by Brian Cox
And so it is that we have demonstrated that the momentum of our electron is quantized in a square well. This is a big deal. However, we do need to take care. The potential in Figure 6.3 is a special case, and for other potentials the standing waves are not generally sine waves. Figure 6.5 shows a photograph of the standing waves on a drum. The drum skin is sprinkled with sand, which collects at the nodes of the standing wave. Because the boundary enclosing the vibrating drum skin is circular, rather than square, the standing waves are no longer sine waves.2 This means that, as soon as we move to the more realistic case of an electron trapped by a proton, its standing waves will likewise not be sine waves. In turn this means that the link between wavelength and momentum is lost. How, then, are we to interpret these standing waves? What is it that is generally quantized for trapped particles, if it isn’t their momentum?
We can get the answer by noticing that in the square well potential, if the electron’s momentum is quantized, then so too is its energy. That is a simple observation and appears to contain no important new information, since energy and momentum are simply related to each other. Specifically, the energy E = p2/2m, where p is the momentum of the trapped electron and m is its mass.3 This is not such a pointless observation as it might appear, because, for potentials that are not as simple as the square well, each standing wave always corresponds to a particle of definite energy.
Figure 6.5. A vibrating drum covered in sand. The sand collects at the nodes of the standing waves.
The important difference between energy and momentum emerges because E = p2/2m is only true when the potential is flat in the region where the particle can exist, allowing the particle to move freely, like a marble on a table top or, more to the point, an electron in a square well. More generally, the particle’s energy will not be equal to E = p2/2m; rather it will be the sum of the energy due to its motion and its potential energy. This breaks the simple link between the particle’s energy and its momentum.
We can illustrate this point by thinking again about the ball in a valley, shown in Figure 6.4. If we start with the ball resting happily on the valley floor, then nothing happens.4 To make it roll up the side of the valley, we’d have to give it a kick, which is equivalent to saying that we need to add some energy to it. The instant after we kick the ball, all of its energy will be in the form of kinetic energy. As it climbs the side of the valley, the ball will slow down until, at some height above the valley floor, it will come to a halt before rolling back down again and up the other side. At the moment it stops, high up the valley side, it has no kinetic energy, but the energy hasn’t just magically vanished. Instead, all of the kinetic energy has been changed into potential energy, equal to mgh, where g is the acceleration due to gravity at the Earth’s surface and h is the height of the ball above the valley floor. As the ball starts to roll back down into the valley, this stored potential energy is gradually converted back into kinetic energy as the ball speeds up again. So as the ball rolls from one side of the valley to the other, the total energy remains constant, but it periodically switches between kinetic and potential. Clearly, the ball’s momentum is constantly changing, but its energy remains constant (we have pretended that there is no friction to slow the ball down. If we did include it then the total energy would still be constant but only after including the energy dissipated via friction).
We are now going to explore the link between standing waves and particles of definite energy in a different way, without appealing to the special case of the square well. We’ll do this using those little quantum clocks.
Figure 6.6: Four snapshots of a standing wave at successively later times. The arrows represent the clock hands and the dotted line is the projection onto the ‘12 o’clock’ direction. The clocks all turn around in unison.
First, notice that, if an electron is described by a standing wave at some instant in time, then it will be described by the same standing wave at some later time. By ‘the same’, we mean that the shape of the wave is unchanged, as was the case for the standing water wave in Figure 6.1. We don’t, of course, mean that the wave does not change at all; the water height does change, but crucially the positions of the peaks and nodes do not. This allows us to figure out what the quantum clock description of a standing wave must look like, and it is illustrated in Figure 6.6 for the case of the fundamental standing wave. The clock sizes along the wave reflect the position of the peaks and nodes, and the clock hands sweep around together at the same rate. We hope you can see why we’ve drawn this particular pattern of clocks. The nodes must always be nodes, the peaks must always be peaks and they must always stay in the same place. This means that the clocks sitting in the vicinity of the nodes must always be very small, and the clocks representing the peaks must always have the longest hands. The only freedom we have, therefore, is to allow the clocks to sit where we put them and rotate in sync.
If we were following the methodology of the earlier chapters, we would now start from the configuration of clocks shown in the top row of Figure 6.6 and use the shrinking and turning rules to generate the bottom three rows at later times. This exercise in clock hopping is a hop too far for this book, but it can be done, and there is a nice twist because to do it correctly it is necessary to include the possibility that the particle ‘bounces off the walls of the box’ before hopping to its destination. Incidentally, because the clocks are bigger in the centre, we can immediately conclude that an electron described by this array of clocks is more likely to be found in the middle of the box than at the edges.
So, we have found that the trapped electron is described by an array of clocks that all whizz around at the same rate. Physicists don’t usually talk like this, and musicians certainly don’t; they both say that standing waves are waves of definite frequency.5 High-frequency waves correspond to clocks that whizz around faster than the clocks of low-frequency waves. You can see this because, if a clock whizzes around faster, then the time it takes a peak to turn into a trough and then rise back again (represented by a single rotation of the clock hand) decreases. In terms of water waves, the high-frequency standing waves move up and down faster than the low-frequency ones. In music, a middle C is said to have a frequency of 262 Hz, which means that, on a guitar, the string vibrates up and down 262 times every second. The A above middle C has a frequency of 440 Hz, so it vibrates more rapidly (this is the agreed tuning standard for most orchestras and musical instruments across the world). As we’ve noted, however, it is only for pure sine waves that these waves of definite frequency also have definite wavelength. Generally speaking, frequency is the fundamental quantity that describes standing waves, and this sentence is probably a pun.
The million-dollar question, then, is ‘What does it mean to speak of an electron of a certain frequency?’ We remind you that these electron states are interesting to us because they are quantized and because an electron in one such state remains in that state for all time (unless something enters the region of the potential and gives the electron a whack).
That last sentence is the big clue we need to establish the significance of ‘frequency’. We encountered the law of energy conservation earlier in the chapter, and it is one of the few non-negotiable laws of physics. Energy conservation dictates that if an electron inside a hydrogen atom (or a square well) has a particular energy, then that energy cannot change until ‘something happens’. In other words, an electron cannot spontaneously change its energy without a reason. This might sound uninteresting, but contrast this with the case of an electron that is known to be located at a point. As we know very well, the electron will leap off across the Universe in an instant, spawning an infinity of clocks. But the standing wave clock pattern is different. It keeps its shape, with all the clocks happily rotating away for ever unless something disturbs them. The unchanging nature of standing waves therefore makes them a clear candidate to describe an electron of definite energy.
Once we make the step of associating the frequency of a standing wave with t
he energy of a particle then we can exploit our knowledge of guitar strings to infer that higher frequencies must correspond to higher energies. That is because high frequency implies short wavelength (since short strings vibrate faster) and, from what we know of the special case of the square well potential, we can anticipate that a shorter wavelength corresponds to a higher-energy particle via de Broglie. The important conclusion, therefore, and all that really needs to be remembered for what follows, is that standing waves describe particles of definite energy and the higher the energy the faster the clocks whizz round.
In summary, we have deduced that when an electron is confined by a potential, its energy is quantized. In the physics jargon, we say that a trapped electron can only exist in certain ‘energy levels’. The lowest energy the electron can have corresponds to its being described by the ‘fundamental’ standing wave alone,6 and this energy level is usually referred to as the ‘ground state’. The energy levels corresponding to standing waves with higher frequencies are referred to as ‘excited states’.
Let us imagine an electron of a particular energy, trapped in a square well potential. We say that it is ‘sitting in a particular energy level’ and its quantum wave will be associated with a single value of n (see page 100). The language ‘sitting in a particular energy level’ reflects the fact that the electron doesn’t, in the absence of any external influence, do anything. More generally, the electron could be described by many standing waves at once, just as the sound of a guitar will be made up of many harmonics at once. This means that the electron will not in general have a unique energy.
Crucially, a measurement of the electron’s energy must always reveal a value equal to that associated with one of the contributing standing waves. In order to compute the probability of finding the electron with a particular energy, we should take the clocks associated with the specific contribution to the total wavefunction coming from the corresponding standing wave, square them all up and add them all together. The resulting number tells us the probability that the electron is in this particular energy state. The sum of all such probabilities (one for each contributing standing wave) must add up to one, which reflects the fact that we will always find that the particle has an energy that corresponds to a specific standing wave.
Let’s be very clear: an electron can have several different energies at the same time, and this is just as weird a statement as saying that it has a variety of positions. Of course, by this stage in the book this ought not to be such a shock, but it is shocking to our everyday sensibilities. Notice that there is a crucial difference between a trapped quantum particle and the standing waves in a swimming pool or on a guitar string. In the case of the waves on a guitar string, the idea that they are quantized is not at all weird, because the actual wave describing the vibrating string is simultaneously composed of many different standing waves, and all those waves physically contribute to the total energy of the wave. Because they can be mixed together in any way, the actual energy of the vibrating string can take on any value at all. For an electron trapped inside an atom, however, the relative contribution of each standing wave describes the probability that the electron will be found with that particular energy. The crucial difference arises because water waves are waves of water molecules but electron waves are most certainly not waves of electrons.
These deliberations have shown us that the energy of an electron inside an atom is quantized. This means that the electron is simply unable to possess any energy intermediate between certain allowed values. This is just like saying that a car can travel at 10 miles per hour or 40 miles per hour, but at no other speeds in between. Immediately, this fantastically bizarre conclusion offers us an explanation for why atoms do not continuously radiate light as the electron spirals into the nucleus. It is because there is no way for the electron to constantly shed energy, bit by bit. Instead, the only way it can shed any energy is to lose a whole chunk in one go.
Figure 6.7. The Balmer series for hydrogen: this is what happens when light from hydrogen gas is passed through a spectroscope.
We can also relate what we have just learnt to the observed properties of atoms, and in particular we can explain the unique colours of light they emit. Figure 6.7 shows the visible light emitted from the simplest atom, hydrogen. The light is composed of five distinct colours, a bright-red line corresponding to light with a wavelength of 656 nanometres, a light-blue line of wavelength 486 nanometres, and three other violet lines which fade away into the ultraviolet end of the spectrum. This series of coloured lines is known as the Balmer series, after the Swiss mathematical physicist Johann Balmer, who wrote down a formula able to describe them in 1885. Balmer had no idea why his formula worked, because quantum theory was yet to be discovered – he simply expressed the regularity behind the pattern in a simple mathematical formula. But we can do better, and it is all to do with the allowed quantum waves that fit inside the hydrogen atom.
We know that light can be thought of as a stream of photons, each of energy E = hc/λ, where λ is the wavelength of the light.7 The observation that atoms only emit certain colours of light therefore means that they only emit photons of very specific energies. We have also learnt that an electron ‘trapped in an atom’ can only possess certain very specific energies. It is a small step now to explain the long-standing mystery of the coloured light emitted from atoms: the different colours correspond to the emission of photons when electrons ‘drop down’ from one allowed energy level to another. This idea implies that the observed photon energies should always correspond to differences between a pair of allowed electron energies. This way of describing the physics nicely illustrates the value of expressing the state of the electron in terms of its allowed energies. If we had instead chosen to talk about the allowed values of the electron’s momentum then the quantum nature would not be so apparent and we would not so easily conclude that the atom can only emit and absorb radiation at specific wavelengths.
The particle-in-a-box model of an atom is not accurate enough to allow us to compute the electron energies in a real atom, which is necessary to check this idea. But accurate calculations can be done if we model more accurately the potential in the vicinity of the proton that traps the electron. It is enough to say that these calculations confirm, without any shadow of doubt, that this really is the origin of those enigmatic spectral lines.
You may have noticed that we have not explained why it is that the electron loses energy by emitting a photon. For the purposes of this chapter, we do not need an explanation. But something must induce the electron to leave the sanctity of its standing wave, and that ‘something’ is the topic of Chapter 10. For now, we are simply saying that ‘in order to explain the observed patterns of light emitted by atoms it is necessary to suppose that the light is emitted when an electron drops down from one energy level to another level of lower energy’. The allowed energy levels are determined by the shape of the confining box and they vary from atom to atom because different atoms present a different environment within which their electrons are confined.
Up until now, we have made a good fist of explaining things using a very simple picture of an atom, but it isn’t really good enough to pretend that electrons move around freely inside some confining box. They are moving around in the vicinity of a bunch of protons and other electrons, and to really understand atoms we must now think about how to describe this environment more accurately.
The Atomic Box
Armed with the notion of a potential, we can be more accurate in our description of atoms. Let’s start with the simplest of all atoms, a hydrogen atom. A hydrogen atom is made up of just two particles: one electron and one proton. The proton is nearly 2,000 times heavier than the electron, so we can assume that it is not doing much and just sits there, creating a potential within which the electron is trapped.
The proton has a positive electric charge and the electron has an equal and opposite negative charge. As an aside, the reason why the electric charges of th
e proton and the electron are exactly equal and opposite is one of the great mysteries of physics. There is probably a very good reason, associated with some underlying theory of subatomic particles that we have yet to discover, but, as we write this book, nobody knows.
What we do know is that, because opposite charges attract, the proton is going to tug the electron towards it and, as far as pre-quantum physics is concerned, it could pull the electron inwards to arbitrarily small distances. How small would depend on the precise nature of a proton; is it a hard ball or a nebulous cloud of something? This question is irrelevant because, as we have seen, there is a minimum energy level that the electron can be in, determined (roughly speaking) by the longest wavelength quantum wave that will fit inside the potential generated by the proton. We’ve sketched the potential created by the proton in Figure 6.8. The deep ‘hole’ functions like the square well potential we met earlier except that the shape is not as simple. It is known as the ‘Coulomb potential’, because it is determined by the law describing the interaction between two electric charges, first written down by Charles-Augustin de Coulomb in 1783. The challenge is the same, however: we must find out what quantum waves can fit inside the potential, and these will determine the allowed energy levels of the hydrogen atom.