## Max Born on Wave Mechanics

 In 1935 Max Born published The Restless Universe. My [EFR] copy of the First Edition cost 8s. 6d. and it is a book I treasure. The title page of the book gives the author as "Max Born, M.A. (Cantab.); Dr. phil. (Göttingen); Sc.D. (Bristol). Stokes Lecturer in Mathematics, University of Cambridge." The Jacket of the book contains the following description of the work:- Epoch-making discoveries and startling adventures along new paths have made the present the most exciting period in the history of science. In 'The Restless Universe' Dr Max Born, one of the foremost scientists of the day, throws light for the ordinary man on such bewildering mysteries as the nature of matter and the structure of the universe. The book is written throughout in language intelligible to young and old. We give an extract to illustrate Born's style. It is a style which, I remember, I found exciting and easy to understand when I read the book as a schoolboy. We have made a few minor changes to the text where reference is made to other parts of the book:

Wave Mechanics and its Statistical Interpretation.

Experiments demonstrate quite clearly that light and matter unite in themselves properties of waves and properties of particles. We therefore cannot say that they are one or the other: they are both, displaying one side of their nature of the other, according to the type of obstruction they meet.

This circumstance raises great difficulties of theoretical interpretation. Bohr has declared outright that there is an incomprehensible irrational factor in physical events. To make the position clear, need only state bluntly what the quantum postulate of Planck and de Broglie means.

Energy and momentum are properties of minute particles. Frequency and wave-number, on the other hand, are properties of simple harmonic waves, whose definition implies that they extend indefinitely in time and space.

Yet it is asserted that - apart from the factor ℏ, which serves to transform the units of measurement - energy and frequency are identified, and also momentum and wave-number.

We see at once that this is not possible unless we sacrifice some fundamental assumption of ordinary thought.

The case is like that of the theory of relativity. There experiments on the behaviour of light in rapidly moving systems forced us to form a new conception of space and time. Here in the quantum theory it is the principle of causality, or more accurately that of determinism, which must be dropped and replaced by something else.

To be quite clear about what this principle means, we use the illustration of a gun firing. A knowledge of the laws of nature is far from being sufficient to enable us to make predictions about future events; we must know the initial conditions as well. In the case of the gun the form of all possible trajectories is determined by a law of nature, which expresses the effect of gravity (and perhaps also air resistance) on the motion of the shell; but the path actually followed by the shell in a prescribed case depends on the direction in which the gun is trained and the muzzle velocity of the shell.

Now in the older physics it was assumed as obvious that these initial conditions can always be stated with any desired degree of accuracy. Then the course of the subsequent phenomena (the trajectory of the shell in the case of the gun) can also be calculated with any desired degree of accuracy. The initial state determines the future according to the laws of nature. From a given state onwards everything goes on like an automatic machine and, provided we know the laws of nature and the initial state, we can predict the future merely by processes of thought and calculation.

This actually does happen. Astronomers, above all, predict the positions of the moon and the planets, the occurrence of eclipses, and other celestial phenomena, with great accuracy. Engineers, too, rely firmly on their machines and structures doing what they have been calculated to do - and successfully.

Nevertheless, modern physics declares that the matter is not so simple as this, whenever we have to deal with the restless universe of atoms and electrons.

Even in the case of gases the determination of phenomena from the initial state may be an excellent idea in theory, but is of no practical consequence. For it is quite impossible to determine the positions and velocities of all the particles at one instant. Instead we have to recourse to statistics. We make an assumption about equally probable cases (the hypothesis that the molecules are arranged at random) and deduce results from this. As these agree with experiment, we are led to the belief that statements about probabilities can be just as good objective laws of physics. This kind of statistical argument, however, has only a loose and superficial connection with the rest of physics.

All this is changed by the discovery of the dual entity of wave and particle. Experiments show that the waves have objective reality just as much as the particles - the interference maxima of the waves can be photographed just as well as the cloud-tracks of the particles. There seems to be only one way out of the dilemma; a way I have proposed, which is now generally accepted, namely, the statistical interpretation of wave mechanics. Briefly it is this: the waves are waves of probability. They determine the "supply" of the particles, that is, their distribution in space and time. It follows that the waves, apart from their objective reality, must have something to do with the subjective act of observation.

Here lies the root of the whole matter.

In the older physics it was assumed that the universe goes on like a machine, independently of whether there is someone observing it or not. For the observer was believed to be capable of making his observations without disturbing the course of events. At all events, an astronomer through his telescope does not disturb the march of the planets.

But the position of a physicist who wishes to observe an electron in its path is not so simple. He is like a craftsman who is trying to set a diamond with a mason's trowel. He has no apparatus available which is smaller and finer than the electron. He can only use other electrons or photons; but these have an intense effect on the particles under observation, and spoil the experiment. We see that a necessary consequence of atomic physics is that we must abandon the idea that it is possible to observe the course of events in the universe without disturbing it.

Now if the steps necessary for making an observation had quite complicated effects on the events, mathematical physics could not exist at all. Happily this is not so. The fundamental laws of the quantum theory see to it that enough is left to enable us to make predictions. But the predictions are no longer "deterministic", in the sense that "the particle observed here today will be at such and such a place tomorrow"; but "statistical": "the probability that the particle will be at such and such a place tomorrow is so and so". In the limiting case of large masses, such as we have in ordinary life, this probability of course becomes certainty; here the principle of causality still holds in its old form.

To penetrate more deeply into the meaning of these statements, we consider an electron and its pilot wave. Physically there is no meaning in regarding this wave as a simple harmonic wave of infinite extent; we must, on the contrary, regard it as a wave packet consisting of a small group of indefinitely close wave-numbers, that is, of great extent in space. Then the group velocity is identical with the velocity of the particle; the wave-packet moves with the particle. But whereabouts in the packet is the particle?

Clearly it is in accordance with the spirit of the probability idea to say that this question has no answer. We can, however, say that the particle has an equal probability of being anywhere in the wave-packet. The wave is just that part of the description of the phenomenon that depends on the intrusion of the observer; it replaces the initial conditions of classical physics. The difference, however, is this: the assumption that the particle has a definite velocity necessarily means that the position of the particle is and must remain largely indeterminate. For it is only in the case of a group of waves of almost equal wave-numbers that we can speak of a group velocity.

The product

Extension in space of wave × Range of wave-number
is approximately 1; hence if the range of wave-number is small, the extension in space must be great.

This rule can be stated in another way, in which it no longer refers to waves but tells us something about the measurability of the position and velocity of the particle.

The extension of the wave in space corresponds to the uncertainty about the position of the particle. We now recall de Broglie's relation:

Wave-number = Momentum ÷ ℏ.
A definite range of wave-number therefore corresponds to a definite uncertainty about the momentum. Thus we obtain the result that the product
Uncertainty of Position × Uncertainty of Momentum
is never less than ℏ. This is the celebrated Uncertainty Principle of Heisenberg, which interprets the irrationality of the quantum laws as a limitation of the accuracy with which various quantities can be measured. There is another similar relation between time and energy.

JOC/EFR February 2017