The Quantum Universe

by Brian Cox

  This might sound very abstract, but you may well use technology based on Fourier’s ideas every day, because the decomposition of a wave into its component sine waves is the foundation of audio and video compression technology. Think about the sound waves that make up your favourite tune. This complicated wave can, as we have just learnt, be broken down into a series of numbers that give the relative contributions of each of a large number of pure sine waves to the sound. It turns out that, although you may need a vast number of individual sine waves to reproduce the original sound wave exactly, you can in fact throw a lot of them away without compromising the perceived audio quality at all. In particular, the sine waves that contribute to sound waves that humans can’t hear are not kept. This vastly reduces the amount of data needed to store an audio file – hence your mp3 player doesn’t need to be too large.

  We might also ask what possible use could this different and even more abstract version of the wavefunction be? Well, think of a particle represented, in position space, by a single clock. This describes a particle located at a certain place in the Universe; the single point where the clock sits. Now think of a particle represented by a single clock, but this time in momentum space. This represents a particle with a single, definite momentum. Describing such a particle using the position space wavefunction would, in contrast, require an infinite number of equally sized clocks, because according to the Uncertainty Principle, a particle with a definite momentum can be found anywhere. As a result, it is sometimes simpler to perform calculations directly in terms of the momentum space wavefunction.

  In this chapter, we have learnt that the description of a particle in terms of clocks is capable of capturing what we ordinarily call ‘movement’. We have learnt that our perception that objects move smoothly from point to point is, from the perspective of quantum theory, an illusion. It is closer to the truth to suppose that particles move from A to B via all possible paths. Only when we add together all of the possibilities does motion as we perceive it emerge. We have also seen explicitly how the clock description manages to encode the physics of waves, even though we only ever deal with point-like particles. It is time now to really exploit the similarity with the physics of waves as we tackle the important question: how does quantum theory explain the structure of atoms?

  6. The Music of the Atoms

  The interior of an atom is a strange place. If you could stand on a proton and gaze outwards into inter-atomic space, you would see only void. The electrons would still be imperceptibly tiny even if they approached close enough for you to touch them, which they very rarely would. The proton is around 10−15 m in diameter, 0.000000000000001 metres, and is a quantum colossus compared to the electrons. If you stand on your proton at the edge of England on the White Cliffs of Dover, the fuzzy edge of the atom lies somewhere amongst the farms of northern France. Atoms are vast and empty, which means the full-size you is vast and empty too. Hydrogen is the simplest atom, comprising a single proton and a single electron. The electron, vanishingly small as far as we can tell, might seem to have a limitless arena within which to roam, but this is not true. It is bound to its proton, trapped by their mutual electromagnetic attraction, and it is the size and shape of this generous prison that gives rise to the characteristic barcode rainbow of light meticulously documented in the Handbuch der Spectroscopie by our old friend and dinner-party guest Professor Kayser.

  We are now in a position to apply the knowledge we have accumulated so far to the question that so puzzled Rutherford, Bohr and others in the early decades of the twentieth century: what exactly is going on inside an atom? The problem, if you recall, was that Rutherford discovered that the atom is in some ways like a miniature solar system, with a dense nucleus Sun at the centre and electrons as planets sweeping around in distant orbits. Rutherford knew that this model couldn’t be right, because electrons in orbit around a nucleus should continually emit light. The result should be catastrophic for the atom, because if the electron continually emits light then it must lose energy and spiral inwards on an inevitable collision course with the proton. This, of course, doesn’t happen. Atoms tend to be stable things, so what is wrong with this picture?

  This chapter marks an important stage in the book, because it is the first time that our theory is to be used to explain real-world phenomena. All our hard work to this point has been concerned with getting the essential formalism worked out so that we have a way to think about a quantum particle. Heisenberg’s Uncertainty Principle and the de Broglie equation represent the pinnacle of our achievements, but in the main we have been modest, thinking about a universe containing just one particle. It is now time to show how quantum theory impacts on the everyday world in which we live. The structure of atoms is a very real and tangible thing. You are made of atoms: their structure is your structure, and their stability is your stability. It would not be unduly hyperbolic to say that understanding the structure of atoms is one of the necessary conditions for understanding our Universe as a whole.

  Inside a hydrogen atom, the electron is trapped in a region surrounding the proton. We are going to start by imagining that the electron is trapped in some sort of box, which is not very far from the truth. Specifically, we’ll investigate to what extent the physics of an electron trapped inside a tiny box captures the salient features of a real atom. We are going to proceed by exploiting what we learnt in the previous chapter about the wave-like properties of quantum particles, because, when it comes to describing atoms, the wave picture really simplifies things and we can make a good deal of progress without having to worry about shrinking, winding and adding clocks. Always bear in mind, though, that the waves are a convenient shorthand for what is going on ‘under the bonnet’.

  Because the framework we’ve developed for quantum particles is extremely similar to that used in the description of water waves, sound waves or the waves on a guitar string, we’ll think first about how these more familiar material waves behave when they are confined in some way.

  Figure 6.1. Six successive snapshots of a standing wave in a tank of water. The time advances from the top left to the bottom right.

  Generally speaking, waves are complicated things. Imagine jumping into a swimming pool full of water. The water will slosh around all over the place, and it would seem to be futile to try to describe what is going on in any simple fashion. Underlying the complexity, however, there is hidden simplicity. The key point is that the water in a swimming pool is confined, which means that all the waves are trapped inside the pool. This gives rise to a phenomenon known as ‘standing waves’. The standing waves are hidden away in the mess when we disturb the pool by jumping into it, but there is a way to make the water move so that it oscillates in the regular, repeating patterns of the standing waves. Figure 6.1 shows how the water surface looks when it is undergoing one such oscillation. The peaks and troughs rise and fall, but most importantly they rise and fall in exactly the same place. There are other standing waves too, including one where the water in the middle of the tank rises and falls rhythmically. We do not usually see these special waves because they are hard to produce, but the key point is that any disturbance of the water at all – even the one we caused by our inelegant dive and subsequent thrashing around – can be expressed as some combination or other of the different standing waves. We’ve seen this type of behaviour before; it is a direct generalization of Fourier’s ideas that we encountered in the last chapter. There, we saw that any wave packet can be built up out of a combination of waves each of definite wavelength. These special waves, representing particle states of definite momentum, are sine waves. In the case of confined water waves, the idea generalizes so that any disturbance can always be described using some combination of standing waves. We’ll see later in this chapter that standing waves have an important interpretation in quantum theory, and in fact they hold the key to understanding the structure of atoms. With this in mind, let’s explore them in a little more detail.

  Figure 6.2. The th
ree longest wavelength waves that can fit on a guitar string. The longest wavelength (at the top) corresponds to the lowest harmonic (fundamental) and the others correspond to the higher harmonics (overtones).

  Figure 6.2 shows another example of standing waves in Nature: three of the possible standing waves on a guitar string. On plucking a guitar string, the note we hear is usually dominated by the standing wave with the largest wavelength – the first of the three waves shown in the figure. This is known in both physics and music as the ‘lowest harmonic’ or ‘fundamental’. Other wavelengths are usually present too, and they are known as overtones or higher harmonics. The other waves in the figure are the two longest-wavelength overtones. The guitar is a nice example because it’s simple enough to see why a guitar string can only vibrate at these special wavelengths. It is because it is held fixed at both ends – by the guitar bridge at one end and your finger pressing against a fret at the other. This means that the string cannot move at these two points, and this determines the allowed wavelengths. If you play the guitar, you’ll know this physics instinctively; as you move your fingers up the fret board towards the bridge, you decrease the length of the string and therefore force it to vibrate with shorter and shorter wavelengths, corresponding to higher-pitched notes.

  The lowest harmonic is the wave that has only two stationary points, or ‘nodes’; it moves everywhere except at the two fixed ends. As you can see from the figure, this note has a wavelength of twice the length of the string. The next smallest wavelength is equal to the length of the string, because we can fit another node in the centre. Next, we can get a wave with wavelength equal to ⅔ times the length of the string, and so on.

  In general, just as in the case of the water confined in a swimming pool, the string will vibrate in some combination of the different possible standing waves, depending on how it is plucked. The actual shape of the string can always be obtained by adding together the standing waves corresponding to each of the harmonics present. The harmonics and their relative sizes give the sound its characteristic tone. Different guitars will have different distributions of harmonics and therefore sound different, but a middle C (a pure harmonic) on one guitar is always the same as a middle C on another. For the guitar, the shape of the standing waves is very simple: they are pure sine waves whose wavelengths are fixed by the length of the string. For the swimming pool, the standing waves are more complicated, as shown in Figure 6.1, but the idea is exactly the same.

  You may be wondering why these special waves are called ‘standing waves’. It is because the waves do not change their shape. If we take two snapshots of a guitar string vibrating in a standing wave, then the two pictures will only differ in the overall size of the wave. The peaks will always be in the same place, and the nodes will always be in the same place because they are fixed by the ends of the string or, in the case of the swimming pool, by the sides of the pool. Mathematically, we could say that the waves in the two snapshots differ only by an overall multiplicative factor. This factor varies periodically with time, and expresses the rhythmical vibration of the string. The same is true for the swimming pool in Figure 6.1, where each snapshot is related to the others by an overall multiplicative factor. For example, the last snapshot can be obtained from the first by multiplying the wave height at every point by minus one.

  In summary, waves that are confined in some way can always be expressed in terms of standing waves (waves that do not change their shape) and, as we have said, there are very good reasons for devoting so much time to understanding them. At the top of the list is the fact that standing waves are quantized. This is very clear for the standing waves on a guitar string: the fundamental has a wavelength of twice the length of the string, and the next longest allowed wavelength is equal to the length of the string. There is no standing wave with a wavelength in between these two and so we can say that the allowed wavelengths on a guitar string are quantized.

  Standing waves therefore make manifest the fact that something gets quantized when we trap waves. In the case of a guitar string, it is clearly the wavelength. For the case of an electron inside a box, the quantum waves corresponding to the electron will also be trapped, and by analogy we should expect that only certain standing waves will be present in the box, and therefore that something will be quantized. Other waves simply cannot exist, just as a guitar string doesn’t play all the notes in an octave at the same time no matter how it is plucked. And just as for the sound of a guitar, the general state of the electron will be described by a blend of standing waves. These quantum standing waves are starting to look very interesting, and, encouraged by this, let’s start our analysis proper.

  To make progress, we must be specific about the shape of the box inside which we place our electron. To keep things simple, we’ll suppose that the electron is free to hop around inside a region of size L, but that it is totally forbidden from wandering outside this region. We do not need to say how we intend to forbid the electron from wandering – but if this is supposed to be a simplified model of an atom then we should imagine that the force exerted by the positively charged nucleus is responsible for its confinement. In the jargon, this is known as a ‘square well potential’. We’ve sketched the situation in Figure 6.3, and the reason for the name should be obvious.

  Figure 6.3. An electron trapped in a square well potential.

  The idea of confining a particle in a potential is a very important one that we’ll use again, so it will be useful to make sure we understand exactly what it means. How do we actually trap particles? That is quite a sophisticated question; to get to the bottom of it we’ll need to learn about how particles interact with other particles, which we will do in Chapter 10. Nevertheless, we can make progress provided we don’t ask too many questions.

  The ability ‘not to ask too many questions’ is a necessary skill in physics because we have to draw the line somewhere in order to answer any questions at all; no system of objects is perfectly isolated. It seems reasonable that if we want to understand how a microwave oven works, we don’t need to worry about any traffic passing by outside. The traffic will have a tiny influence on the operation of the oven. It will induce vibrations in the air and ground which will shake the oven a little bit. There may also be stray magnetic fields that influence the internal electronics of the oven, no matter how well they are shielded. It is possible to make mistakes in ignoring things because there might be some crucial detail that we miss. If this is the case, we’ll simply get the wrong answer and have to reconsider our assumptions. This is very important, and goes to the heart of the success of science; all assumptions are ultimately validated or negated by experiment. Nature is the arbiter, not human intuition. Our strategy here is to ignore the details of the mechanism that traps the electron and model it by something called a potential. The word ‘potential’ really just means ‘an effect on the particle due to some physics or other that I will not bother to explain in detail’. We will bother to describe in detail how particles interact later on, but for now we’ll talk in the language of potentials. If this sounds a bit cavalier, let us give an example to illustrate how potentials are used in physics.

  Figure 6.4. A ball sitting on a valley floor. The height of the ground above sea-level is directly proportional to the potential that the particle experiences when it rolls around.

  Figure 6.4 illustrates a ball trapped in a valley. If we give the ball a kick then it can roll up the valley, but only so far, and then it will roll back down again. This is an excellent example of a particle trapped by a potential. In this case, the Earth’s gravitational field generates the potential and a steep hill makes a steep potential. It should be clear that we could calculate the details of how a ball rolls around in a valley without knowing the precise details of how the valley floor interacts with the ball – for this we’d have to know about the theory of quantum electrodynamics. If it turned out that the details of the inter-atomic interactions between the atoms in the ball and the atoms in the valley
floor affected the motion of the ball too much, then the predictions we make would be wrong. In fact, the inter-atomic interactions are important because they give rise to friction, but we can also model this without getting into Feynman diagrams. But we digress.

  This example is very tangible because we can literally see the shape of the potential1. However, the idea is more general and works for potentials other than those created by gravity and valleys. An example is the electron trapped in a square well. Unlike the case of the ball in a valley, the height of the walls is not the actual height of anything; rather it represents how fast the electron needs to be moving before it can escape from the well. For the case of a valley, this would be analogous to rolling the ball so fast that it climbed up the walls and out of the valley. If the electron is moving slowly enough then the actual height of the potential won’t matter much, and we can safely assume that the electron is confined to the interior of the well.

  Let us now focus on the electron trapped inside a box described by a square well potential. Since it cannot escape from the box, the quantum waves must fall to zero at the edges of the box. The three possible quantum waves with the largest wavelengths are then entirely analogous to the guitar-string waves illustrated in Figure 6.2: the longest possible wavelength is twice the size of the box, 2L; the next longest wavelength is equal to the size of the box, L; and the next again has a wavelength of 2L/3. Generally, we can fit electron waves with wavelength 2L/n in the box where n = 1, 2, 3, 4, etc.

  Specifically for the square box, therefore, the electron waves are precisely the same shape as the waves on a guitar string; they are sine waves with a very particular set of allowed wavelengths. Now we can go ahead and invoke the de Broglie equation from the last chapter to relate the wavelength of these sine waves to the momentum of the electron via p = h/λ. In which case, the standing waves describe an electron that is only allowed to have certain momenta, given by the formula p = nh/(2L), where all we did here was to insert the allowed wavelengths into the de Broglie equation.