The Quantum Universe

by Brian Cox

  Newton’s laws provide us with a very intuitive picture of the world. As we have seen, they take the form of equations – mathematical relationships between measurable quantities – that allow us to predict with precision how objects move around. Inherent in the whole framework is the assumption that objects are, at any instant, located somewhere and that, as time passes, objects move smoothly around from place to place. This seems so self-evidently true that it is hardly worth commenting upon, but we need to recognize that this is a prejudice. Can we really be sure that things are definitely here or there, and that they are not actually in two different places at the same time? Of course, your garden shed is not in any noticeable sense sitting in two distinctly different places at once – but how about an electron in an atom? Could that be both ‘here’ and ‘there’? Right now that kind of suggestion sounds crazy, mainly because we can’t picture it in our mind’s eye, but it will turn out to be the way things actually work. At this stage in our narrative, all we are doing in making this strange-sounding statement is pointing out that Newton’s laws are built on intuition, and that is like a house built on sand as far as fundamental physics is concerned.

  There is a very simple experiment, first conducted by Clinton Davisson and Lester Germer at Bell Laboratories in the United States and published in 1927, which shows that Newton’s intuitive picture of the world is wrong. Although apples, planets and people certainly appear to behave in a ‘Newtonian’ way, gliding from place to place in a regular and predictable fashion as time unfolds, their experiment showed that the fundamental building blocks of matter do not behave at all like this.

  Davisson and Germer’s paper begins: ‘The intensity of scattering of a homogeneous beam of electrons of adjustable speed incident upon a single crystal of nickel has been measured as a function of direction.’ Fortunately, there is a way to appreciate the key content of their findings using a simplified version of their experiment, known as the double-slit experiment. The experiment consists of a source that sends electrons towards a barrier with two small slits (or holes) cut into it. On the other side of the barrier, there is a screen that glows when an electron hits it. It doesn’t matter what the source of electrons is, but practically speaking one can imagine a length of hot wire stretched out along the side of the experiment.2 We’ve sketched the double-slit experiment in Figure 2.2.

  Imagine pointing a camera at the screen and leaving the shutter open to take a long-exposure photograph of the little flashes of light emitted as, one by one, the electrons hit it. A pattern will build up, and the simple question is, what is the pattern? Assuming electrons are simply little particles that behave rather like apples or planets, we might expect the emergent pattern to look something like that shown in Figure 2.2. Some electrons go through the slits, most don’t. The ones that make it through might bounce off the edge of the slits a bit, which will spread them out, but the most hits, and therefore the brightest bits of the photograph, will surely appear directly aligned with the two slits.

  Figure 2.2: An electron-gun source fires electrons towards a pair of slits and, if the electrons behaved like ‘regular’ particles, we would expect to see hits on the screen that build up a pair of stripes, as illustrated. Remarkably, this is not what happens.

  Figure 2.3: In reality the electrons do not hit the screen aligned with the slits. Instead they form a stripy pattern: electron by electron, the stripes build up over time.

  This isn’t what happens. Instead, the picture looks like Figure 2.3. A pattern like this is what Davisson and Germer published in their 1927 paper. Davisson subsequently received the 1937 Nobel Prize for the ‘experimental discovery of electron diffraction by crystals’. He shared the prize, not with Germer, but with George Paget Thomson, who saw the same pattern independently in experiments at the University of Aberdeen. The alternating stripes of light and dark are known as an interference pattern, and interference is more usually associated with waves. To understand why, let’s imagine doing the double-slit experiment with water waves instead of electrons.

  Imagine a water tank with a wall midway down with two slits cut into it. The screen and camera could be replaced with a wave-height detector, and the hot wire with something that makes waves: a plank of wood along the side of the tank attached to a motor that keeps it dipping in and out of the water would do. The waves from the plank will travel across the surface of the water until they reach the wall. When a wave hits the wall, most of it will bounce back, but two small pieces will pass through the slits. These two new waves will spread outwards from the slits towards the wave-height detector. Notice that we used the term ‘spread out’ here, because the waves don’t just carry on in a straight line from the slits. Instead, the slits act as two sources of new waves, each issuing forth in ever increasing semi-circles. Figure 2.4 illustrates what happens.

  Figure 2.4. An aerial view of water waves emanating from two points in a tank of water (they are located at the top of the picture). The two circular waves overlap and interfere with each other. The ‘spokes’ are the regions where the two waves have cancelled each other out and the water there remains undisturbed.

  The figure provides a striking visual demonstration of the behaviour of waves in water. There are regions where there are no waves at all, which seem to radiate out from the slits like the spokes of a wheel, whilst other regions are still filled with the peaks and troughs of the waves. The parallels with the pattern seen by Davisson, Germer and Thomson are striking. For the case of electrons hitting the screen, the regions where few electrons are detected correspond to the places in the tank where the water surface remains flat – the spokes you can see radiating outwards in the figure.

  In a tank of water it is quite easy to understand how these spokes emerge: it is in the mixing and merging of the waves as they spread out from the slits. Because waves have peaks and troughs, when two waves meet they can either add or subtract. If two waves meet such that the peak of one is aligned with the trough of the other, they will cancel out and there will be no wave at that point. At a different place, the waves might arrive with their peaks in perfect alignment, and here they will add to produce a bigger wave. At each point in the water tank, the distance between it and the two slits will be a little different, which means that at some places the two waves will arrive with peaks together, at others with peaks and troughs aligned and, in most places, with some combination of these two extremes. The result will be an alternating pattern; an interference pattern.

  In contrast to water waves, the experimentally observed fact that electrons also produce an interference pattern is very difficult to understand. According to Newton and common sense, the electrons emerge from the source, travel in straight lines towards the slits (because there are no forces acting on them – remember Newton’s first law), pass through with perhaps a slight deflection if they glance off the edge of the slit, and continue in a straight line until they hit the screen. But this would not result in an interference pattern – it would give the pair of stripes as shown in Figure 2.2. Now we could suppose that there is some clever mechanism whereby the electrons exert a force on each other so as to deflect themselves from straight lines as they stream through the slits. But this can be ruled out because we can set the experiment up such that we send just one electron at a time from source to screen. You would have to wait, but, slowly and surely, as the electrons hit the screen one after the other, the stripy pattern would build up. This is very surprising because the stripy pattern is absolutely characteristic of waves interfering with each other, yet it emerges one electron at a time – dot by dot. It’s a good mental exercise to try to imagine how it could be that, particle by particle, an interference pattern builds up as we fire tiny bullet-like particles at a pair of slits in a screen. It’s a good exercise because it’s futile, and a few hours of brain racking should convince you that a stripy pattern is inconceivable. Whatever those particles are that hit the screen, they are not behaving like ‘regular’ particles. It is as if the
electrons are in some sense ‘interfering with themselves’. The challenge for us is to come up with a theory that can explain what that means.

  There is an interesting historical coda to this story, which provides a glimpse into the intellectual challenge raised by the double-slit experiment. George Paget Thomson was the son of J. J. Thomson, who himself received a Nobel Prize for his discovery of the electron in 1899. J. J. Thomson showed that the electron is a particle, with a particular electric charge and a particular mass; a tiny, point-like grain of matter. His son received the Nobel Prize forty years later for showing that the electron doesn’t behave as his father might have expected. Thomson senior was not wrong; the electron does have a well-defined mass and electric charge, and every time we see one it appears as a little point of matter. It just doesn’t seem to behave exactly like a regular particle, as Davisson, Germer and Thomson junior discovered. Importantly, though, it doesn’t behave exactly like a wave either because the pattern is not built up as a result of some smooth deposition of energy; rather it is built out of many tiny dots. We always detect Thomson senior’s single, point-like electrons.

  Perhaps you can already see the need to engage with Heisenberg’s way of thinking. The things we observe are particles, so we had better construct a theory of particles. Our theory must also be able to predict the appearance of the interference pattern that builds up as the electrons, one after another, pass through the slits and hit the screen. The details of how the electrons travel from source to slits to screen are not something we observe, and therefore need not be in accord with anything we experience in daily life. Indeed, the electron’s ‘journey’ need not even be something we can talk about at all. All we have to do is find a theory capable of predicting that the electrons hit the screen in the pattern observed in the double-slit experiment. This is what we will do in the next chapter.

  Lest we lapse into thinking that this is merely a fascinating piece of micro-physics that has little relevance to the world at large, we should say that the quantum theory of particles we develop to explain the double-slit experiment will also turn out to be capable of explaining the stability of atoms, the coloured light emitted from the chemical elements, radioactive decay, and indeed all of the great puzzles that perplexed scientists at the turn of the twentieth century. The fact that our framework describes the way electrons behave when locked away inside matter will also allow us to understand the workings of quite possibly the most important invention of the twentieth century: the transistor.

  In the very final chapter of this book, we will meet a striking application of quantum theory that is one of the great demonstrations of the power of scientific reasoning. The more outlandish predictions of quantum theory usually manifest themselves in the behaviour of small things. But, because large things are made of small things, there are certain circumstances in which quantum physics is required to explain the observed properties of some of the most massive objects in the Universe – the stars. Our Sun is fighting a constant battle with gravity. This ball of gas a third of a million times more massive than our planet has a gravitational force at its surface that is almost twenty-eight times that at the Earth, which provides a powerful incentive for it to collapse in on itself. The collapse is prevented by the outward pressure generated by nuclear fusion reactions deep within the solar core as 600 million tonnes of hydrogen are converted into helium every second. Vast though our star is, burning fuel at such a ferocious rate must ultimately have consequences, and one day the Sun’s fuel source will run out. The outward pressure will then cease and the force of gravity will reassert its grip unopposed. It would seem that nothing in Nature could stop a catastrophic collapse.

  In reality, quantum physics steps in and saves the day. Stars that have been rescued by quantum effects in this way are known as white dwarves, and such will be the final fate of our Sun. At the end of this book we will employ our understanding of quantum mechanics to determine the maximum mass of a white dwarf star. This was first calculated, in 1930, by the Indian astrophysicist Subrahmanyan Chandrasekhar, and it turns out to be approximately 1.4 times the mass of our Sun. Quite wonderfully, that number can be computed using only the mass of a proton and the values of the three constants of Nature we have already met: Newton’s gravitational constant, the speed of light, and Planck’s constant.

  The development of the quantum theory itself and the measurement of these four numbers could conceivably have been achieved without ever looking at the stars. It is possible to imagine a particularly agoraphobic civilization confined to deep caves below the surface of their home planet. They would have no concept of a sky, but they could have developed quantum theory. Just for fun, they may even decide to calculate the maximum mass of a giant sphere of gas. Imagine that, one day, an intrepid explorer chooses to venture above ground for the first time and gaze in awe at the spectacle above: a sky full of lights; a galaxy of a hundred billion suns arcing from horizon to horizon. The explorer would find, just as we have found from our vantage point here on Earth, that out there amongst the many fading remnants of dying stars there is not a single one with a mass exceeding the Chandrasekhar limit.

  3. What Is a Particle?

  Our approach to quantum theory was pioneered by Richard Feynman, the Nobel Prize-winning, bongo-playing New Yorker described by his friend and collaborator Freeman Dyson as ‘half genius, half buffoon’. Dyson later changed his opinion: Feynman could be more accurately described as ‘all genius, all buffoon’. We will follow his approach in our book because it is fun, and probably the simplest route to understanding our Quantum Universe.

  As well as being responsible for the simplest formulation of quantum mechanics, Richard Feynman was also a great teacher, able to transfer his deep understanding of physics to the page or lecture theatre with unmatched clarity and a minimum of fuss. His style was contemptuous of those who might seek to make physics more complicated than it need be. Even so, at the beginning of his classic undergraduate textbook series The Feynman Lectures on Physics, he felt the need to be perfectly honest about the counterintuitive nature of the quantum theory. Subatomic particles, Feynman wrote, ‘do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything that you have ever seen’. Let’s get on with building a model for exactly how they do behave.

  As our starting point we will assume that the elemental building blocks of Nature are particles. This has been confirmed not only by the double-slit experiment, where the electrons always arrive at specific places on the screen, but by a whole host of other experiments. Indeed ‘particle physics’ is not called that for nothing. The question we need to address is: how do particles move around? Of course, the simplest assumption would be that they move in nice straight lines, or curved lines when acted upon by forces, as dictated by Newton. This cannot be correct though, because any explanation of the double-slit experiment requires that the electrons ‘interfere with themselves’ when they pass through the slits, and to do that they must in some sense be spread out. This therefore is the challenge: build a theory of point-like particles such that those same particles are also spread out. This is not as impossible as it sounds: we can do it if we let any single particle be in many places at once. Of course, that may still sound impossible, but the proposition that a particle should be in many places at once is actually a rather clear statement, even if it sounds silly. From now on, we’ll refer to these counterintuitive, spread-out-yet-point-like particles as quantum particles.

  With this ‘a particle can be in more than one place at once’ proposal, we are moving away from our everyday experience and into uncharted territory. One of the major obstacles to developing an understanding of quantum physics is the confusion this kind of thinking can engender. To avoid confusion, we should follow Heisenberg and learn to feel comfortable with views of the world that run counter to tangible experience. Feeling ‘uncomfortable’ can be mistaken for ‘confusion’, and very
often students of quantum physics continue to attempt to understand what is happening in everyday terms. It is the resistance to new ideas that actually leads to confusion, not the inherent difficulty of the ideas themselves, because the real world simply doesn’t behave in an everyday way. We must therefore keep an open mind and not be distressed by all the weirdness. Shakespeare had it right when Hamlet says, ‘And therefore as a stranger give it welcome. There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy.’

  A good way to begin is to think carefully about the double-slit experiment for water waves. Our aim will be to work out just what it is about waves that causes the interference pattern. We should then make sure that our theory of quantum particles is capable of encapsulating this behaviour, so that we can have a chance of explaining the double-slit experiment for electrons.