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Eddington 1927: Development of the New Quantum Theory

September 14, 2018 – 2:14 PM
quantization

Reference: The Nature of the Physical World

This paper presents Chapter X (section 3) 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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Development of the New Quantum Theory

The “New Quantum Theory” originated in a remarkable paper by Heisenberg in the autumn of 1925. I am writing the first draft of this lecture just twelve months after the appearance of the paper. That does not give long for development; nevertheless the theory has already gone through three distinct phases associated with the names of Born and Jordan, Dirac, Schrodinger. My chief anxiety at the moment is lest another phase of reinterpretation should be reached before the lecture can be delivered. In an ordinary way we should describe the three phases as three distinct theories. The pioneer work of Heisenberg governs the whole, but the three theories show wide differences of thought. The first entered on ‘the new road in a rather matter-of-fact way; the second was highly transcendental, almost mystical; the third seemed at first to contain a reaction towards classical ideas, but that was probably a false impression. You will realise the anarchy of this branch of physics when three successive pretenders seize the throne in twelve months; but you will not realise the steady progress made in that time unless you turn to the mathematics of the subject. As regards philosophical ideas the three theories are poles apart; as regards mathematical content they are one and the same. Unfortunately the mathematical content is just what I am forbidden to treat of in these lectures.

Heisenberg’s paper was about the Uncertainty Principle where two related quantities could not be determined accurately even on a theoretical basis. This uncertainty was entering the picture because space and time were being treated as absolute, and the variability in the “substantialness of substance” was not fully understood. 

As the substance became less substantial, the nature of space and time also became more diffused inside the atom. This phenomenon of quantization was simply bypassed when material units of space and time were used. This amounted to treating space and time as absolute.

I am, however, going to transgress to the extent of writing down one mathematical formula for you to contemplate; I shall not be so unreasonable as to expect you to understand it. All authorities seem to be agreed that at, or nearly at, the root of everything in the physical world lies the mystic formula

qp—pq = ih/2π

We do not yet understand that; probably if we could understand it we should not think it so fundamental. Where the trained mathematician has the advantage is that he can use it, and in the past year or two it has been used in physics with very great advantage indeed. It leads not only to those phenomena described by the older quantum laws such as the h rule, but to many related phenomena which the older formulation could not treat.

On the right-hand side, besides h (the atom of action) and the merely numerical factor 2π, there appears i (the square root of -1) which may seem rather mystical. But this is only a well-known subterfuge; and far back in the last century physicists and engineers were well aware that √-1i in their formulae was a kind of signal to look out for waves or oscillations. The right-hand side contains nothing unusual, but the left-hand side baffles imagination. We call q and p co-ordinates and momenta, borrowing our vocabulary from the world of space and time and other coarse-grained experience; but that gives no real light on their nature, nor does it explain why qp is so ill-behaved as to be unequal to pq.

It is here that the three theories differ most essentially. Obviously q and p cannot represent simple numerical measures, for then qp—pq would be zero. For Schrodinger p is an operator. His “momentum” is not a quantity but a signal to us to perform a certain mathematical operation on any quantities which may follow. For Born and Jordan p is a matrix—not one quantity, nor several quantities, but an infinite number of quantities arranged in systematic array. For Dirac p is a symbol without any kind of numerical interpretation; he calls it a q-number, which is a way of saying that it is not a number at all.

I venture to think that there is an idea implied in Dirac’s treatment which may have great philosophical significance, independently of any question of success in this particular application. The idea is that in digging deeper and deeper into that which lies at the base of physical phenomena we must be prepared to come to entities which, like many things in our conscious experience, are not measurable by numbers in any way; and further it suggests how exact science, that is to say the science of phenomena correlated to measure-numbers, can be founded on such a basis.

One of the greatest changes in physics between the nineteenth century and the present day has been the change in our ideal of scientific explanation. It was the boast of the Victorian physicist that he would not claim to understand a thing until he could make a model of it; and by a model he meant something constructed of levers, geared wheels, squirts, or other appliances familiar to an engineer. Nature in building the universe was supposed to be dependent on just the same kind of resources as any human mechanic; and when the physicist sought an explanation of phenomena his ear was straining to catch the hum of machinery. The man who could make gravitation out of cog-wheels would have been a hero in the Victorian age.

Nowadays we do not encourage the engineer to build the world for us out of his material, but we turn to the mathematician to build it out of his material. Doubtless the mathematician is a loftier being than the engineer, but perhaps even he ought not to be entrusted with the Creation unreservedly. We are dealing in physics with a symbolic world, and we can scarcely avoid employing the mathematician who is the professional wielder of symbols; but he must rise to the full opportunities of the responsible task entrusted to him and not indulge too freely his own bias for symbols with an arithmetical interpretation. If we are to discern controlling laws of Nature not dictated by the mind it would seem necessary to escape as far as possible from the cut-and-dried framework into which the mind is so ready to force everything that it experiences.

I think that in principle Dirac’s method asserts this kind of emancipation. He starts with basal entities inexpressible by numbers or number-systems and his basal laws are symbolic expressions unconnected with arithmetical operations. The fascinating point is that as the development proceeds actual numbers are exuded from the symbols. Thus although p and q individually have no arithmetical interpretation, the combination qp—pq has the arithmetical interpretation expressed by the formula above quoted. By furnishing numbers, though itself non-numerical, such a theory can well be the basis for the measure-numbers studied in exact science. The measure-numbers, which are all that we glean from a physical survey of the world, cannot be the whole world; they may not even be so much of it as to constitute a self-governing unit. This seems the natural interpretation of Dirac’s procedure in seeking the governing laws of exact science in a non-arithmetical calculus.

I am afraid it is a long shot to predict anything like this emerging from Dirac’s beginning; and for the moment Schrodinger has rent much of the mystery from the p’s and q’s by showing that a less transcendental interpretation is adequate for present applications. But I like to think that we may have not yet heard the last of the idea.

Schrodinger’s theory is now enjoying the full tide of popularity, partly because of intrinsic merit, but also, I suspect, partly because it is the only one of the three that is simple enough to be misunderstood. Rather against my better judgment I will try to give a rough impression of the theory. It would probably be wiser to nail up over the door of the new quantum theory a notice, “Structural alterations in progress—No admittance except on business”, and particularly to warn the doorkeeper to keep out prying philosophers. I will, however, content myself with the protest that, whilst Schrodinger’s theory is guiding us to sound and rapid progress in many of the mathematical problems confronting us and is indispensable in its practical utility, I do not see the least likelihood that his ideas will survive long in their present form.

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By vinaire | Tagged | Comments (0)

Eddington 1927: Transition to a New Theory

September 14, 2018 – 10:01 AM
Geometry of space

Reference: The Nature of the Physical World

This paper presents Chapter X (section 2) 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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Transition to a New Theory

By 1925 the machinery of current theory had developed another flaw and was urgently calling for reconstruction; Bohr’s model of the atom had quite definitely broken down. This is the model, now very familiar, which pictures the atom as a kind of solar system with a central positively charged nucleus and a number of electrons describing orbits about it like planets, the important feature being that the possible orbits are limited by the rules referred to on p. 190. Since each line in the spectrum of the atom is emitted by the jump of an electron between two particular orbits, the classification of the spectral lines must run parallel with the classification of the orbits by their quantum numbers in the model. When the spectroscopists started to unravel the various series of lines in the spectra they found it possible to assign an orbit jump for every line—they could say what each line meant in terms of the model. But now questions of finer detail have arisen for which this correspondence ceases to hold. One must not expect too much from a model, and it would have been no surprise if the model had failed to exhibit minor phenomena or if its accuracy had proved imperfect. But the kind of trouble now arising was that only two orbit jumps were provided in the model to represent three obviously associated spectral lines; and so on. The model which had been so helpful in the interpretation of spectra up to a point, suddenly became altogether misleading; and spectroscopists were forced to turn away from the model and complete their classification of lines in a way which ignored it. They continued to speak of orbits and orbit jumps but there was no longer a complete one-to- one correspondence with the orbits shown in the model.*

*Each orbit or state of the atom requires three (or, for later refinements, four) quantum numbers to define it. The first two quantum numbers are correctly represented in the Bohr model ; but the third number which discriminates the different lines forming a doublet or multiplet spectrum is represented wrongly—a much more serious failure than if it were not represented at all.

Bohr’s model of atom is inconsistent with the atomic spectra. The classification of the orbits by their quantum numbers in the model ceases to explain the finer details of the spectral lines.

There are neither electrons nor any orbits within the atom. There is only rotating field-substance with its various quantization levels. There are field particles as a result of quantization. The spectral lines relate to the absorption and emission of these field-particles.

The time was evidently ripe for the birth of a new theory. The situation then prevailing may be summarised as follows:

(1) The general working rule was to employ the classical laws with the supplementary proviso that whenever anything of the nature of action appears it must be made equal to h, or sometimes to an integral multiple of h.

(2) The proviso often led to a self-contradictory use of the classical theory. Thus in the Bohr atom the acceleration of the electron in its orbit would be governed by classical electrodynamics whilst its radiation would be governed by the h rule. But in classical electrodynamics the acceleration and the radiation are indissolubly connected.

(3) The proper sphere of classical laws was known. They are a form taken by the more general laws in a limiting case, viz. when the number of quanta concerned is very large. Progress in the investigation of the complete system of more general laws must not be hampered by classical conceptions which contemplate only the limiting case.

(4) The present compromise involved the recognition that light has both corpuscular and wave properties. The same idea seems to have been successfully extended to matter and confirmed by experiment. But this success only renders the more urgent some less contradictory way of conceiving these properties.

(5) Although the above working rule had generally been successful in its predictions, it was found to give a distribution of electron orbits in the atom differing in some essential respects from that deduced spectroscopically. Thus a reconstruction was required not only to remove logical objections but to meet the urgent demands of practical physics.

Electrons do not exist as such within the atom. They are only formed out of atomic reactions. The structure of atom is explained by certain quantum numbers. These quantum numbers needs to be explained in terms of quantization of field-substance.

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Eddington 1927: The New Quantum Theory

September 14, 2018 – 8:12 AM
New-Quantum-Theory

Reference: The Nature of the Physical World

This paper presents Chapter X (section 1) 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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The New Quantum Theory

The conflict between quantum theory and classical theory becomes especially acute in the problem of the propagation of light. Here in effect it becomes a conflict between the corpuscular theory of light and the wave theory.

In the early days it was often asked, How large is a quantum of light? One answer is obtained by examining a star image formed with the great 100-inch reflector at Mt. Wilson. The diffraction pattern shows that each emission from each atom must be filling the whole mirror. For if one atom illuminates one part only and another atom another part only, we ought to get the same effect by illuminating different parts of the mirror by different stars (since there is no particular virtue in using atoms from the same star) ; actually the diffraction pattern then obtained is not the same. The quantum must be large enough to cover a 100-inch mirror.

But if this same star-light without any artificial concentration falls on a film of potassium, electrons will fly out each with the whole energy of a quantum. This is not a trigger action releasing energy already stored in the atom, because the amount of energy is fixed by the nature of the light, not by the nature of the atom. A whole quantum of light energy must have gone into the atom and blasted away the electron. The quantum must be small enough to enter an atom.

I do not think there is much doubt as to the ultimate origin of this contradiction. We must not think about space and time in connection with an individual quantum; and the extension of a quantum in space has no real meaning. To apply these conceptions to a single quantum is like reading the Riot Act to one man. A single quantum has not travelled 50 billion miles from Sirius; it has not been 8 years on the way. But when enough quanta are gathered to form a quorum there will be found among them statistical properties which are the genesis of the 50 billion miles’ distance of Sirius and the 8 years’ journey of the light.

The contradiction about the size of light quantum comes about when we consider it in terms of material-space and material-time. According to Einstein’s papers on quantization and relativity, the space and time for light quanta are much less condensed than the material-space and material-time.

The classical laws are based on the material substance, material-space and material-time. Even when light is not material (it is physical), it was treated only in context of material substance.

As science went deeper into the properties of light and electromagnetic phenomena, it ran into the property of quantization. The electromagnetic spectrum revealed a new substance, which may be called “field-substance”. The field-substance acted as continuous wave at lower frequencies, but with increased frequency it became condensed and acted more like a particle. Ultimately, the field-substance condensed to form the material-substance as in the nucleus of the atom.

Classical mechanics did not have to deal with quantization because it did not deal with field-substance. The New Quantum Theory was then developed to deal with field-substance.

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Wave-Theory of Matter

It is comparatively easy to realise what we have got to do. It is much more difficult to start to do it. Before we review the attempts in the last year or two to grapple with this problem we shall briefly consider a less drastic method of progress initiated by De Broglie. For the moment we shall be content to accept the mystery as a mystery. Light, we will say, is an entity with the wave property of spreading out to fill the largest object glass and with all the well-known properties of diffraction and interference; simultaneously it is an entity with the corpuscular or bullet property of expending its whole energy on one very small target. We can scarcely describe such an entity as a wave or as a particle; perhaps as a compromise we had better call it a “wavicle”.

We misunderstand light by defining its wavelength, period and cycles in material units. A light quantum is the energy per cycle in light-units.

There is nothing new under the sun, and this latest. volte-face almost brings us back to Newton’s theory of light—a curious mixture of corpuscular and wave-theory. There is perhaps a pleasing sentiment in this “return to Newton”. But to suppose that Newton’s scientific reputation is especially vindicated by De Broglie’s theory of light, is as absurd as to suppose that it is shattered by Einstein’s theory of gravitation. There was no phenomenon known to Newton which could not be amply covered by the wave-theory; and the clearing away of false evidence for a partly corpuscular theory, which influenced Newton, is as much a part of scientific progress as the bringing forward of the (possibly) true evidence, which influences us to-day. To imagine that Newton’s great scientific reputation is tossing up and down in these latter-day revolutions is to confuse science with omniscience.

The wave-particle confusion with respect to light is resolved by the property of quantization discovered by Einstein.

To return to the wavicle.—If that which we have commonly regarded as a wave partakes also of the nature of a particle, may not that which we have commonly regarded as a particle partake also of the nature of a wave? It was not until the present century that experiments were tried of a kind suitable to bring out the corpuscular aspect of the nature of light; perhaps experiments may still be possible which will bring out a wave aspect of the nature of an electron.

So, as a first step, instead of trying to clear up the mystery we try to extend it. Instead of explaining how anything can possess simultaneously the incongruous properties of wave and particle we seek to show experimentally that these properties are universally associated. There are no pure waves and no pure particles.

The discovery that there are no pure waves and no pure particles, as made by de Broglie, supports the fundamental perspective of “continuum of substance”.  One special case of this broad perspective that applies only to material-substance is the “particles in void” perspective.

The characteristic of a wave-theory is the spreading of a ray of light after passing through a narrow aperture —a well-known phenomenon called diffraction. The scale of the phenomenon is proportional to the wavelength of the light. De Broglie has shown us how to calculate the lengths of the waves (if any) associated with an electron, i.e. considering it to be no longer a pure particle but a wavicle. It appears that in some circumstances the scale of the corresponding diffraction effects will not be too small for experimental detection. There are now a number of experimental results quoted as verifying this prediction. I scarcely know whether they are yet to be considered conclusive, but there does seem to be serious evidence that in the scattering of electrons by atoms phenomena occur which would not be produced according to the usual theory that electrons are purely corpuscular. These effects analogous to the diffraction and interference of light carry us into the stronghold of the wave-theory. Long ago such phenomena ruled out all purely corpuscular theories of light; perhaps to-day we are finding similar phenomena which will rule out all purely corpuscular theories of matter.*

*The evidence is much stronger now than when the lectures were delivered.

One cycle in light units shall appear as many cycles in material units. Using de Broglie’s method to calculate wavelengths from diffraction of waves, we may be able to find the ratio of light-units to material units for lengths. This shall reveal how much length shrinks from light frequency to material frequency.

A similar idea was entertained in a “new statistical mechanics” developed by Einstein and Bose—at least that seems to be the physical interpretation of the highly abstract mathematics of their theory. As so often happens the change from the classical mechanics, though far-reaching in principle, gave only insignificant corrections when applied to ordinary practical problems. Significant differences could only be expected in matter much denser than anything yet discovered or imagined. Strange to say, just about the time when it was realised that very dense matter might have strange properties different from those expected according to classical conceptions, very dense matter was found in the universe. Astronomical evidence seems to leave practically no doubt that in the so-called white dwarf stars the density of matter far transcends anything of which we have terrestrial experience; in the Companion of Sirius, for example, the density is about a ton to the cubic inch. This condition is explained by the fact that the high temperature and correspondingly intense agitation of the material breaks up (ionises) the outer electron systems of the atoms, so that the fragments can be packed much more closely together. At ordinary temperatures the minute nucleus of the atom is guarded by outposts of sentinel electrons which ward off other atoms from close approach even under the highest pressures; but at stellar temperatures the agitation is so great that the electrons leave their posts and run all over the place. Exceedingly tight packing then becomes possible under high enough pressure. R. H. Fowler has found that in the white dwarf stars the density is so great that classical methods are inadequate and the new statistical mechanics must be used. In particular he has in this way relieved an anxiety which had been felt as to their ultimate fate; under classical laws they seemed to be heading towards an intolerable situation—the star could not stop losing heat, but it would have insufficient energy to be able to cool down!**

** The energy is required because on cooling down the matter must regain a more normal density and this involves a great expansion of volume of the star. In the expansion work has to be done against the force of gravity.

The matter inside white dwarf stars is much denser than ordinary matter because it involves more dense packing of atomic nuclei. Such dense matter shall have higher quantization level compared to ordinary matter.

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By vinaire | Tagged , | Comments (1)

Eddington 1927: Relation of Classical Laws to Quantum Laws

September 13, 2018 – 9:10 PM
Classical-Definition-of-Kno.svg

Reference: The Nature of the Physical World

This paper presents Chapter IX (section 5) 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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Relation of Classical Laws to Quantum Laws

To follow up the verification and successful application of the quantum laws would lead to a detailed survey of the greater part of modern physics—specific heats, magnetism, X-rays, radioactivity, and so on. We must leave this and return to a general consideration of the relation between classical laws and quantum laws. For at least fifteen years we have used classical laws and quantum laws alongside one another notwithstanding the irreconcilability of their conceptions. In the model atom the electrons are supposed to traverse their orbits under the classical laws of electrodynamics; but they jump from one orbit to another in a way entirely inconsistent with those laws. The energies of the orbits in hydrogen are calculated by classical laws; but one of the purposes of the calculation is to verify the association of energy and period in the unit h, which is contrary to classical laws of radiation. The whole procedure is glaringly contradictory but conspicuously successful.

In my observatory there is a telescope which condenses the light of a star on a film of sodium in a photoelectric cell. I rely on the classical theory to conduct the light through the lenses and focus it in the cell; then I switch on to the quantum theory to make the light fetch out electrons from the sodium film to be collected in an electrometer. If I happen to transpose the two theories, the quantum theory convinces me that the light will never get concentrated in the cell and the classical theory shows that it is powerless to extract the electrons if it does get in. I have no logical reason for not using the theories this way round; only experience teaches me that I must not. Sir William Bragg was not overstating the case when he said that we use the classical theory on Mondays, Wednesday and Fridays, and the quantum theory on Tuesdays, Thursdays and Saturdays. Perhaps that ought to make us feel a little sympathetic towards the man whose philosophy of the universe takes one form on weekdays and another form on Sundays.

In the last century—and I think also in this—there must have been many scientific men who kept their science and religion in watertight compartments. One set of beliefs held good in the laboratory and another set of beliefs in church, and no serious effort was made to harmonise them. The attitude is defensible. To discuss the compatibility of the beliefs would lead the scientist into regions of thought in which he was inexpert; and any answer he might reach would be undeserving of strong confidence. Better admit that there was some truth both in science and religion; and if they must fight, let it be elsewhere than in the brain of a hard-working scientist. If we have ever scorned this attitude, Nemesis has overtaken us. For ten years we have had to divide modern science into two compartments; we have one set of beliefs in the classical compartment and another set of beliefs in the quantum compartment. Unfortunately our compartments are not watertight.

Classical and quantum laws must be consistent with each other. If they are not then we are unaware of some truth.

We must, of course, look forward to an ultimate reconstruction of our conceptions of the physical world which will embrace both the classical laws and the quantum laws in harmonious association. There are still some who think that the reconciliation will be effected by a development of classical conceptions. But the physicists of what I may call “the Copenhagen school” believe that the reconstruction has to start at the other end, and that in the quantum phenomena we are getting down to a more intimate contact with Nature’s way of working than in the coarse-grained experience which has furnished the classical laws. The classical school having become convinced of the existence of these uniform lumps of action, speculates on the manufacture of the chopper necessary to carve off uniform lumps; the Copenhagen school on the other hand sees in these phenomena the insubstantial pageant of space, time and matter crumbling into grains of action. I do not think that the Copenhagen school has been mainly influenced by the immense difficulty of constructing a satisfactory chopper out of classical material; its view arises especially from a study of the meeting point of quantum and classical laws.

The classical laws are the limit to which the quantum laws tend when states of very high quantum number are concerned.

This is the famous Correspondence Principle enunciated by Bohr. It was at first a conjecture based on rather slight hints; but as our knowledge of quantum laws has grown, it has been found that when we apply them to states of very high quantum number they converge to the classical laws, and predict just what the classical laws would predict.

I find the Correspondence Principle as stated above quite logical.

For an example, take a hydrogen atom with its electron in a circular orbit of very high quantum number, that is to say far away from the proton. On Monday, Wednesday and Friday it is governed by classical laws. These say that it must emit a feeble radiation continuously, of strength determined by the acceleration it is undergoing and of period agreeing with its own period of revolution. Owing to the gradual loss of energy it will spiral down towards the proton. On Tuesday, Thursday and Saturday it is governed by quantum laws and jumps from one orbit to another. There is a quantum law that I have not mentioned which prescribes that (for circular orbits only) the jump must always be to the circular orbit next lower, so that the electron comes steadily down the series of steps without skipping any. Another law prescribes the average time between each jump and therefore the average time between the successive emissions of light. The small lumps of energy cast away at each step form light-waves of period determined by the h rule. “Preposterous! You cannot seriously mean that the electron does different things on different days of the week!”

The higher is the quantum number the lower is the quantization. This is inconsistent with the Correspondence Principle. It is the higher quantization that leads to material-substance and to classical laws.

But did I say that it does different things? I used different words to describe its doings. I run down the stairs on Tuesday and slide down the banisters on Wednesday; but if the staircase consists of innumerable infinitesimal steps, there is no essential difference in my mode of progress on the two days. And so it makes no difference whether the electron steps from one orbit to the next lower or comes down in a spiral when the number of steps is innumerably great. The succession of lumps of energy cast overboard merges into a continuous outflow. If you had the formulae before you, you would find that the period of the light and the strength of radiation are the same whether calculated by the Monday or the Tuesday method—but only when the quantum number is infinitely great. The disagreement is not very serious when the number is moderately large; but for small quantum numbers the atom cannot sit on the fence. It has to decide between Monday (classical) and Tuesday (quantum) rules. It chooses Tuesday rules.

If, as we believe, this example is typical, it indicates one direction which the reconstruction of ideas must take. We must not try to build up from classical conceptions, because the classical laws only become true and the conceptions concerned in them only become defined in the limiting case when the quantum numbers of the system are very large. We must start from new conceptions appropriate to low as well as to high numbered states; out of these the classical conceptions should emerge, first indistinctly, then definitely, as the number of the state increases, and the classical laws become more and more nearly true. I cannot foretell the result of this remodelling, but presumably room must be found for a conception of “states”, the unity of a state replacing the kind of tie expressed by classical forces. For low numbered states the current vocabulary of physics is inappropriate; at the moment we can scarcely avoid using it, but the present contradictoriness of our theories arises from this misuse. For such states space and time do not exist—at least I can see no reason to believe that they do. But it must be supposed that when high numbered states are considered there will be found in the new scheme approximate counterparts of the space and time of current conception—something ready to merge into space and time when the state numbers are infinite. And simultaneously the interactions described by transitions of states will merge into classical forces exerted across space and time. So that in the limit the classical description becomes an available alternative. Now in practical experience we have generally had to deal with systems whose ties are comparatively loose and correspond to very high quantum numbers; consequently our first survey of the world has stumbled across the classical laws and our present conceptions of the world consist of those entities which only take definite shape for high quantum numbers. But in the interior of the atom and molecule, in the phenomena of radiation, and probably also in the constitution of very dense stars such as the Companion of Sirius, the state numbers are not high enough to admit this treatment. These phenomena are now forcing us back to the more fundamental conceptions out of which the classical conceptions (sufficient for the other types of phenomena) ought to emerge as one extreme limit.

Higher quantum states must parallel higher quantization of field-substance. In other words, Quantum numbers should be increasing from periphery toward the center of the atom, but they do not. This is inconsistent with the Correspondence Principle.

For an example I will borrow a quantum conception from the next chapter. It may not be destined to survive in the present rapid evolution of ideas, but at any rate it will illustrate my point. In Bohr’s semi-classical model of the hydrogen atom there is an electron describing a circular or elliptic orbit. This is only a model; the real atom contains nothing of the sort. The real atom contains something which it has not entered into the mind of man to conceive, which has, however, been described symbolically by Schrodinger. This “something” is spread about in a manner by no means comparable to an electron describing an orbit. Now excite the atom into successively higher and higher quantum states. In the Bohr model the electron leaps into higher and higher orbits. In the real atom Schrodinger’s “something” begins to draw itself more and more together until it begins sketchily to outline the Bohr orbit and even imitates a condensation running round. Go on to still higher quantum numbers, and Schrodinger’s symbol now represents a compact body moving round in the same orbit and the same period as the electron in Bohr’s model, and moreover radiating according to the classical laws of an electron. And so when the quantum number reaches infinity, and the atom bursts, a genuine classical electron flies out. The electron, as it leaves the atom, crystallises out of Schrodinger’s mist like a genie emerging from his bottle.

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By vinaire | Tagged | Comments (0)

Eddington 1927: Theory of the Atom

September 13, 2018 – 11:50 AM
atom3

Reference: The Nature of the Physical World

This paper presents Chapter IX (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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Theory of the Atom

We return now to further experimental knowledge of quanta. The mysterious quantity h crops up inside the atom as well as outside it. Let us take the simplest of all atoms, namely, the hydrogen atom. This consists of a proton and an electron, that is to say a unit charge of positive electricity and a unit charge of negative electricity. The proton carries nearly all the mass of the atom and remains rock-like at the centre, whilst the nimble electron moves round in a circular or elliptic orbit under the inverse square-law of attraction between them. The system is thus very like a sun and a planet. But whereas in the solar system the planet’s orbit may be of any size and any eccentricity, the electron’s orbit is restricted to a definite series of sizes and shapes. There is nothing in the classical theory of electromagnetism to impose such a restriction; but the restriction exists, and the law imposing it has been discovered. It arises because the atom is arranging to make something in its interior equal to h. The intermediate orbits are excluded because they would involve fractions of h, and h cannot be divided.

The significance of the mysterious quantity ‘h’ (Planck’s constant) is that it is energy per cycle at the center of the atom. Here the frequency is near infinite, and the energy per cycle is the lowest. Actually, ‘h’ is the limiting value as frequency goes to infinity.

An atom is a whirlpool of field-substance, much like a galaxy. The rotating field-substance is increasing in substantiality as it approaches the center. At the center it condenses into a nucleus. The nucleus anchors the atom.

The rotating field-substance within the atom is diffused at the periphery but it increases in frequency and quantization as it approaches the center. Increasingly discrete field-particles appear closer to the nucleus. In case of the simplest hydrogen atom, the whirlpool-like field-substance is identified as an “electron”, and the condensed nucleus at the center is identified as a “proton”. The field-substance and field-particles have charge instead of mass. The property of mass belongs to the whole atom.

This field-substance has many quantization levels. Each quantization level has a unique energy per cycle. It acquires the lowest value ‘h’ at the center. The value ‘h’ appears to be constant and indivisible only because it is a limiting value for infinite frequency.

But there is one relaxation. When wave-energy is sent out from or taken into the atom, the amount and period must correspond exactly to h. But as regards its internal arrangements the atom has no objection to 2h, 3h, 4h, etc.; it only insists that fractions shall be excluded. That is why there are many alternative orbits for the electron corresponding to different integral multipliers of h. We call these multipliers quantum numbers, and speak of 1 -quantum orbits, 2-quantum orbits, etc. I will not enter here into the exact definition of what it is that has to be an exact multiple of h; but it is something which, viewed in the four-dimensional world, is at once seen to be action though this may not be so apparent when we view it in the ordinary way in three-dimensional sections. Also several features of the atom are regulated independently by this rule, and accordingly there are several quantum numbers—one for each feature; but to avoid technical complication I shall refer only to the quantum numbers belonging to one leading feature.

Within an atom the highest quantization level exist at the center where the frequency is the highest and energy per cycle is the lowest. As one moves towards the periphery of the atom, the quantization decreases and the energy per cycle increases.

At lower quantization levels, the space and time units are larger because of lesser substantiality. The energy per cycle at these levels is identified as the wave-energy sent from or taken into the atom. The values of energy per cycle appear to be unique and as strict multiples of ‘h’.

Bohr’s atom seems to identify different quantum-orbits filled with electrons. Instead there seems to be different quantization levels manifested as unique field-particles for that level. These field-particles are not completely discrete.

According to this picture of the atom, which is due to Niels Bohr, the only possible change of state is the transfer of an electron from one quantum orbit to another. Such a jump must occur whenever light is absorbed or emitted. Suppose then that an electron which has been travelling in one of the higher orbits jumps down into an orbit of less energy. The atom will then have a certain amount of surplus energy that must be got rid of. The lump of energy is fixed, and it remains to settle the period of vibration that it shall have when it changes into aether-waves. It seems incredible that the atom should get hold of the aether and shake it in any other period than one of those in which it is itself vibrating. Yet it is the experimental fact that, when the atom by radiating sets the aether in vibration, the periods of its electronic circulation are ignored and the period of the aether-waves is settled not by any picturable mechanism but by the seemingly artificial h-rule. It would seem that the atom carelessly throws overboard a lump of energy which, as it glides into the aether, moulds itself into a quantum of action by taking on the period required to make the product of energy and period equal to h. If this unmechanical process of emission seems contrary to our preconceptions, the exactly converse process of absorption is even more so. Here the atom has to look out for a lump of energy of the exact amount required to raise an electron to the higher orbit. It can only extract such a lump from aether-waves of particular period—not a period which has resonance with the structure of the atom, but the period which makes the energy into an exact quantum.

There are no electrons jumping from one quantum orbit to another. Instead there are field-particles being added or subtracted at different quantization levels due to interactions. Each field-particle constitutes a cycle, which is absorbed or emitted as light.

There is no aether. There is only field-substance quantized as field-particle, and which may de-quantize back to field-substance (light).

As the adjustment between the energy of the orbit jump and the period of the light carrying away that energy so as to give the constant quantity h is perhaps the most striking evidence of the dominance of the quantum, it will be worthwhile to explain how the energy of an orbit jump in an atom can be measured. It is possible to impart to a single electron a known amount of energy by making it travel along an electric field with a measured drop of potential. If this projectile hits an atom it may cause one of the electrons circulating in the atom to jump to an upper orbit, but, of course, only if its energy is sufficient to supply that required for the jump; if the electron has too little energy it can do nothing and must pass on with its energy intact. Let us fire a stream of electrons all endowed with the same known energy into the midst of a group of atoms. If the energy is below that corresponding to an orbit jump, the stream will pass through without interference other than ordinary scattering. Now gradually increase the energy of the electrons; quite suddenly we find that the electrons are leaving a great deal of their energy behind. That means that the critical energy has been reached and orbit jumps are being excited. Thus we have a means of measuring the critical energy which is just that of the jump—the difference of energy of the two states of the atom. This method of measurement has the advantage that it does not involve any knowledge of the constant h, so that there is no fear of a vicious circle when we use the measured energies to test the h rule.* Incidentally this experiment provides another argument against the collection-box theory. Small contributions of energy are not thankfully received, and electrons which offer anything less than the full contribution for a jump are not allowed to make any payment at all.

* Since the h rule is now well established the energies of different states of the atoms are usually calculated by its aid; to use these to test the rule would be a vicious circle.

There are no electrons in the atom jumping orbits. There are only the field-particles condensing and de-condensing at certain energies of quantization levels.

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