The Quantum Universe

by Brian Cox

  There are two reasons why waves journeying through two slits can interfere with themselves. The first is that the wave travels through both of the slits at once, creating two new waves that head off and mix together. It’s obvious that a wave can do this. We have no problem visualizing one long, ocean wave rolling to the shore and crashing on to a beach. It is a wall of water; an extended, travelling thing. We are therefore going to need to decide how to make our quantum particle ‘an extended, travelling thing’. The second reason is that the two new waves heading out from the slits are able either to add or to subtract from each other when they mix. This ability for two waves to interfere is clearly crucial in explaining the interference pattern. The extreme case is when the peak of one wave coincides with the trough of another, in which case they completely cancel each other out. So we are also going to need to allow our quantum particle to interfere somehow with itself.

  Figure 3.1. How the wave describing an electron moves from source to screen and how it should be interpreted as representing all of the ways that the electron travels. The paths A to C to E and B to D to F illustrate just two of the infinity of possible paths the single electron does take.

  The double-slit experiment connects the behaviour of electrons and the behaviour of waves, so let us see how far we can push the connection. Take a look at Figure 3.1 and, for the time being, ignore the lines joining A to E and B to F and concentrate on the waves. The figure could then describe a water tank, with the wavy lines representing, from left to right, how a water wave rolls its way across the tank. Imagine taking a photograph of the tank just after a plank of wood has splashed in on the left-hand side to make a wave. The snapshot would reveal a newly formed wave that extends from top to bottom in the picture. All the water in the rest of the tank would be calm. A second snapshot taken a little later reveals that the water wave has moved towards the slits, leaving flat water behind it. Later still, the water wave passes through the pair of slits and generates the stripy interference pattern illustrated by the wavy lines on the far right.

  Now let us reread that last paragraph but replace ‘water wave’ with ‘electron wave’, whatever that may mean. An electron wave, suitably interpreted, has the potential to explain the stripy pattern we want to understand as it rolls through the experiment like a water wave. But we do need to explain why the electron pattern is made up of tiny dots as the electrons hit the screen one by one. At first sight that seems in conflict with the idea of a smooth wave, but it is not. The clever bit is to realize that we can offer an explanation if we interpret the electron wave not as a real material disturbance (as is the case with a water wave), but rather as something that simply informs us where the electron is likely to be found. Notice we said ‘the’ electron because the wave is to describe the behaviour of a single electron – that way we have a chance of explaining how those dots emerge. This is an electron wave, and not a wave of electrons: we must never fall into the trap of thinking otherwise. If we imagine a snapshot of the wave at some instant in time, then we want to interpret it such that where the wave is largest the electron is most likely to be found, and where the wave is smallest the electron is least likely to be found. When the wave finally reaches the screen, a little spot appears and informs us of the location of the electron. The sole job of the electron wave is to allow us to compute the odds that the electron hits the screen at some particular place. If we do not worry about what the electron wave actually ‘is’, then everything is straightforward because once we know the wave then we can say where the electron is likely to be. The fun comes next, when we try to understand what this proposal for an electron wave implies for the electron’s journey from slit to screen.

  Before we do this, it might be worth reading the above paragraph again because it is very important. It’s not supposed to be obvious and it is certainly not intuitive. The ‘electron wave’ proposal has all the necessary properties to explain the appearance of the experimentally observed interference pattern, but it is something of a guess as to how things might work out. As good physicists we should work out the consequences and see if they correspond to Nature.

  Returning to Figure 3.1, we have proposed that at each instant in time the electron is described by a wave, just as in the case of water waves. At an early time, the electron wave is to the left of the slits. This means that the electron is in some sense located somewhere within the wave. At a later time, the wave will advance towards the slits just as the water wave did, and the electron will now be somewhere in the new wave. We are saying that the electron ‘could be first at A and then at C’, or it ‘could be first at B and then at D’, or it ‘could be at A and then at D’, and so on. Hold that thought for a minute, and think about an even later time, after the wave has passed through the slits and reached the screen. The electron could now be found at E or perhaps at F. The curves that we have drawn on the diagram represent two possible paths that the electron could have taken from the source, through the slits and onto the screen. It could have gone from A to C to E, and it could have gone from B to D to F. These are just two out of an infinite number of possible paths that the electron could have taken.

  The crucial point is that it makes no sense to say that ‘the electron could have ventured along each of these routes, but really it went along only one of them’. To say that the electron really ventured along one particular path would be to give ourselves no more of a chance of explaining the interference pattern than if we had blocked up one of the slits in the water wave experiment. We need to allow the wave to go through both slits in order to get an interference pattern, and this means that we must allow all the possible paths for the electron to travel from source to screen. Put another way, when we said that the electron is ‘somewhere within the wave’ we really meant to say that it is simultaneously everywhere in the wave! This is how we must think because if we suppose the electron is actually located at some specific point, then the wave is no longer spread out and we lose the water wave analogy. As a result, we cannot explain the interference pattern.

  Again, it might be worth rereading the above piece of reasoning because it motivates much of what follows. There is no sleight of hand: what we are saying is that we need to describe a spread-out wave that is also a point-like electron, and one possible way to achieve this is to say that the electron sweeps from source to screen following all possible paths at once.

  This suggests that we should interpret an electron wave as describing a single electron that travels from source to screen by an infinity of different routes. In other words, the correct answer to the question ‘how did that electron get to the screen’ is ‘it travelled by an infinity of possible routes, some of which went through the upper slit and some of which went though the lower one’. Clearly the ‘it’ that is the electron is not an ordinary, everyday particle. This is what it means to be a quantum particle.

  Having decided to seek a description of an electron that mimics in many ways the behaviour of waves, we need to develop a more precise way to talk about waves. We shall begin with a description of what is happening in a water tank when two waves meet, mix and interfere with each other. To do this, we must find a convenient way of representing the positions of the peaks and troughs of each wave. In the technical jargon, these are known as phases. Colloquially things are described as ‘in phase’ if they reinforce one another in some way, or ‘out of phase’ if they cancel each other out. The word is also used to describe the Moon: over the course of around twenty-eight days, the Moon passes from new to full and back again in a continuous waxing and waning cycle. The etymology of the word ‘phase’ stems from the Greek phasis, which means the appearance and disappearance of an astronomical phenomenon, and the regular appearance and disappearance of the bright lunar surface seems to have led to its twentieth-century usage, particularly in science, as a description of something cyclical. And this is a clue as to how we might find a pictorial representation of the positions of the peaks and troughs of water waves.


  Figure 3.2. The phases of the Moon.

  Have a look at Figure 3.2. One way to represent a phase is as a clock face with a single hand rotating around. This gives us the freedom to represent visually a full 360 degrees worth of possibilities: the clock hand can point to 12 o’clock, 3 o’clock, 9 o’clock and all points in between. In the case of the Moon, you could imagine a new Moon represented by a clock hand pointing to 12 o’clock, a waxing crescent at 1:30, the first quarter at 3, the waxing gibbous at 4:30, the full Moon at 6 and so on. What we are doing here is using something abstract to describe something concrete; a clock face to describe the phases of the Moon. In this way we could draw a clock with its hand pointing to 12 o’clock and you’d immediately know that the clock represented a new Moon. And even though we haven’t actually said it, you’d know that a clock with the hand pointing to 5 o’clock would mean that we are approaching a full Moon. The use of abstract pictures or symbols to represent real things is absolutely fundamental in physics – this is essentially what physicists use mathematics for. The power of the approach comes when the abstract pictures can be manipulated using simple rules to make firm predictions about the real world. As we’ll see in a moment, the clock faces will allow us to do just this because they are able to keep track of the relative positions of the peaks and troughs of waves. This in turn will allow us to calculate whether they will cancel or reinforce one another when they meet.

  Figure 3.3 shows a sketch of two water waves at an instant in time. Let’s represent the peaks of the waves by clocks reading 12 o’clock and the troughs by clocks reading 6 o’clock. We can also represent places on the waves intermediate between peaks and troughs with clocks reading intermediate times, just as we did for the phases of the Moon between new and full. The distance between the successive peaks and troughs of the wave is an important number; it is known as the wavelength of the wave.

  The two waves in Figure 3.3 are out of phase with each other, which means that the peaks of the top wave are aligned with the troughs of the bottom wave, and vice versa. As a result it is pretty clear that they will entirely cancel each other out when we add them together. This is illustrated at the bottom of the figure, where the ‘wave’ is flat-lining. In terms of clocks, all of the 12 o’clock clocks for the top wave, representing its peaks, are aligned with the 6 o’clock clocks for the bottom wave, representing its troughs. In fact, everywhere you look, the clocks for the top wave are pointing in the opposite direction to the clocks for the bottom wave.

  Using clocks to describe waves does, at this stage, seem like we are over-complicating matters. Surely if we want to add together two water waves, then all we need to do is add the heights of each of the waves and we don’t need clocks at all. This is certainly true for water waves, but we are not being perverse and we have introduced the clocks for a very good reason. We will discover soon enough that the extra flexibility they allow is absolutely necessary when we come to use them to describe quantum particles.

  Figure 3.3. Two waves arranged such that they cancel out completely. The top wave is out of phase with the second wave, i.e. peaks align with troughs. When the two waves are added together they cancel out to produce nothing – as illustrated at the bottom where the ‘wave’ is flat-lining.

  With this in mind, we shall now spend a little time inventing a precise rule for adding clocks. In the case of Figure 3.3, the rule must result in all the clocks ‘cancelling out’, leaving nothing behind: 12 o’clock cancels 6 o’clock, 3 o’clock cancels 9 o’clock and so on. This perfect cancellation is, of course, for the special case when the waves are perfectly out of phase. Let’s search for a general rule that will work for the addition of waves of any alignment and shape.

  Figure 3.4 shows two more waves, this time aligned in a different way, such that one is only slightly offset against the other. Again, we have labelled the peaks, troughs and points in between with clocks. Now, the 12 o’clock clock of the top wave is aligned with the 3 o’clock clock of the bottom wave. We are going to state a rule that allows us to add these two clocks together. The rule is that we take the two hands and stick them together head to tail. We then complete the triangle by drawing a new hand joining the other two hands together. We have sketched this recipe in Figure 3.5. The new hand will be a different length to the other two, and point in a different direction; it is a new clock face, which is the sum of the other two.

  Figure 3.4. Two waves offset relative to each other. The top and middle waves add together to produce the bottom wave.

  We can be more precise now and use simple trigonometry to calculate the effect of adding together any specific pair of clocks. In Figure 3.5 we are adding together the 12 o’clock and 3 o’clock clocks. Let’s suppose that the original clock hands are of length 1 cm (corresponding to water waves of peak height equal to 1 cm). When we place the hands head-to-tail we have a right-angled triangle with two sides each of length 1 cm. The new clock hand will be the length of the third side of the triangle: the hypotenuse. Pythagoras’ Theorem tells us that the square of the hypotenuse is equal to the sum of the squares of the other two sides: h2 = x2 + y2. Putting the numbers in, h2 = 12 + 12 = 2. So the length of the new clock hand h is the square root of 2, which is approximately 1.414 cm. In what direction will the new hand point? For this we need to know the angle in our triangle, labelled θ in the figure. For the particular example of two hands of equal length, one pointing to 12 o’clock and one to 3 o’clock, you can probably work it out without knowing any trigonometry at all. The hypotenuse obviously points at an angle of 45 degrees, so the new ‘time’ is half way between 12 o’clock and 3 o’clock, which is half past one. This example is a special case, of course. We chose the clocks so that the hands were at right angles and of the same length to make the mathematics easy. But it is obviously possible to work out the length of the hand and time resulting from the addition of any pair of clock faces.

  Figure 3.5. The rule for adding clocks.

  Now look again at Figure 3.4. At every point along the new wave, we can compute the wave height by adding the clocks together using the recipe we just outlined and asking how much of the new clock hand points in the 12 o’clock direction. When the clock points to 12 o’clock this is obvious – the height of the wave is simply the length of the clock hand. Similarly at 6 o’clock, it’s obvious because the wave has a trough with a depth equal to the length of the hand. It’s also pretty obvious when the clock reads 3 o’clock (or 9 o’clock) because then the wave height is zero, since the clock hand is at right angles to the 12 o’clock direction. To compute the wave height described by any particular clock we should multiply the length of the hand, h, by the cosine of the angle the hand makes with the 12 o’clock direction. For example, the angle that a 3 o’clock makes with 12 o’clock is 90 degrees and the cosine of 90 degrees is zero, which means the wave height is zero. Similarly, a time of half-past-one corresponds to an angle of 45 degrees with the 12 o’clock direction and the cosine of 45 degrees is approximately 0.707, so the height of the wave is 0.707 times the length of the hand (notice that 0.707 is 1/√2). If your trigonometry is not up to those last few sentences then you can safely ignore the details. It’s the principle that matters, which is that, given the length of a clock hand and its direction you can go ahead and calculate the wave height – and even if you don’t understand trigonometry you could make a good stab at it by carefully drawing the clock hands and projecting on to the 12 o’clock direction using a ruler. (We would like to make it very clear to any students reading this book that we do not recommend this course of action: sines and cosines are useful things to understand.)

  That’s the rule for adding clocks, and it works a treat, as illustrated in the bottom of the three pictures in Figure 3.4, where we have repeatedly applied the rule for various points along the waves.

  Figure 3.6. Three different clocks all with the same projection in the 12 o’clock direction.

  In this description of water waves, all that ever matters is the project
ion of the ‘time’ in the 12 o’clock direction, corresponding to just one number: the wave height. That is why the use of clocks is not really necessary when it comes to describing water waves. Take a look at the three clocks in Figure 3.6: they all correspond to the same wave height and so they provide equivalent ways of representing the same height of water. But clearly they are different clocks and, as we shall see, these differences do matter when we come to use them to describe quantum particles because, for them, the length of the clock hand (or equivalently the size of the clock) has a very important interpretation.

  At some points in this book and at this point especially, things are abstract. To keep ourselves from succumbing to dizzying confusion, we should remember the bigger picture. The experimental results of Davisson, Germer and Thomson, and their similarity with the behaviour of water waves, have inspired us to make an ansatz: we should represent a particle by a wave, and the wave itself can be represented by lots of clocks. We imagine that the electron wave propagates ‘like a water wave’, but we haven’t explained how that works in any detail. But then we never said how the water wave propagates either. All that matters for the moment is that we recognize the analogy with water waves, and the notion that the electron is described at any instant by a wave that propagates and interferes like water waves do. In the next chapter we will do better than this and be more precise about how an electron actually moves around as time unfolds. In doing that we will be led to a host of treasures, including Heisenberg’s famous Uncertainty Principle.

  Before we move on to that, we want to spend a little time talking about the clocks that we are proposing to represent the electron wave. We emphasize that these clocks are not real in any sense, and their hour hand has absolutely nothing to do with what time of day it is. This idea of using an array of little clocks to describe a real physical phenomenon is not so bizarre a concept as it may seem. Physicists use similar techniques to describe many things in Nature, and we have already seen how they can be used to describe water waves.