This paper presents Chapter X (section 4) from the book THE NATURE OF THE PHYSICAL WORLD by A. S. EDDINGTON. The contents of this book are based on the lectures that Eddington delivered at the University of Edinburgh in January to March 1927.
The paragraphs of original material are accompanied by brief comments in color, based on the present understanding. Feedback on these comments is appreciated.
The heading below links to the original materials.
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Outline of Schrodinger’s Theory
Imagine a sub-aether whose surface is covered with ripples. The oscillations of the ripples are a million times faster than those of visible light—too fast to come within the scope of our gross experience. Individual ripples are beyond our ken; what we can appreciate is a combined effect—when by convergence and coalescence the waves conspire to create a disturbed area of extent large compared with individual ripples but small from our own Brobdingnagian point of view. Such a disturbed area is recognised as a material particle; in particular it can be an electron.
The Disturbance theory starts with the concept of NO SUBSTANCE, which we may refer to as “emptiness”. This emptiness differs from void in that it neither consists of space nor time because space and time are characteristics of the substance.
In that emptiness the SUBSTANCE appears as a continuum of complex cyclic motion. This is a field of disturbance that has no limit as to its frequency and complexity. We may identify it as field-substance. This is the substance of electromagnetic radiation. The variation of its frequency and complexity produces the electromagnetic spectrum.
As the field-substance increases in frequency and complexity, it becomes more substantial and discrete, though it maintains continuity at the fundamental level. This inherent property of field-substance is called QUANTIZATION.
This quantization occurs in the atom from periphery to the center. It ends up as the nucleus at the center of the atom. Thus the limiting effect of quantization is to condense the field-substance into material substance.
There is no sub-aether as postulated in Schrodinger’s Theory. There is only field-substance that quantizes into material substance.
The sub-aether is a dispersive medium, that is to say the ripples do not all travel with the same velocity; like water-ripples their speed depends on their wave-length or period. Those of shorter period travel faster. Moreover the speed may be modified by local conditions. This modification is the counterpart in Schrodinger’s theory of a field of force in classical physics. It will readily be understood that if we are to reduce all phenomena to a propagation of waves, then the influence of a body on phenomena in its neighbourhood (commonly described as the field of force caused by its presence) must consist in a modification of the propagation of waves in the region surrounding it.
The greater is the quantization the slower is the speed. The quantization increases with increasing frequency and shortening wavelength and period. Therefore, ripples of shorter period travel slower and not faster. Modification of speed implies change in quantization of substance.
We have to connect these phenomena in the sub-aether with phenomena in the plane of our gross experience. As already stated, a local stormy region is detected by us as a particle; to this we now add that the frequency (number of oscillations per second) of the waves constituting the disturbance is recognised by us as the energy of the particle. We shall presently try to explain how the period manages to manifest itself to us in this curiously camouflaged way; but however it comes about, the recognition of a frequency in the sub-aether as an energy in gross experience gives at once the constant relation between period and energy which we have called the h rule.
Field-substance of higher quantization takes the appearance of field-particles. Each field-particle is formed out of a single cycle of that quantization level. Space and time condenses with increasing quantization and these cycles become shorter in wavelength and period. At the level of material-substance these cycles achieve the limiting condition of infinitesimal size. The energy per cycle at this limiting condition is the Planck constant ‘h’. The h-rule says that the energy per cycle at lower quantization levels is a multiple of h.
Generally the oscillations in the sub-aether are too rapid for us to detect directly; their frequency reaches the plane of ordinary experience by affecting the speed of propagation, because the speed depends (as already stated) on the wave-length or frequency. Calling the frequency v, the equation expressing the law of propagation of the ripples will contain a term in v. There will be another term expressing the modification caused by the “field of force” emanating from the bodies present in the neighbourhood. This can be treated as a kind of spurious v, since it emerges into our gross experience by the same method that v does. If v produces those phenomena which make us recognise it as energy, the spurious v will produce similar phenomena corresponding to a spurious kind of energy. Clearly the latter will be what we call potential energy, since it originates from influences attributable to the presence of surrounding objects.
The speed of propagation depends on the quantization of the field-substance. We come to know of this quantization from the atomic spectra. The quantization is related to the frequency of light absorbed or emitted. The equation expressing the law of propagation of the ripples shall contain a term in frequency. There will be another term expressing the gradient of quantization.
Assuming that we know both the real v and the spurious or potential v for our ripples, the equation of wave-propagation is settled, and we can proceed to solve any problem concerning wave-propagation. In particular we can solve the problem as to how the stormy areas move about. This gives a remarkable result which provides the first check on our theory. The stormy areas (if small enough) move under precisely the same laws that govern the motions of particles in classical mechanics. The equations for the motion of a wave-group with given frequency and potential frequency are the same as the classical equations of motion of a particle with the corresponding energy and potential energy.
As quantization increases the equations of wave-propagation start to approximate the classical equations of motion of a particle.
It has to be noticed that the velocity of a stormy area or group of waves is not the same as the velocity of an individual wave. This is well known in the study of water-waves as the distinction between group-velocity and wave-velocity. It is the group-velocity that is observed by us as the motion of the material particle.
The motion of a particle is similar to the motion of a wave-group having a group velocity.
We should have gained very little if our theory did no more than re-establish the results of classical mechanics on this rather fantastic basis. Its distinctive merits begin to be apparent when we deal with phenomena not covered by classical mechanics. We have considered a stormy area of so small extent that its position is as definite as that of a classical particle, but we may also consider an area of wider extent. No precise delimitation can be drawn between a large area and a small area, so that we shall continue to associate the idea of a particle with it; but whereas a small concentrated storm fixes the position of the particle closely, a more extended storm leaves it very vague. If we try to interpret an extended wave-group in classical language we say that it is a particle which is not at any definite point of space, but is loosely associated with a wide region.
Schrodinger’s stormy area of small extent is a particle of high quantization. A more extended storm shall represent a “particle” of low quantization. Here we have quantization of space itself. Space becomes more concentrated at higher quantization.
Perhaps you may think that an extended stormy area ought to represent diffused matter in contrast to a concentrated particle. That is not Schrodinger’s theory. The spreading is not a spreading of density; it is an indeterminacy of position, or a wider distribution of the probability that the particle lies within particular limits of position. Thus if we come across Schrodinger waves uniformly filling a vessel, the interpretation is not that the vessel is filled with matter of uniform density, but that it contains one particle which is equally likely to be anywhere.
Here we have the very unit of space expanding with lower quantization. This is captured by Schrodinger’s equation.
The first great success of this theory was in representing the emission of light from a hydrogen atom— a problem far outside the scope of classical theory. The hydrogen atom consists of a proton and electron which must be translated into their counterparts in the sub-aether. We are not interested in what the proton is doing, so we do not trouble about its representation by waves; what we want from it is its field of force, that is to say, the spurious v which it provides in the equation of wave-propagation for the electron. The waves travelling in accordance with this equation constitute Schrodinger’s equivalent for the electron; and any solution of the equation will correspond to some possible state of the hydrogen atom. Now it turns out that (paying attention to the obvious physical limitation that the waves must not anywhere be of infinite amplitude) solutions of this wave-equation only exist for waves with particular frequencies. Thus in a hydrogen atom the sub-aethereal waves are limited to a particular discrete series of frequencies. Remembering that a frequency in the sub-aether means an energy in gross experience, the atom will accordingly have a discrete series of possible energies. It is found that this series of energies is precisely the same as that assigned by Bohr from his rules of quantization (p. 191). It is a considerable advance to have determined these energies by a wave-theory instead of by an inexplicable mathematical rule. Further, when applied to more complex atoms Schrodinger’s theory succeeds on those points where the Bohr model breaks down; it always gives the right number of energies or “orbits” to provide one orbit jump for each observed spectral line.
The Disturbance theory views the hydrogen atom as a single entity. The “proton” as the nucleus serves to anchor the atom and it provides a boundary condition of infinite frequency or quantization. The “electron” then constitutes a series of quantization levels that are decreasing away from the nucleus. There is a high gradient of quantization between the electronic region and the nucleus. Schrodinger’s equation may be modified for Disturbance theory.
It is, however, an advantage not to pass from wave-frequency to classical energy at this stage, but to follow the course of events in the sub-aether a little farther. It would be difficult to think of the electron as having two energies (i.e. being in two Bohr orbits) simultaneously; but there is nothing to prevent waves of two different frequencies being simultaneously present in the sub-aether. Thus the wave-theory allows us easily to picture a condition which the classical theory could only describe in paradoxical terms. Suppose that two sets of waves are present. If the difference of frequency is not very great the two systems of waves will produce “beats”. If two broadcasting stations are transmitting on wave-lengths near together we hear a musical note or shriek resulting from the beats of the two carrier waves; the individual oscillations are too rapid to affect the ear, but they combine to give beats which are slow enough to affect the ear. In the same way the individual wave-systems in the sub-aether are composed of oscillations too rapid to affect our gross senses ; but their beats are sometimes slow enough to come within the octave covered by the eye. These beats are the source of the light coming from the hydrogen atom, and mathematical calculation shows that their frequencies are precisely those of the observed light from hydrogen. Heterodyning of the radio carrier waves produces sound; heterodyning of the sub-aethereal waves produces light. Not only does this theory give the periods of the different lines in the spectra, but it also predicts their intensities —a problem which the older quantum theory had no means of tackling. It should, however, be understood that the beats are not themselves to be identified with light-waves; they are in the sub-aether, whereas light-waves are in the aether. They provide the oscillating source which in some way not yet traced sends out light-waves of its own period.
Schrodinger’s sub-aether is the gamma range of electromagnetic spectrum, which determines the energy of the quantization level itself. The difference between two adjacent quantization levels is related to the frequency of light absorbed or emitted.
What precisely is the entity which we suppose to be oscillating when we speak of the waves in the sub-aether? It is denoted by ψ, and properly speaking we should regard it as an elementary indefinable of the wave-theory. But can we give it a classical interpretation of any kind? It seems possible to interpret it as a probability. The probability of the particle or electron being within a given region is proportional to the amount of ψ in that region. So that if ψ is mainly concentrated in one small stormy area, it is practically certain that the electron is there; we are then able to localise it definitely and conceive of it as a classical particle. But the ip-waves of the hydrogen atom are spread about all over the atom; and there is no definite localisation of the electron, though some places are more probable than others.*
* The probability is often stated to be proportional to ψ2, instead of ψ, as assumed above. The whole interpretation is very obscure, but it seems to depend on whether you are considering the probability after you know what has happened or the probability for the purposes of prediction. The ψ2 is obtained by introducing two symmetrical systems of ψ-waves travelling in opposite directions in time; one of these must presumably correspond to probable inference from what is known (or is stated) to have been the condition at a later time. Probability necessarily means “probability in the light of certain given information”, so that the probability cannot possibly be represented by the same function in different classes of problems with different initial data.
The significance of the wave-function ψ in Schrodinger’s equation seems to be the quantization value of the substance.
Attention must be called to one highly important consequence of this theory. A small enough stormy area corresponds very nearly to a particle moving about under the classical laws of motion; it would seem therefore that a particle definitely localised as a moving point is strictly the limit when the stormy area is reduced to a point. But curiously enough by continually reducing the area of the storm we never quite reach the ideal classical particle; we approach it and then recede from it again. We have seen that the wave-group moves like a particle (localised somewhere within the area of the storm) having an energy corresponding to the frequency of the waves; therefore to imitate a particle exactly, not only must the area be reduced to a point but the group must consist of waves of only one frequency. The two conditions are irreconcilable. With one frequency we can only have an infinite succession of waves not terminated by any boundary. A boundary to the group is provided by interference of waves of slightly different length, so that while reinforcing one another at the centre they cancel one another at the boundary. Roughly speaking, if the group has a diameter of 1000 wavelengths there must be a range of wave-length of o-i per cent., so that 1000 of the longest waves and 1001 of the shortest occupy the same distance. If we take a more concentrated stormy area of diameter 10 wave- lengths the range is increased to 10 per cent.; 10 of the longest and 1 1 of the shortest waves must extend the same distance. In seeking to make the position of the particle more definite by reducing the area we make its energy more vague by dispersing the frequencies of the waves. So our particle can never have simultaneously a perfectly definite position and a perfectly definite energy; it always has a vagueness of one kind or the other unbefitting a classical particle. Hence in delicate experiments we must not under any circumstances expect to find particles behaving exactly as a classical particle was supposed to do—a conclusion which seems to be in accordance with the modern experiments on diffraction of electrons already mentioned.
A classical particle is assumed to be 100% discrete. Since the substance fundamentally forms a continuum, there is no 100% discreteness. Therefore, no field-particle is 100% discrete, even when discreteness increases with quantization.
We remarked that Schrodinger’s picture of the hydrogen atom enabled it to possess something that would be impossible on Bohr’s theory, viz. two energies at once. For a particle or electron this is not merely permissive, but compulsory—otherwise we can put no limits to the region where it may be. You are not asked to imagine the state of a particle with several energies; what is meant is that our current picture of an electron as a particle with single energy has broken down, and we must dive below into the sub-aether if we wish to follow the course of events. The picture of a particle may, however, be retained when we are not seeking high accuracy; if we do not need to know the energy more closely than 1 per cent., a series of energies ranging over 1 per cent, can be treated as one definite energy.
There are no electrons within the atom but quantization levels made up of field-particles, which are not completely discrete.
Hitherto I have only considered the waves corresponding to one electron; now suppose that we have a problem involving two electrons. How shall they be represented? “Surely, that is simple enough! We have only to take two stormy areas instead of one.” I am afraid not. Two stormy areas would correspond to a single electron uncertain as to which area it was located in. So long as there is the faintest probability of the first electron being in any region, we cannot make the Schrodinger waves there represent a probability belonging to a second electron. Each electron wants the whole of three-dimensional space for its waves; so Schrodinger generously allows three dimensions for each of them. For two electrons he requires a six-dimensional sub-aether. He then successfully applies his method on the same lines as before. I think you will see now that Schrodinger has given us what seemed to be a comprehensible physical picture only to snatch it away again. His sub-aether does not exist in physical space; it is in a “configuration space” imagined by the mathematician for the purpose of solving his problems, and imagined afresh with different numbers of dimensions according to the problem proposed. It was only an accident that in the earliest problems considered the configuration space had a close correspondence with physical space, suggesting some degree of objective reality of the waves. Schrodinger’s wave-mechanics is not a physical theory but a dodge—and a very good dodge too.
The Schrodinger’s equation may make more sense if we replace the idea of sub-aether by the gamma region of the electromagnetic spectrum, and replace the idea of electron by quantization levels made up of field-particles.
The fact is that the almost universal applicability of this wave-mechanics spoils all chance of our taking it seriously as a physical theory. A delightful illustration of this occurs incidentally in the work of Dirac. In one of the problems, which he solves by Schrodinger waves, the frequency of the waves represents the number of systems of a given kind. The wave-equation is formulated and solved, and (just as in the problem of the hydrogen atom) it is found that solutions only exist for a series of special values of the frequency. Consequently the number of systems of the kind considered must have one of a discrete series of values. In Dirac’s problem the series turns out to be the series of integers. Accordingly we infer that the number of systems must be either 1, 2, 3, 4, …, but can never be 2¾ r example. It is satisfactory that the theory should give a result so well in accordance with our experience! But we are not likely to be persuaded that the true explanation of why we count in integers is afforded by a system of waves.
Hopefully, the Disturbance theory may be able to provide the true explanation.
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