The Faber Book of Science (36 page)

Let us start with the old problem of the constitution of light. At the beginning of the scientific epoch two rival theories were proposed: the corpuscular theory by Newton, the wave theory by Huygens. About a hundred years elapsed before experiments were found deciding in favour of one of them, the wave theory, by the discovery of interference. When two trains of waves are superposed, and a crest of one wave coincides with a valley of the other, they annihilate one another; this effect creates the well-known patterns which you can observe on any pond on which swimming ducks or gulls excite
water-waves
. Exactly the same kind of pattern can be observed when two beams of light cross one another, the only difference being that you need a magnifying-lens to see them; the inference is that a beam of light is a train of waves of short wave-length. This conclusion has been supported by innumerable experiments.

But about a hundred years later, during my student days, another set of observations began to indicate with equal cogency that light consists of corpuscles. This type of evidence can best be explained by analogy with two types of instruments of war, mines and guns. When a mine explodes you will be killed if you are near it, by the energy transferred to you as a wave of compressed air. But if you are some hundred yards away you are absolutely safe; the explosion-wave has lost its dangerous energy by continuously spreading out over a large area. Now imagine that the same amount of explosive is used as the
propellant in a machine-gun which is rapidly fired, turning round in all directions. If you are near it you will almost certainly be shot, unless you hastily run away. When you have reached a distance of some hundred yards you will feel much safer, but certainly not quite safe. The probability of being hit has dropped enormously, but if you are hit the effect is just as fatal as before.

Here you have the difference between energy spread out from a centre in the form of a continuous wave-motion, and a discontinuous rain of particles. Planck discovered, in 1900, the first indication of this discontinuity of light in the laws governing the heat radiated from hot bodies. In his celebrated paper of 1905, Einstein pointed out that experiments on the energetic effect of light, the so-called photoelectric effect, could be interpreted in the way indicated as showing unambiguously the corpuscular constitution of light. These corpuscles are called quanta of light or photons.

This dual aspect of the luminous phenomenon has been confirmed by many observations of various types. The most important step was made by Bohr, who showed that the enormous amount of
observations
on spectra collected by the experimentalists could be interpreted and understood with the help of the conception of light quanta. For this purpose he had also to apply the idea of discontinuous behaviour to the motion of material particles, the atoms, which are the source of light.

I cannot follow out here the historical development of the quantum idea which led step by step to the recognition that we have here to do with a much more general conception. Light is not the only ‘radiation’ we know; I may remind you of the cathode rays which appear when electric currents pass through evacuated bulbs, or the rays emitted by radium and other radioactive substances. These rays are certainly not light. They are beams of fast-moving electrons, i.e. atoms of electricity, or ordinary atoms of matter like helium. In the latter case this has been proved directly by Rutherford, who caught the beam (a so-called α-ray of radium) in an evacuated glass vessel and showed that it was finally filled with helium gas. Today one can actually photograph the tracks of these particles of radiating matter in their passage through other substances.

In this case the corpuscular evidence was primary. But in 1924 de Broglie, from theoretical reasoning, suggested the idea that these radiations should show interference and behave like waves under
proper conditions. This idea was actually confirmed by experiments a short time later. Not only electrons, but real atoms of ordinary matter like hydrogen or helium have all the properties of waves if brought into the form of rays by giving them a rapid motion.

This is a most exciting result, revolutionizing all our ideas of matter and motion. But when it became known, theoretical physics was already prepared to treat it by proper mathematical methods, the
so-called
quantum mechanics, initiated by Heisenberg, worked out in collaboration with Jordan and myself, and quite independently by Dirac; and another form of the same theory, the wave-mechanics, worked out by Schrödinger in close connection with de Broglie’s suggestion. The mathematical formalism is a wonderful invention for describing complicated things. But it does not help much towards a real understanding. It took several years before this understanding was reached, even to a limited extent. But it leads right amidst philosophy, and this is the point about which I have to speak.

The difficulty arises if we consider the fundamental discrepancy in describing one and the same process sometimes as a rain of particles, and at other times as a wave. One is bound to ask, what is it really? You see here the question of reality appears. The reason why it appears is that we are talking about particles or waves, things considered as well known; but which expression is adequate depends on the method of observation. We thus meet a situation similar to that in relativity, but much more complicated. For here the two representations of the same phenomenon are not only different but contradictory. I think everyone feels that a wave and a particle are two types of motion which cannot easily be reconciled. But if we take into account the simple quantitative law relating energy and frequency already discovered by Planck, the case becomes very serious. It is clear that the properties of a given ray when appearing as a rain of particles must be connected with its properties when appearing as a train of waves. This is indeed the case, and the connecting law is extremely simple when all the particles of the beam have exactly the same velocity. Experiment then shows that the corresponding train of waves has the simplest form possible, which is called harmonic, and is characterized by a definite sharp frequency and wave-length. The law of Planck states that the kinetic energy of the particles is exactly proportional to the frequency of vibration of the wave; the factor of proportionality, called Planck’s constant, and denoted by the letter
h
,
has a definite
numerical value which is known from experiment with fair accuracy.

There you have the logical difficulty: a particle with a given velocity is,
qua
particle, a point, existing at any instant without extension in space. A train of waves is by definition harmonic only if it fills the whole of space and lasts from eternity to eternity! (The latter point may not appear so evident; but a mathematical analysis made by Fourier more than a hundred years ago has clearly shown that every train of waves finite in space and time has to be considered as a superposition of many infinite harmonic waves of different frequencies and wave-lengths which are arranged in such a way that the outer parts destroy one another by interference; and it can be shown that every finite wave can be decomposed into its harmonic components.) Bohr has emphasized this point by saying that Planck’s principle introduces an irrational feature into the description of nature.

Indeed the difficulty cannot be solved unless we are prepared to sacrifice one or other of those principles which were assumed as fundamental for science. The principle to be abandoned now is that of causality as it has been understood ever since it could be formulated exactly. I can indicate this point only very shortly. The laws of mechanics as developed by Galileo and Newton allow us to predict the future motion of a particle if we know its position and velocity at a given instant. More generally, the future behaviour of a system can be predicted from a knowledge of proper initial conditions. The world from the standpoint of mechanics is an automaton, without any freedom, determined from the beginning. I never liked this extreme determinism, and I am glad that modern physics has abandoned it. But other people do not share this view.

To understand how the quantum idea and causality are connected, we must explain the second fundamental law relating particles and waves. This can be readily understood with the help of our example of the exploding mine and the machine-gun. If the latter fires not only horizontally but equally in all directions, the number of bullets, and therefore the probability of being hit, will decrease with distance in exactly the same ratio as the surface of the concentric spheres, over which the bullets are equally distributed, increases. But this
corresponds
exactly to the decrease of energy of the expanding wave of the exploding mine. If we now consider light spreading out from a small source, we see immediately that in the corpuscular aspect the number of photons will decrease with the distance in exactly the same way as
does the energy of the wave in the undulatory aspect. I have generalized this idea for electrons and any other kind of particles by the statement that we have to do with ‘waves of probability’ guiding the particles in such a way that the intensity of the wave at a point is always proportional to the probability of finding a particle at that point. This suggestion has been confirmed by a great number of direct and indirect experiments. It has to be modified if the particles do not move independently, but act on one another; for our purpose, however, the simple case is sufficient.

Now we can analyse the connection between the quantum laws and causality.

Determining the position of a particle means restricting it physically to a small part of space. The corresponding probability wave must also be restricted to this small part of space, according to our second quantum law. But we have seen that by Fourier’s analysis such a wave is a superposition of a great number of simple harmonic waves with wave-lengths and frequencies spread over a wide region. Using now the first quantum law stating the proportionality of frequency and energy, we see that this geometrically well-defined state must contain a wide range of energies. The opposite holds just as well. We have derived qualitatively the celebrated uncertainty law of Heisenberg: exact determination of position and velocity exclude one another; if one is determined accurately the other becomes indefinite.

The quantitative law found by Heisenberg states that for each direction in space the product of the uncertainty interval of space and that of momentum (equal to mass times velocity) is always the same, being given by Planck’s quantum constant
h.

Here we have the real meaning of this constant as an absolute limit of simultaneous measurement of position and velocity. For more complicated systems there are other pairs or groups of physical quantities which are not measurable at the same instant.

Now we remember that the knowledge of position and velocity at one given time was the supposition of classical mechanics for determining the future motion. The quantum laws contradict this supposition, and this means the break-down of causality and determinism. We may say that these propositions are not just wrong, but empty: the premise is never fulfilled.

Light, we all know, travels in straight lines. If it could bend round corners, the world would be harder to discern. It would be like listening to it instead of seeing it. We would be immersed in a symphony of colour from objects that could be vaguely located but only hazily scrutinized. There would be no night; the symphony would be endless.

But saying that light travels in straight lines is not quite right. It conflicts with observation. Light bends at the junction of different media. Your leg in your bath looks broken even if it isn’t. A lens bends light, and is shaped to focus the image on a film or on an eye. We therefore have to find a rule that captures both the straightness of the path when the medium is uniform and its bending when it passes from one medium to another.

The rule that captures both turns out to be elegantly simple (like all acceptable rules prior to their elimination): light travels by the path that takes the least time.

This succinct rule obviously accounts for the motion of light through air or any other uniform medium, because a straight line is then also the briefest path for anything travelling with a uniform speed. The rule also accounts for light’s bending at the junction of media. Light travels at different speeds in different substances; the briefest path is then no longer the straightest, as can be understood by thinking about drowning.

Suppose the victim is out to sea, and you are on the shore. What path brings you to him in the shortest time, bearing in mind that you can run faster than you can swim? One possibility is for you to select a geometrically straight path from your deckchair to where he is sinking: that involves a certain amount of running and swimming.
Alternatively
you could run to a point on the water’s edge directly opposite him and swim out straight from there. That is greater in distance but it may be briefer in duration if you can run very much faster than you can swim. By trial and error, or trigonometry, you would find that the path involving the least time is one where you run at some angle across the beach, then change direction and swim at another angle in a straight line towards your target (if it is not too late by now). This is exactly the behaviour of light passing into a denser medium.

But how does light know, apparently in advance, which is the briefest path? And, anyway, why should it care? The only way of discovering the briefest path appears to be to try them all, and then to eliminate all traces of having done so. There must be something about the nature of light which entails that it naturally tries all paths, and then eliminates all but the briefest.

The essential property is that light travels as a wave. Once that is realized, its other properties fall into place: light cannot help travelling by the briefest path.

A wave is an undulation, a series of peaks and troughs. Two or more waves of disturbance may spread into the same region. If it happens that the peaks of one coincide with the troughs of the other, then they tend to cancel, and an observer sees less disturbance, and perhaps no disturbance at all if they happen to cancel completely. That is basically all the information we need in order to see how light’s character determines its destiny.

We are taking the view that things happen if they are not expressly forbidden, and that an infinitely lazy creator does not trouble to forbid. Think, then, of a light ray that happens to travel from A to B along some meandering path.
W
e
know that light doesn’t travel like that, but light doesn’t. If that path is permissible, then so too, as far as the light is concerned, is one that lies very close to it. So the light also travels by that path. Whereas the light that snaked by the first path may have reached B with a peak, the light that snakes by the second might reach B with a trough, or something in between. There are very many paths lying close to the first, and an observer at B sees the total
disturbance arising from the waves that explore them all: many are troughs at B, many are peaks, and many are all the possibilities in between. The total disturbance at B is consequently zero, because there is always a neighbour to wash its neighbour out. In other words, by letting light travel by any path, it appears to be unable to travel at all. But light does travel.

One step was too hasty. Think of a ray that happens to go straight from A to B. Now think of a neighbouring path and the ray that takes it. If that path lies close to the first, it will have a trough at B if the first had a trough, and a peak if the first had a peak. There are very many almost-straight lines from A to B, and they all give disturbances at B differing only slightly from the disturbance due to the straight path. These paths therefore do not wash each other out, and an observer at B sees the light. He observes that the light travelled to him by lines that are straight, or very nearly straight.

The extent to which the nearly but not quite straight rays contribute to the overall disturbance at B depends on their wavelength (the separation of successive peaks). If the wavelength is short, then only rays correspondingly close to the straight line survive, all the others having sufficiently destructive neighbours. As the wavelength increases the waves get out of step less quickly, and the eliminating power of neighbours declines. Then even quite bent paths survive and can deliver their disturbance. That is the reason why radio transmissions (which use long-wavelength waves) can circumvent houses, and why we cannot see round corners. We can hear round corners: sound waves’ wavelengths are long.

The wave nature of light accounts for the inevitability of its selection of straight lines. That, though, is true of uniform media, as in air. When light passes from one medium into something denser it travels more slowly. As a result, the positions of its peaks and troughs are modified. Still it explores all possible paths, but no longer is the geometrically straight line the one without neighbours that annihilate. Now the surviving path, because of the shift of peaks and troughs, is the one that bends at the junction. The surviving path also happens to be the briefest path. That rule therefore turns out to be merely a distant commentary on deeper purposelessness. Light automatically discovers briefest paths by trying all paths, and automatically eradicates all traces of its explorations; this presents itself to us as a behaviour, which we summarize as a rule.

We see in this example how perfect freedom generates its own constraint. As well as accounting for observed behaviour, everything we have said accords with the common-sense view that inanimate things are innately simple. That is one more step along the path to the view that animate things, being innately inanimate, are innately simple too.

The next step in the development involves noticing a similar observation about another thing. Since the behaviour is similar, we can suspect that the explanation is similar too. I should like you to notice that particles of matter also travel in straight lines unless subject to a force. Why?

According to the view we are taking, they do so because it is their intrinsic nature. But what can be this intrinsic nature that determines such behaviour? It must be that particles are distributed as waves.

In a single leap, impelled by common sense, we have gone from the old-fashioned original physics of Newton to the modern theory of matter, quantum theory, which regards the qualities of ‘particle’ and ‘wave’ as inseparable. Many feel at home with classical physics and regard quantum theory, being less familiar, as contrary to common sense. In my view, though, common sense drives us to accept quantum theory in place of classical physics as more consistent with common sense. I hold that the mind-shutting familiarities of classical physics actually conceal its incomprehensibility, except as a commentary and a mode of calculation. When they are inspected, the explanations of classical physics fall apart, and are seen to be mere superficial delusions, like film-sets.

There is much more to quantum theory than the assertion that particles are intrinsically wavelike, but that remark is at its core, and is what we develop here, first by seeking the actual rule that appears to govern the classical mechanics of particles, and then by looking for an explanation.

The rule that appears to govern the propagation of particles is remarkably, and therefore suspiciously, like the rule that appears to govern the propagation of light: particles follow trajectories between A and B that involve least action. Never mind the technical meaning of action; it is good enough, and truthful enough, to think of action as having its everyday meaning. In particular, if the particle is not subject to a force then the path that involves least action – no meandering and no acceleration – is uniform and straight.

Now, how does a particle know, before it tries, which of the infinity of possible paths from A to B corresponds to the one of least action? And why should it care?

As soon as we take the view that particles are distributed like waves, both questions are eliminated by the same reasoning that eliminates them in the case of light. The intrinsic nature of particles, their wavelike character, ensures that they travel in straight lines of least action, because all other paths, which they are perfectly free to explore, are eliminated automatically. The reason why particles like pigs and people do not normally seem to be waves is simply that their wavelengths are normally so short as to be undetectable. Nevertheless, distributed as waves they are, and that attribute provides explanations which are totally beyond the reach of classical physics.

This picture accounts for motion in straight lines, because in the absence of forces such paths are survivors; the wave nature lets them survive.