This paper presents Chapter X (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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Principle of Indeterminacy
My apprehension lest a fourth version of the new quantum theory should appear before the lectures were delivered was not fulfilled; but a few months later the theory definitely entered on a new phase. It was Heisenberg again who set in motion the new development in the summer of 1927, and the consequences were further elucidated by Bohr. The outcome of it is a fundamental general principle which seems to rank in importance with the principle of relativity. I shall here call it the “principle of indeterminacy”.
The gist of it can be stated as follows : a particle may have position or it may have velocity but it cannot in any exact sense have both.
Both the principles of relativity and indeterminacy come about because of quantization as discussed earlier. Material-substance is highly quantized. The field-substance in “empty” space and within the atom is quantized to a much lesser degree. As quantization becomes less, the substance, space and time become less substantial. The space and time expand and become more diffused.
If we are content with a certain margin of inaccuracy and if we are content with statements that claim no certainty but only high probability, then it is possible to ascribe both position and velocity to a particle. But if we strive after a more accurate specification of position a very remarkable thing happens; the greater accuracy can be attained, but it is compensated by a greater inaccuracy in the specification of the velocity. Similarly if the specification of the velocity is made more accurate the position becomes less determinate.
Science addresses this diffusion of space by means of probability of location.
Suppose for example that we wish to know the position and velocity of an electron at a given moment. Theoretically it would be possible to fix the position with a probable error of about 1/1000 of a millimetre and the velocity with a probable error of 1 kilometre per second. But an error of 1/1000 of a millimetre is large compared with that of some of our space measurements; is there no conceivable way of fixing the position to 1/10,000 of a millimetre? Certainly; but in that case it will only be possible to fix the velocity with an error of 10 kilometres per second.
The error comes about because material units of highly quantized compact space are being used to express measurement.
The conditions of our exploration of the secrets of Nature are such that the more we bring to light the secret of position the more the secret of velocity is hidden. They are like the old man and woman in the weather-glass; as one comes out of one door, the other retires behind the other door. When we encounter unexpected obstacles in finding out something which we wish to know, there are two possible courses to take. It may be that the right course is to treat the obstacle as a spur to further efforts; but there is a second possibility—that we have been trying to find something which does not exist. You will remember that that was how the relativity theory accounted for the apparent concealment of our velocity through the aether.
The hidden influence is that of quantization.
When the concealment is found to be perfectly systematic, then we must banish the corresponding entity from the physical world. There is really no option. The link with our consciousness is completely broken. When we cannot point to any causal effect on anything that comes into our experience, the entity merely becomes part of the unknown—undifferentiated from the rest of the vast unknown. From time to time physical discoveries are made; and new entities, coming out of the unknown, become connected to our experience and are duly named. But to leave a lot of unattached labels floating in the as yet undifferentiated unknown in the hope that they may come in useful later on, is no particular sign of prescience and is not helpful to science. From this point of view we assert that the description of the position and velocity of an electron beyond a limited number of places of decimals is an attempt to describe something that does not exist; although curiously enough the description of position or of velocity if it had stood alone might have been allowable.
The “electron” is used to describe the lesser quantized substance within the atom. The space and time within the atom is diffused compared to the material space and time. Both position and velocity within the atom cannot be measured precisely using material units.
Ever since Einstein’s theory showed the importance of securing that the physical quantities which we talk about are actually connected to our experience, we have been on our guard to some extent against meaningless terms. Thus distance is defined by certain operations of measurement and not with reference to nonsensical conceptions such as the “amount of emptiness” between two points. The minute distances referred to in atomic physics naturally aroused some suspicion, since it is not always easy to say how the postulated measurements could be imagined to be carried out. I would not like to assert that this point has been cleared up; but at any rate it did not seem possible to make a clean sweep of all minute distances, because cases could be cited in which there seemed no natural limit to the accuracy of determination of position. Similarly there are ways of determining momentum apparently unlimited in accuracy. What escaped notice was that the two measurements interfere with one another in a systematic way, so that the combination of position with momentum, legitimate on the large scale, becomes indefinable on the small scale. The principle of indeterminacy is scientifically stated as follows: if q is a co-ordinate and p the corresponding momentum, the necessary uncertainty of our knowledge of q multiplied by the uncertainty of p is of the order of magnitude of the quantum constant h.
A general kind of reason for this can be seen without much difficulty. Suppose it is a question of knowing the position and momentum of an electron. So long as the electron is not interacting with the rest of the universe we cannot be aware of it. We must take our chance of obtaining knowledge of it at moments when it is interacting with something and thereby producing effects that can be observed. But in any such interaction a complete quantum is involved; and the passage of this quantum, altering to an important extent the conditions at the moment of our observation, makes the information out of date even as we obtain it.
Einstein’s theory is interpreted subjectively in terms of the experience of moving observer. This is unscientific and leads to errors. The same theory can be interpreted objectively in terms of quantization (substantialness of substance). This is the basis of Disturbance theory.
The quantum constant ‘h’ is the limiting energy per cycle for material substance. This is the accuracy we measure by. Material position is accurate within the material cycle of infinitesimal wavelength. The quantum of energy for an “electron” is larger and more spread out because wavelength increases at lower quantization. This leads to lesser accuracy in measurements.
Suppose that (ideally) an electron is observed under a powerful microscope in order to determine its position with great accuracy. For it to be seen at all it must be illuminated and scatter light to reach the eye. The least it can scatter is one quantum. In scattering this it receives from the light a kick of unpredictable amount; we can only state the respective probabilities of kicks of different amounts. Thus the condition of our ascertaining the position is that we disturb the electron in an incalculable way which will prevent our subsequently ascertaining how much momentum it had. However, we shall be able to ascertain the momentum with an uncertainty represented by the kick, and if the probable kick is small the probable error will be small. To keep the kick small we must use a quantum of small energy, that is to say, light of long wave-length. But to use long wave-length reduces the accuracy of our microscope. The longer the waves, the larger the diffraction images. And it must be remembered that it takes a great many quanta to outline the diffraction image; our one scattered quantum can only stimulate one atom in the retina of the eye, at some haphazard point within the theoretical diffraction image. Thus there will be an uncertainty in our determination of position of the electron proportional to the size of the diffraction image. We are in a dilemma. We can improve the determination of the position with the microscope by using light of shorter wave-length, but that gives the electron a greater kick and spoils the subsequent determination of momentum.
A picturesque illustration of the same dilemma is afforded if we imagine ourselves trying to see one of the electrons in an atom. For such finicking work it is no use employing ordinary light to see with; it is far too gross, its wave-length being greater than the whole atom. We must use fine-grained illumination and train our eyes to see with radiation of short wave-length— with X-rays in fact. It is well to remember that X-rays have a rather disastrous effect on atoms, so we had better use them sparingly. The least amount we can use is one quantum. Now, if we are ready, will you watch, whilst I flash one quantum of X-rays on to the atom? I may not hit the electron the first time; in that case, of course, you will not see it. Try again; this time my quantum has hit the electron. Look sharp, and notice where it is. Isn’t it there? Bother! I must have blown the electron out of the atom.
This is not a casual difficulty; it is a cunningly arranged plot—a plot to prevent you from seeing something that does not exist, viz. the locality of the electron within the atom. If I use longer waves which do no harm, they will not define the electron sharply enough for you to see where it is. In shortening the wavelength, just as the light becomes fine enough its quantum becomes too rough and knocks the electron out of the atom.
We cannot do direct experimentation, as proposed above, to see reactions within the atom.
Other examples of the reciprocal uncertainty have been given, and there seems to be no doubt that it is entirely general. The suggestion is that an association of exact position with exact momentum can never be discovered by us because there is no such thing in Nature. This is not inconceivable. Schrodinger’s model of the particle as a wave-group gives a good illustration of how it can happen. We have seen (p. 217) that as the position of a wave-group becomes more defined the energy (frequency) becomes more indeterminate, and vice versa. I think that that is the essential value of Schrodinger’s theory; it refrains from attributing to a particle a kind of determinacy which does not correspond to anything in Nature. But I would not regard the principle of indeterminacy as a result to be deduced from Schrodinger’s theory; it is the other way about. The principle of indeterminacy, like the principle of relativity, represents the abandonment of a mistaken assumption which we never had sufficient reason for making. Just as we were misled into untenable ideas of the aether through trusting to an analogy with the material ocean, so we have been misled into untenable ideas of the attributes of the microscopic elements of world-structure through trusting to analogy with gross particles.
The missing concept here is quantization.
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