicists to do better than this. After all, we are well into the last half of the twentieth century. It is over 2000 years since Democritus and his colleagues first advanced the atomic concept, and by this time if we do not have a true picture of the atom, we should at least have an atom-model that we can all use, rather than having one for the educators, another for the experimental physicists, and a third for the theorists. But we have gone astray because we have allowed the distinction between a model and a picture to become obscured. If we had a reliable picture of the general structure of the atom, definite knowledge with accurate factual confirmation, then there would be some justification for the enormous amount of time that is being spent investigating the specific details of this structure. H. A. Bethe tells us, for example, that the study of the so-called “nuclear force” has consumed “probably more man-hours than have been given to any other scientific question in the history of mankind.”76 But if we realize that the nuclear atom is only a model, a purely imaginary concept invented to explain a part of the behavior of the atom, the expenditure of this much time and effort on any specific detail is completely out of order. When and if we meet an unusually difficult obstacle of this kind in the development of the consequences of a model, the logical conclusion is that the model itself is inadequate, and we should direct our attention to the necessary reconstruction or replacement of the model, rather than wasting our time and resources on futile efforts to discover by experiment features of a model that actually have no counterparts in the physical world.
II
A model becomes a picture when it meets the requirements for proof outlined in Chapter 1; that is, when it can be shown that the model and the consequences thereof are consistent with the observed facts in a large number of random cases throughout the areas involved, without exception, and without the use of contrived methods of evading contradictions and inconsistencies. The atomic theory itself—a model of the structure of matter—has qualified under these strict requirements and, as Bridgman pointed out in the statement previously quoted, we now regard this theory as a picture of the situation that actually exists. This model has graduated into the more advanced class.
It is obvious that no atom-model yet proposed comes anywhere near meeting these requirements. Even if we were to take the modern theories at their face value, without regard to the skeletons in the closet that have been exposed in the previous chapters of this book, no such model even attempts to cover the whole field, and every one is beset with inconsistencies which cannot be overcome except through the use of unprecedented numbers of ad hoc assumptions. All of the present-day atom-models are therefore purely models; none of them has the slightest claim to the status of a true picture of the actual physical atom. As Schrödinger reminds us, the goal of a real understanding of atomic structure is still far away.
A recognition of this point is all that is needed to clear up a large part of the confusion that now exists in atomic circles. An outside observer cannot fail to marvel at a situation in which the leading theorists insist that the atom has “no immediate and direct properties,” while at the same time their colleagues in the laboratories are enjoying remarkable success in observing and measuring these properties which the theorists say do not exist. But if we realize that the theorists are dealing with models whereas the experimenters are dealing with actual physical atoms, the anomaly disappears. When Heisenberg tells us that the atom has no properties and can only be symbolized by a mathematical equation, he is not talking about an actual physical atom, the kind of an atom that is being studied in the laboratories. He may think that he is, but he is not speaking on the authority of experimental facts—the observed facts flatly contradict him—and he has no picture of the atom that he can use as a basis for conclusions that will be applicable to the physical atom.
In effect, he concedes this point when he says, “…the new mathematical formulae no longer describe nature itself but our knowledge of nature.”77 If our knowledge of nature is complete and correct, then a description of our knowledge is a description of nature, and by making a distinction between the two, Heisenberg is admitting the obvious fact that the “knowledge” of which he speaks, the incomplete knowledge of the atom represented by the atom-model which he champions, is not a true picture of the atom as it actually exists. What he is actually telling us, then, regardless of the kind of language that he uses, is that the Copenhagen atom-model has no properties and can only be symbolized by a mathematical equation. This is all that he can tell us in this connection, because his conclusions are based on an atom-model, and a model cannot tell us anything about the physical atom; it can only tell us about itself. It may suggest something about the atom and thereby give us a hint as to where to look for additional knowledge, indeed that is the primary purpose of a model, but this additional knowledge does not exist unless and until we verify it experimentally.
The statement that the Copenhagen atom-model cannot be conceived in anything but mathematical terms has exactly the same kind of significance as the statement that the billiard-ball atom-model has no internal structure. Neither of these statements has any applicability to the actual physical atom; they are statements about the models, not about the atoms which the models are intended to represent. The billiard-ball model was designed to fit the behavior of gases and similar phenomena, and it is applicable only where the presence or absence of internal structure is immaterial. The lack of any internal structure in the model does not signify, or even infer, that there is no internal structure in the physical atom; it simply means that the billiard-ball atom is incomplete, as models usually are, and gives us no information as to whether the physical atom has an internal structure or not. Similarly, the statement that the Copenhagen atom-model is purely mathematical and has no physical properties does not signify that the physical atom has no such properties; it merely means that the Copenhagen model is likewise incomplete. This model was originally designed to fit certain properties which are primarily mathematical in nature, and because it has been successful in the mathematical applications and unsuccessful elsewhere, it has gradually been modified to eliminate the non-mathematical aspects, until it is now almost entirely mathematical in character. The result is that it is now applicable only to the mathematical aspects, particularly to the quantum aspects, of the atomic behavior, and it has no relevance to anything else. This does not signify that nothing else exists; it merely means that the Copenhagen model is unable to deal with those other aspects.
The historical record of the development that has culminated in this Copenhagen model shows very clearly what has happened. Bohr found himself confronted with some serious contradictions in his first efforts to set up a nuclear atom-model in detail and, as the textbook author expresses it, solved the problem by postulating that it did not exist. As might be expected, this kind of a “solution” led to further difficulties of an equally serious nature and in casting about for means of overcoming these obstacles, Bohr hit upon the expedient of eliminating the recalcitrant aspects of the situation by assumptions and “principles” and confining the theoretical development to the more tractable remainder. “…Adequate tools were found,” he says, “in highly developed mathematical abstractions.” This set the pattern for future revisions and modifications. When new problems were encountered, they must inevitably be handled in the same way. “…We are here confronted with new problems,” says Bohr, “whose solution obviously demands further abstractions.”78 By this means Bohr and his fellow theorists have simply excluded everything which resisted treatment in this particular manner, and have restricted their atom-model to a representation of only a small portion of the behavior of the actual physical atom. What they did, says Herbert Dingle, “was to establish the fact that the hypothetical atoms were pure conceptions, that they belonged essentially to a different category from the facts of observation. They were creatures of the imagination, to be formed into the image of our fancies and restricted by whatever laws we cared to prescribe, provided only that when they behaved in accordance with those laws they should produce phenomena.” This, Dingle continues, “is the essence of the famous ’quantum theory’ though it is not the aspect under which it was first revealed and from which it derives its name.”79
Under these circumstances it is simply preposterous to contend that the Copenhagen atom-model is in any sense a true picture of the physical atom. This has been recognized by many observers. Ernest Nagel tells us, for instance, “Many physicists have therefore concluded that… the (quantum) theory must be regarded simply as a conceptual schema or a policy for guiding and coordinating experiments.”80 The only surprising thing about this statement is that it should be necessary to say “many” physicists rather than “all” physicists. Whatever one may think about the Copenhagen atom-model, whether he agrees with “the proponents of the Copenhagen interpretation” who, according to David Bohm, “regard the development of any alternative to their point of view as logically impossible”81; whether he is cautiously skeptical in the manner of Louis de Broglie, who says, “…quantum physics has found itself for several years tackling problems which it has not been able to solve and seems to have arrived at a dead end,”82 or Norwood R. Hanson, who goes a little farther and suggests, “The whole (quantum) theory may topple; in places the foundations seem far from secure”83; or whether he is definitely disillusioned in the manner of P.A.M. Dirac, who states, “One is thus led to doubt the validity of the whole structure of quantum field theory”84, the fact still remains that all of these opinions refer to an atom-model; indeed the very existence of such diversity of opinion is further evidence that we are dealing with a model, not a picture, of the atom.
Let us now take Jordan’s statement that the atom, as we know it today (that is, the Copenhagen atom-model), “can only be characterized by a system of mathematical formulae,” and set this up against the following from Paul F. Schmidt; “Mathematical propositions tell us nothing about the character of nature; they are uninterpreted formalisms.”85 Here we have the whole situation in a nutshell. Bohr’s atom-model attempted to explain the observed facts, but the attempt was unsuccessful and this model has been succeeded by the Copenhagen model, which no longer attempts to explain anything. This new model merely provides us with a mathematical system and a set of rules for operating it.
When we attempt to ascertain the meaning of the operations prescribed by the Copenhagen theory, we are thus in the same position as when we attempt to ascertain the internal structure of the billiard ball atom-model; the billiard-ball model has no internal structure and the Copenhagen model has no meaning. There are reasons to believe that the actual physical atom has some kind of internal structure and that there are logical explanations for the mathematical relationships which this physical atom follows, but the respective models have been so constructed as to exclude these aspects of the physical atom. Even the original Bohr atom had only a very tenuous tie with physical reality. Mendelssohn speaks of “…the strange emptiness of this ingenious and successful model.”86 In the revisions that followed, the few explainable features that did exist were eliminated and as Lanczos puts it, only “one feature of the theory remained unaltered up to our days: its incomprehensible character.”19
Essentially the same comment has been made by other observers. Herbert Dingle, for instance, asserts, “If there is one word that more aptly than another describes modern intellectual activity in its widest generality, that word is 'unintelligibility.'”87 Presumably these remarks by Lanczos and Dingle are intended as criticism, but if we recognize that the present-day descendant of the Bohr atom is an atom-model, not a picture of the physical atom, and that, according to the frontline theorists, this model does not purport to represent anything but the mathematical properties of the atom, these particular criticisms, so far as they apply to the atomic theory, are actually unjustified. After all, we cannot logically criticize a theory for being exactly what it claims to be. If the Copenhagen atom-model is “the solution of a wave equation and nothing more,”88 as Andrade puts it, we cannot expect it to be understandable in any other terms.
But although we cannot legitimately criticize the Copenhagen atom-model for being incomprehensible, since it is deliberately designed to be understandable only in mathematical terms, this merely emphasizes the fact that it is only a model, a creature of the imagination, not a picture of the actual physical atom, and whatever conclusions may be drawn from it, whether they are legitimate or not, are conclusions about the model, not about the physical atom.
As long as no true picture of the atom is yet available, a reasonably good model serves a useful purpose, and the foregoing discussion is not intended to imply anything to the contrary. The preceding chapters have demonstrated that the currently popular atom-models are incomplete and inconsistent with experimental knowledge, and are based on a completely erroneous concept of the nature of the atomic structure. Their usefulness is severely limited by these considerations, but if they are recognized for what they actually are, and utilized accordingly, they can be of considerable value, just as the earlier “billiard-ball” model played an important role in its day. On the other hand, the present tendency to look upon today’s models as “final and ultimate” pictures of the true physical situation, just because they happen to be the best explanations currently available, is a very serious impediment to scientific progress, not only because it diverts a huge amount of research activity into unproductive channels, but also because it prevents recognition of the correct answers if and when they do appear.