by Brian Cox
Figure 7.4. The upper part of the figure illustrates that adding clocks 1 and 2 together after winding clock 1 by 90 degrees is not the same as adding them together after winding clock 2 by 90 degrees. The lower part illustrates the interesting possibility that we could wind one of the clocks by 180 degrees before adding.
This is not a benign observation – it has a very important consequence, because there are only two possible ways of playing around with the winding and shrinking of clocks before adding them together that will deliver a final clock with the property that it does not depend upon which of the original clocks gets the treatment.
This is illustrated in Figure 7.4. The top half of the figure illustrates that, if we wind clock 1 by 90 degrees and add it to clock 2 then the resultant clock is not of the same size as the resultant we would get if we instead wound clock 2 by 90 degrees and add it to clock 1. We can see this because, if we first wind clock 1, the new hand, represented by the dotted arrow, points in the opposite direction to clock 2’s hand, and therefore partly cancels it out. Winding clock 2 instead leaves its hand pointing in the same direction as clock 1’s, and now the hands will add together to form a larger hand.
It should be clear that 90 degrees is not special, and that other angles will also give resultant clocks that depend upon which of clocks 1 and 2 we decided to wind.
The obvious exception is a clock wind of zero degrees, because winding clock 1 by zero degrees before adding to clock 2 is obviously exactly the same as winding clock 2 by zero degrees before adding to clock 1. This means that adding clocks together without any wind is a viable possibility. Similarly, winding both clocks by the same amount would work, but that really is just the same as the ‘no winding’ situation and corresponds simply to redefining what we call ‘12 o’clock’. This is tantamount to saying that we are always free to wind every clock around by some amount, as long as we do that to every clock. This will never impact on the probabilities we are trying to compute.
The lower part of Figure 7.4 illustrates that there is, perhaps surprisingly, one other way we can combine the clocks: we could turn one of them through 180 degrees before adding them together. This does not produce exactly the same clock in the two cases but it does produce the same size of clock, and that means it leads to the same probability to find one electron at A and a second at B.
A similar line of reasoning rules out the possibility of shrinking or expanding one of the clocks before adding, because if we shrink clock 1 by some fraction before adding to clock 2 then this will not usually be the same as shrinking clock 2 by that same amount before adding it to clock 1, and there are no exceptions to that rule.
So, we have an interesting conclusion to draw. Even though we started out by allowing ourselves complete freedom, we have discovered that, because there is no way of telling the particles apart, there are in fact only two ways we can combine the clocks: we can either add them or we can add them after first winding one or the other by 180 degrees. The truly delightful thing is that Nature exploits both possibilities.
For electrons, we have to incorporate the extra twist before adding the clocks. For particles like photons, or Higgs bosons, we have to add clocks without the twist. And so it is that Nature’s particles come in two types: those which need the twist are called fermions and those without the twist are called bosons. What determines whether a particular particle is a fermion or a boson? It is the spin.
The spin is, as the name suggests, a measure of the angular momentum of a particle and it is a matter of fact that fermions always have a spin equal to some half-integer value3 while bosons always have integer spin. We say that the electron has spin-half, the photon has spin-one and the Higgs boson has spin-zero. We have been avoiding dealing with the details of spin in this book, because it is a technical detail most of the time. However, we did need the result that electrons can come in two types, corresponding to the two possible values of their angular momentum (spin up and spin down), when we were discussing the periodic table. This is an example of a general rule that says particles of spin s generally come in 2s + 1 types, e.g. spin ½ particles (like electrons) come in two types, spin 1 particles come in three types and spin 0 particles come in one type. The relationship between the angular momentum of a particle and the way we are to combine clocks is known as the spin-statistics theorem, and it emerges when quantum theory is formulated so that it is consistent with Einstein’s Theory of Special Relativity. More specifically, it is a direct result of making sure that the law of cause and effect is not violated. Unfortunately, deriving the spin-statistics theorem is beyond the level of this book – actually it is beyond the level of many books. In The Feynman Lectures on Physics, Richard Feynman has this to say:
We apologise for the fact that we cannot give you an elementary explanation. An explanation has been worked out by Pauli from complicated arguments of quantum field theory and relativity. He has shown that the two must necessarily go together, but we have not been able to find a way of reproducing his arguments at an elementary level. It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation.
Bearing in mind that Richard Feynman wrote this in a university-level textbook, we must hold up our hands and concur. But the rule is simple, and you must take our word for it that it can be proved: for fermions, you have to give a twist, and for bosons you don’t. It turns out that the twist is the reason for the Exclusion Principle, and therefore for the structure of atoms; and, after all our hard work, this is now something that we can explain very simply.
Imagine moving points A and B in Figure 7.3 closer and closer together. When they are very close together, clock 1 and clock 2 must be of nearly the same size and read nearly the same time. When A and B are right on top of each other then the clocks must be identical. That should be obvious, because clock 1 corresponds to particle 1 ending up at point A and clock 2 is, in this special case, representing exactly the same thing because points A and B are on top of each other. Nevertheless, we do still have two clocks, and we must still add them together. But here is the catch: for fermions, we must give one of the clocks a twist, winding it first by 180 degrees. This means that the clocks will always read exactly ‘opposite’ times when A and B are in the same place – if one reads 12 o’clock then the other will read 6 o’clock – so adding them together always gives a resultant clock of zero size. That is a fascinating result, because it means that there is always zero chance of finding the two electrons at the same place: the laws of quantum physics are causing them to avoid each other. The closer they get to each other, the smaller the resultant clock, and the less likely that is to happen. This is one way to articulate Pauli’s famous principle: electrons avoid each other.
Originally, we set out to demonstrate that no two identical electrons can be in the same energy level in a hydrogen atom. We have not quite shown this to be true yet, but the notion that electrons avoid each other clearly has implications for atoms and for why we do not fall through the floor. Now we can see that not only do the electrons in the atoms in our shoes push against the electrons in the floor because like-charges repel; they also push against each other because they naturally avoid each other, according to the Pauli Exclusion Principle. It turns out that, as Dyson and Lenard proved, it is the electron avoidance that really keeps us from falling through the floor, and it also forces the electrons to occupy the different energy levels inside atoms, giving them a structure, and ultimately leading to the variety of chemical elements we see in Nature. This is clearly a piece of physics with very significant consequences for everyday life. In the final chapter of this book, we will show how Pauli’s principle also plays a crucial role in preventing some stars from collapsing under the influence of their own gravity.
To finish, we should explain how it is that, if no two electrons can be at the same place at the same time, then it also follows that no two electrons in an atom can have th
e same quantum numbers, which means that they cannot have the same energy and spin. If we consider two electrons of the same spin, then we want to show that they cannot be in the same energy level. If they were in the same energy level then necessarily each electron would be described by exactly the same array of clocks distributed through space (corresponding to the relevant standing wave). For each pair of points in space – let’s denote them X and Y – there are then two clocks. Clock 1 corresponds to ‘electron 1 at X’ and ‘electron 2 at Y’, whilst clock 2 corresponds to ‘electron 1 at Y’ and ‘electron 2 at X’. We know from our previous deliberations that these clocks should be added together after winding one of them by 6 hours in order to deduce the probability to find one electron at X and a second at Y. But if the two electrons have the same energies, then clocks 1 and 2 must be identical to each other before the crucial extra wind. After the wind, they will read ‘opposite’ times and, as before, add together to make a clock of no size. That happens for any particular locations X and Y, and so there is absolutely zero chance of ever finding a pair of electrons in the same standing wave configuration, and therefore with the same energy. That, ultimately, is responsible for the stability of the atoms in your body.
8. Interconnected
So far we have been paying close attention to the quantum physics of isolated particles and atoms. We have learnt that electrons sit inside atoms in states of definite energy, known as stationary states, although the atom may be in a superposition of different such states. We have also learnt that it is possible for an electron to make a transition from one energy state to another with the concurrent emission of a photon. The emission of photons in this way makes tangible the energy states in an atom; we see the characteristic colours of atomic transitions everywhere. Our physical experience, though, is of vast assemblies of atoms stuck together in clumps, and for that reason alone it is time to start pondering what happens when we stick atoms together.
The contemplation of atomic clusters is going to lead us along a road that will take in chemical bonding, the differences between conductors and insulators and, eventually, to semiconductors. These interesting materials have properties that can be exploited to build tiny devices capable of carrying out operations in basic logic. They are known as transistors, and by stringing many millions of them together we can build microchips. As we shall see, the theory of transistors is deeply quantum. It is difficult to see how they could have been invented and exploited without quantum theory, and difficult to imagine the modern world without them. They are a prime example of serendipity in science; the curiosity-led exploration of Nature that we’ve spent so much time describing in all its counterintuitive detail, eventually led to a revolution in our everyday lives. The dangers in trying to classify and control scientific research is beautifully summarized in the words of William Shockley, one of the inventors of the transistor and head of the Solid State Physics Group at Bell Telephone Laboratories:1
I would like to express some viewpoints about words often used to classify types of research in physics; for example, pure, applied, unrestricted, fundamental, basic, academic, industrial, practical, etc. It seems to me that all too frequently some of these words are used in a derogatory sense, on the one hand to belittle the practical objectives of producing something useful and, on the other hand, to brush off the possible long-range value of explorations into new areas where a useful outcome cannot be foreseen. Frequently, I have been asked if an experiment I have planned is pure or applied research; to me it is more important to know if the experiment will yield new and probably enduring knowledge about nature. If it is likely to yield such knowledge, it is, in my opinion, good fundamental research; and this is much more important than whether the motivation is purely esthetic satisfaction on the part of the experimenter on the one hand or the improvement of the stability of a high-power transistor on the other. It will take both types to confer the greatest benefit on mankind.
Since that comes from the inventor of perhaps the most useful device since the invention of the wheel, policy-makers and managers throughout the world would do well to pay attention. Quantum theory changed the world, and whatever new theories emerge from the cutting-edge physics being done today, they will almost certainly change our lives again.
As ever, we’ll start at the beginning and extend our study of a universe containing just one particle to a universe of two. Imagine, in particular, a simple universe containing two isolated hydrogen atoms; two electrons bound in orbit around two protons that are very far apart. In a few pages we are going to start bringing the two atoms closer together to see what happens, but for now we are to suppose that they are very distant from each other.
The Pauli Exclusion Principle says that the two electrons cannot be in the same quantum state, because electrons are indistinguishable fermions. You might at first be tempted to say that, if the atoms are far apart, then the two electrons must be in very different quantum states and there is not much more to be said on the matter. But things are vastly more interesting than that. Imagine putting electron number 1 in atom number 1 and electron number 2 in atom number 2. After waiting a while it doesn’t make sense to say that ‘electron number 1 is still in atom number 1’. It might be in atom number 2 now because there is always the chance that the electron did a quantum hop. Remember, everything that can happen does happen, and electrons are free to roam the Universe from one instant to the next. In the language of little clocks, even if we started out with clocks describing one of the electrons clustered only in the vicinity of one of the protons, we would be forced to introduce clocks in the vicinity of the other proton at the next instant. And even if the orgy of quantum interference meant that the clocks near the other proton are very tiny, they would not be of zero size, and there would always be a finite probability that the electron could be there. The way to think more clearly about the implications of the Exclusion Principle is to stop thinking in terms of two isolated atoms and think instead of the system as a whole: we have two protons and two electrons and our task is to understand how they organize themselves. Let us simplify the situation by neglecting the electromagnetic interaction between the two electrons – this won’t be a bad approximation if the protons are far apart, and it doesn’t affect our argument in any important way.
What do we know about the allowed energies for the electrons in the two atoms? We don’t need to do a calculation to get a rough idea; we can use what we know already. For protons that are far apart (imagine they are many miles apart), the lowest allowed energies for the electrons must surely correspond to the situation where they are bound to the protons to make two isolated hydrogen atoms. In this case, we might be tempted to conclude that the lowest energy state for the entire two-proton, two-electron system would correspond to two hydrogen atoms sitting in their lowest energy states, ignoring each other completely. But although this sounds right, it cannot be right. We must think of the system as a whole, and just like an isolated hydrogen atom, this four-particle system must have its own unique spectrum of allowed electron energies. And because of the Pauli principle, the electrons cannot both be in exactly the same energy level around each proton, blissfully ignorant of the existence each other.2
It seems that we must conclude that the pair of identical electrons in two distant hydrogen atoms cannot have the same energy but we have also said that we expect the electrons to be in the lowest energy level corresponding to an idealized, perfectly isolated hydrogen atom. Both those things cannot be true and a little thought indicates that the way out of the problem is for there to be not one but two energy levels for each level in an idealized, isolated hydrogen atom. That way we can accommodate the two electrons without violating the Exclusion Principle. The difference in the two energies must be very small indeed for atoms that are far apart, so that we can pretend the atoms are oblivious to each other. But really, they are not oblivious, because of the tendril-like reaches of the Pauli principle: if one of the two electrons is in one energy state th
en the other must be in the second, different energy state and this intimate link between the two atoms persists regardless of how far apart they are.
This logic extends to more than two atoms – if there are twenty-four hydrogen atoms scattered far apart across the Universe, then for every energy state in a single-atom universe there are now twenty-four energy states, all taking on almost but not quite the same values. When an electron in one of the atoms settles into a particular state it does so in full ‘knowledge’ of the states of each of the other twenty-three electrons, regardless of their distance away. And so, every electron in the Universe knows about the state of every other electron. We need not stop there – protons and neutrons are fermions too, and so every proton knows about every other proton and every neutron knows about every other neutron. There is an intimacy between the particles that make up our Universe that extends across the entire Universe. It is ephemeral in the sense that for particles that are far apart the different energies are so close to each other as to make no discernible difference to our daily lives.
This is one of the weirdest-sounding conclusions we’ve been led to so far in the book. Saying that every atom in the Universe is connected to every other atom might seem like an orifice through which all sorts of holistic drivel can seep. But there is nothing here that we haven’t met before. Think about the square well potential we thought about in Chapter 6. The width of the well determines the allowed spectrum of energy levels, and as the size of the well is changed, the energy level spectrum changes. The same is true here in that the shape of the well inside which our electrons are sitting, and therefore the energy levels they are allowed to occupy, is determined by the positions of the protons. If there are two protons, the energy spectrum is determined by the position of both of them. And if there are 1080 protons forming a universe, then the position of every one of them affects the shape of the well within which 1080 electrons are sitting. There is only ever one set of energy levels and when anything changes (e.g. an electron changes from one energy level to another) then everything else must instantaneously adjust itself such that no two fermions are ever in the same energy level.