by John Gribbin
But if individual particles (such as electrons) are fired, one at a time, through the experiment with two holes, you would expect, from everyday experience, that they would pile up in two heaps, one behind each of the holes. A suitable detector screen on the other side (essentially the same as a TV screen) ought, if electrons are particles, to show two blobs, corresponding to the trajectories of electrons going through either of the two holes. But it doesn’t. Here’s what happens. Each individual particle starts out on one side, passes through the experiment, and strikes the detector screen. Surely, you would think, each particle can only go through one hole or the other. And, to be sure, each particle makes just one spark of light on the detector screen, indicating that it arrives as a particle. But after thousands of particles have been fired through the experiment one after the other, a pattern of sparks of light builds up on the detector screen. Not the two blobs behind the two holes that you might expect from your everyday experience of how particles behave, but the familiar interference pattern for waves! (See Figure 4b.) We stress that this experiment really has been done, and both electrons and photons behave in this way. It is as if each particle flies through both holes at once, interferes with itself, decides where it belongs in the interference pattern, and goes there to make its own individual contribution to the emerging pattern. Quantum entities seem to travel as waves, but to arrive (and depart) as particles.
Figure 4. (a) When light spreads out from a pinhole in a screen to pass through two pinholes in a second screen, the pattern made up by the light from the second two holes shows alternating dark and light bands, exactly as if the light is behaving as waves which interfere with one another. (b) When electrons (or, indeed, photons) are fired through a similar set-up with one hole open, they pile up, like particles, in one heap behind the open hole. But if both holes are open, the ‘particles’ somehow interfere with each other to produce a pattern exactly equivalent to the pattern produced when waves interfere. This is the central mystery of quantum mechanics, the experiment with two holes. How do the electrons know in advance whether one or both holes are open, and adjust their behaviour accordingly?
As well as wave–particle duality, this example highlights another aspect of the quantum world – the role of probability. Nothing is certain in the quantum world. For example, before an individual electron is fired through the experiment with two holes it is impossible for the experimenter to say exactly where on the screen on the other side it will arrive. You can only calculate, in accordance with the rules of quantum probability, the chance of it ending up in a particular part of the interference pattern. It is likely to turn up in one of the bright parts of the pattern and unlikely to appear in one of the dark stripes in the pattern, but that is all you can say. Quantum processes obey the same rules of chance as the dice at a craps table in Las Vegas, which prompted Einstein to express his own disgust with the theory with his comment, ‘I cannot believe that God plays dice.’
So how should we think of an electron ‘in’ an atom, where it is ‘travelling’ in its ‘orbit’, rather than ‘arriving’ at a detector? The standard picture, used by physicists for the past 70 years, says that the electron cannot be located at any one point in space near the nucleus, but that the location of each electron is spread out over a region of space surrounding the nucleus – not just stretched out along a single orbit (like the orbits of the planets around the Sun), but spread out in a shell literally surrounding the nucleus, a shell called an ‘orbital’. The orbital is thought of as a ‘cloud of probability’, representing the likelihood of finding the electron. If a measurement were carried out that was accurate enough to locate the precise position of the electron, for that instant it would indeed ‘arrive’ at some definite position within the orbital, and manifest itself as a particle. The position it would arrive at would be subject entirely to chance, selected at random from the options open to it. But as soon as the observation had been completed, the electron would dissolve once again into a haze of probabilities. And this kind of behaviour is supposed to represent the behaviour of all quantum entities.
The way in which an entity such as an electron manifests itself as a particle when it is measured is called the ‘collapse of the wave function’, and all quantum systems are supposed to exist in some sort of state of probabilistic uncertainty until an observation or measurement is made and the wave function collapses.
This has given rise to all kinds of debate about what constitutes a measurement, and when exactly the wave function collapses, which, happily, we do not need to go into here.5 It sounds bizarre. And yet, it works. Quantum theory says that at the level of atoms and subatomic ‘particles’, entities have to be thought of as having properties of both wave and particles, that nothing is certain, and the outcome of an experiment depends on chance, in the strict mathematical sense. But all of this strange mixture has practical applications. Since the face shown by an atom to the world – to other atoms – is its electron cloud, and chemistry depends on the way the electron clouds of different atoms interact with one another, it is this quantum mechanical view of the behaviour of electrons that underpins, among other things, the extremely successful modern understanding of chemistry, developed in the wake of these discoveries.6
In spite of its weirdness, the new quantum mechanics worked. The point was made, in forceful terms, by the greatest single triumph of this period, when, in 1928, Dirac published an equation which incorporated the requirements of the Special Theory of Relativity into the quantum theory to provide a complete description of the electron, in terms of relativistic quantum mechanics. The Dirac equation described everything there was to know about the electron, and made predictions which matched the results of all the experiments. It also made another prediction, which even Dirac did not immediately interpret correctly, and which would have a profound influence on the career of Richard Feynman, who was just ten years old when Dirac came up with his equation.
Dirac’s equation not only explained everything there was to explain about an electron, but it did so in duplicate. The point is, there were two sets of solutions to the equation. Now, there is nothing unusual about this. If you see an equation such as x2 = 4, you know that the solution to the equation is x = 2, because 2 × 2 = 4. But there is, in fact another solution as well. Because two negative numbers multiplied together make a positive number (just as in language, where a ‘double negative’ makes an affirmative), (–2) × (–2) = 4, as well. So –2 is a perfectly good solution to the equation x2 = 4, if you are a mathematician. Such ‘negative roots’ often crop up in equations, and the question is whether they mean anything in practical terms. The second solution to Dirac’s equation describes particles identical to electrons but with negative energy. Most people would probably have dismissed this as a meaningless mathematical quirk. But Dirac’s genius led him to wonder ‘what if’ – what if these negative-energy electrons really existed?
The big snag was that if you allow electrons to have negative energy, at first sight it seems that they all ought to have negative energy. Like water running downhill, any physical system seeks out its lowest possible energy level. If there were ‘negative-energy levels’ for electrons, then obviously even the highest of these levels would be below the lowest positive-energy level, and all electrons would fall down into the negative levels, radiating a blaze of electromagnetic energy as they did so. But suppose, Dirac argued, all of the negative-energy levels were full up, just as the sea is full up with water. Water running downhill would carry on running down to the bottom of what is now the sea, if there were no sea in the way; but in the real world rivers only run down to discharge their water into the top of the sea, because the sea is already full up. If all of the negative-energy ‘sea’ were full of electrons, the only openings available for any more electrons would be the positive-energy levels above. The negative-energy electron sea would be completely undetectable, or invisible, because it was the same everywhere.
But now Dirac went a step further
. In the everyday world, an object in a low energy state can be kicked up to a higher energy state by an input of energy – literally kicked, perhaps, like a ball being kicked up a flight of stairs. What if the negative-energy electron sea were not quite the same everywhere? Suppose an energetic interaction of some kind – perhaps the arrival of a cosmic ray from space – gave energy to one of the invisible electrons in the negative-energy sea, and kicked it up into a state with positive energy? Now, the electron would be detectable (‘visible’) to physicists as a normal electron. But it would have left behind a ‘hole’ in the negative-energy sea. Electrons have negative electrical charge, so, as Dirac pointed out at the end of the 1920s, the hole in a sea of negative charge would behave exactly like a particle with positive charge (absence, of negative being the same as presence of positive). If the hole were near a detectable visible electron, for example, negative-energy electrons in the sea would be repelled from the visible electron and would try to escape by hopping in turn into the hole; as one neighbouring invisible electron hopped in, the hole would fill up, leaving a hole where that invisible electron had been, and so on. The effect would be that the hole would move towards the visible electron, behaving just like a positively charged particle. To see what happens when the hole meets the visible electron, read on.
At this point, Dirac had a failure of nerve. Taking his equation at face value, the only physical meaning you could reasonably give to the hole would be as a particle exactly like the electron except for its positive charge. But in 1928, remember, physicists only knew two kinds of particle, the electron (with negative charge) and the proton (much more massive, but with a positive charge the same size as the electron’s negative charge). Even the neutron had not yet been discovered. So Dirac suggested in his paper that the holes in the negative-energy electron sea could be identified with protons. This really didn’t make sense, and partly as a result nobody really quite knew what to make of the notion of the negative-energy electron sea and its holes at first. But then, in 1932 the American Carl Anderson discovered traces of particles which behaved exactly like electrons but with positive charge, in cosmic ray experiments (cosmic rays are particles that arrive at the Earth from space). He concluded that the ‘new’ kind of particle was a positively charged counterpart to the electron, and gave it the name positron (an example of what is known as antimatter); it had exactly the right properties to match the behaviour of Dirac’s holes. The same year, James Chadwick, in Britain, identified the neutron.
Almost overnight, the number of kinds of individual particles known to physicists had doubled, from two to four, and their view of the physical world was transformed. You can get an idea of the dramatic impact of these discoveries on the physics community by the speed with which the Nobel committee responded to them. In 1933, Dirac received the Nobel Prize in Physics (he deserved it anyway, but the successful ‘prediction’ of positrons clinched it); in 1934, there was no award (an astonishing decision to modern eyes!); in 1935 it was Chadwick’s turn; and in 1936 Anderson received the prize.
Since then, a wealth of other subatomic particles have been discovered, and each variety has its own antimatter counterpart. All of this can be explained by variations on the hole theory, and that theory does still provide one of the best mental pictures of how energy is liberated when a particle (such as an electron) meets its antiparticle counterpart (in this case a positron) and annihilates, leaving nothing behind but a puff of energy. The electron has fallen into the positron hole, releasing energy as it does so, and both hole and electron simply disappear from the everyday world, cancelling each other out. Or, if energy is available (perhaps from an energetic photon) a negative-energy invisible electron can be kicked out of its hole and promoted into visibility, creating, along with the hole it left behind, an electron-positron pair.
But although the physical picture is simple and rather appealing (if you can live with the idea of a sea of negative-energy invisible electrons), the mathematics of the hole theory turned out to be rather cumbersome as a means of describing particle interactions. By the time Dirac received his Nobel Prize, the person who would demonstrate a much simpler way of describing interactions involving electrons and protons was just starting his final year in high school in Far Rockaway. Even though details of all the new discoveries had not yet filtered down into the standard textbooks and courses taught at universities (not even at MIT), Richard Feynman was exactly of the generation to be brought up on the new physics as an undergraduate, and to be prepared to carry things a stage (or two) further when it became time for him to make his own contributions to science. It helped, of course, to be a genius. A genius like Feynman would have made a mark on science whenever he had been born; but the accident of the timing of his birth decided the kind of mark he would make. As a member of the first generation to be brought up on quantum mechanics, he carried the triumphant, but still incomplete, theory through to its greatest fruition.
Even though the standard undergraduate textbooks might not yet tell the full story of quantum mechanics, Dirac himself had written a definitive account in his book The Principles of Quantum Mechanics, first published by Oxford University Press in 1930, which was the first comprehensive textbook on the subject. It came out in a new edition the year that Feynman set out for MIT, and the book (which later went through further revisions) is still the best introduction for serious scientists. The 1935 edition would have a profound influence on the young physicist at MIT – but at the time he started his undergraduate courses there, he didn’t even know that he was a physicist.
Notes
1. The full story of the development of Einstein’s ideas is told in Einstein: A Life in Science, by Michael White & John Gribbin (Simon & Schuster, London, 1993; Dutton, New York, 1994).
2. James Clerk Maxwell, A Dynamical Theory of the Electromagnetic Field, 1864; see, for example, Ralph Baierlein, Newton to Einstein (Cambridge University Press, 1992), p. 122.
3. See note 1.
4. If you do want the details, see John Gribbin, In Search of Schrödinger’s Cat (Bantam, New York & London, 1984).
5. But see Schrödinger’s Kittens.
6. Largely by the American Linus Pauling, who summed up the work in his book The Nature of the Chemical Bond (Cornell University Press) in 1939, and received the Nobel Prize for his work in 1954.
* Einstein’s second great theory, the General Theory of Relativity, is a field theory of gravity, and is quite different (in spite of the similarity of names) from the Special Theory of Relativity. It comes into our story later, but it had little bearing on the mainstream of physics research in the 1930s and 1940s.
3 College boy
New students at MIT had to find a fraternity which they could join, to provide them with a home and a social group within which they would fit into the college community. This system was basically a good one, in which senior students would look after freshmen in their own fraternity, teaching them the college ropes and looking out for their interests; occasionally, rivalry between fraternities and ragging of younger students by older ones got out of hand, but this doesn’t seem to have been a problem at MIT in Feynman’s time there.
For many students, the process of joining a fraternity would involve offering themselves to different fraternities, and trying to persuade them that you were a desirable prospective member of the group. For the best students, like Feynman, it worked the other way around. The fraternities sought you out, and tried to persuade you to join them. In fact, in Feynman’s case the choice (or competition) was limited. There were only two Jewish fraternities at MIT, and there was no way, in those days, that Richard could join a non-Jewish fraternity. This ‘Jewishness’ had nothing to do with religion, which Feynman had long since abandoned; it simply had to do with your family background. Both these fraternities were on the lookout for bright students, and held gatherings called ‘smokers’ to get to know boys from New York who were going to MIT.
Feynman, who still thought of himself as a mat
hematician at this time, went to both these smokers. At one, for the fraternity Phi Beta Delta, he discussed science and maths with two older students, who told him that since he knew so much maths already he could take examinations at MIT as soon as he arrived there, which would allow him to skip the first-year course and go straight on to the second-year work in the subject. Both Phi Beta Delta and the rival fraternity, Sigma Alpha Mu, were eager to enrol Feynman, who was obviously going to be the kind of student that would add lustre to their groups (but don’t run away with the idea that fraternities were only interested in academic ability; they were just as eager to attract students with other talents, such as sportsmen). Partly on the strength of the good advice he had already received from them, Feynman agreed to join Phi Beta Delta.
When the time came to leave Far Rockaway for MIT, however, some of the students from Sigma Alpha Mu called round. They would be driving up to college, and offered Feynman a ride, which he happily accepted. Like all mothers in such circumstances, Lucille watched with mixed feelings when the day came and, as arranged, her son drove off with a bunch of strangers on the journey to Boston, on what became a snowy day with tricky driving conditions. But Feynman was elated that he was being treated like an adult: ‘it was a big deal; you are grown up!’1
But the deal wasn’t quite that simple. What Sigma Alpha Mu had done, in effect, was to kidnap Feynman, hoping to enrol him with their fraternity before their rivals at Phi Beta Delta realized what was going on. They suggested, having arrived late in Boston, that he stay the night in their house, and he agreed, not realizing that he was the subject of this tug of war. In the morning, two of the seniors from Phi Beta Delta turned up to claim their own, and after some discussion Feynman finally did become a pledge at Phi Beta Delta, feeling a warm glow at being the centre of all this attention; partly as a result, he immediately began to overcome his old self-consciousness about being a sissy that everybody laughed at. The other fraternity members soon helped to develop his social skills further, although he never became what you would call a conformist in social matters.