by Brian Cox
Let’s allow time to tick forward and work out what happens to this line of clocks. We’ll start off by thinking about one particular point a large distance away from the initial cluster, marked X in the figure. We’ll be more quantitative about what a ‘large distance’ means later on, but for now it simply means that we need to do a lot of clock winding.
Applying the rules of the game, we should take each clock in the initial cluster and transport it to point X, winding the hand around and shrinking it accordingly. Physically, this corresponds to the particle hopping from that point in the cluster to point X. There will be many clocks arriving at X, one from each initial clock in the line, and we should add them all up. When all this is done, the square of the length of the resulting clock hand at X will give the probability that we will find the particle at X.
Now let’s see how this all pans out and put some numbers in. Let’s say that the point X is a distance of ‘10 units’ away from clock 1, and that the initial cluster is ‘0.2 units’ wide. Answering the obvious question: ‘How far is 10 units?’ is where Planck’s constant enters our story, but for now we shall deftly side-step that issue and simply specify that 1 unit of distance corresponds to 1 complete (twelve-hour) wind of the clock. This means that the point X is approximately 102 = 100 complete windings away from the initial cluster (remember the winding rule). We shall also assume that the clocks in the initial cluster started out of equal size, and that they all point to 12 o’clock. Assuming they are of equal size is simply the assumption that the particle is equally likely to be anywhere in between points 1 and 3 in the figure and the significance of them all reading the same time will emerge in due course.
To transport a clock from point 1 to point X, we have to rotate the clock hand anti-clockwise 100 complete times, as per our rule. Now let’s move across to point 3, which is a further 0.2 units away, and transport that clock to X. This clock has to travel 10.22 units, so we have to wind its hand back a little more than before, i.e. by 10.22, which is very close to 104, complete winds.
We now have two clocks landing at X, corresponding to the particle hopping from 1 to X and from 3 to X, and we must add them together in order to start the task of computing the final clock. Because they both got wound around by very close to a whole number of winds, they will both end up pointing roughly to 12 o’clock, and they will add up to form a clock with a bigger hand also pointing to 12 o’clock. Notice that it is only the final direction of the clock hands that matters. We do not need to keep track of how often they wind around. So far so good, but we haven’t finished because there are many other little clocks in between the right- and left-hand edges of the cluster.
And so our attention now turns to the clock midway between the two edges, i.e. at point 2. That clock is 10.1 units away from X, which means that we have to wind it 10.12 times. This is very close to 102 complete rotations – again a whole number of winds. We need to add this clock to the others at X and, as before, this will make the hand at X even longer. Continuing, there is also a point midway between points 1 and 2 and the clock hopping from there will get 101 complete rotations, which will add to the size of the final hand again. But here is the important point. If we now go midway again between these two points, we get to a clock that will be wound 100.5 rotations when it reaches X. This corresponds to a clock with a hand pointing to 6 o’clock, and when we add this clock we will reduce the length of the clock hand at X. A little thought should convince you that, although the points labelled 1, 2 and 3 each produce clocks at X reading 12 o’clock, and although the points midway between 1, 2 and 3 also produce clocks that read 12 o’clock, the points that are ¼ and ¾ of the way between points 1 and 3 and points 2 and 3 each generate clocks that point to 6 o’clock. In total that is five clocks pointing up and four clocks pointing down. When we add all these clocks together, we’ll get a resultant clock at X that has a tiny hand because nearly all of the clocks will cancel each other out.
This ‘cancellation of clocks’ obviously extends to the realistic case where we consider every possible point lying in the region between points 1 and 3. For example, the point that lies ⅛ of the way along from point 1 contributes a clock reading 9 o’clock, whilst the point lying ⅜ of the way reads 3 o’clock – again the two cancel each other out. The net effect is that the clocks corresponding to all of the ways that the particle could have travelled from somewhere in the cluster to point X cancel each other out. This cancellation is illustrated on the far right of the figure. The arrows indicate the clock hands arriving at X from various points in the initial cluster. The net effect of adding all these arrows together is that they all cancel each other out. This is the crucial ‘take home’ message.
To reiterate, we have just shown that, provided the original cluster of clocks is large enough and that point X is far enough away, then for every clock that arrives at X pointing to 12 o’clock, there will be another that arrives pointing to 6 o’clock to cancel it out. For every clock that arrives pointing to 3 o’clock, there will be another that arrives pointing to 9 o’clock to cancel it out, and so on. This wholesale cancellation means that there is effectively no chance at all of finding the particle at X. This really is very encouraging and interesting, because it looks rather like a description of a particle that isn’t moving. Although we started out with the ridiculous-sounding proposal that a particle can go from being at a single point in space to anywhere else in the Universe a short time later, we have now discovered that this is not the case if we start out with a cluster of clocks. For a cluster, because of the way all the clocks interfere with each other, the particle has effectively no chance of being far away from its initial position. This conclusion has come about as a result of an ‘orgy of quantum interference’, in the words of Oxford professor James Binney.
For the orgy of quantum interference and corresponding cancellation of clocks to happen, point X needs to be far enough away from the initial cluster so that the clocks can rotate around many times. Why? Because if point X is too close then the clock hands won’t necessarily have the chance to go around at least once, which means they will not cancel each other out so effectively. Imagine, for example, that the distance from the clock at point 1 to point X is 0.3 instead of 10. Now the clock at the front of the cluster gets a smaller wind than before, corresponding to 0.32 = 0.09 of a turn, which means it is pointing just past 1 o’clock. Likewise, the clock from point 3, at the back of the cluster, now gets wound by 0.52 = 0.25 of a turn, which means it reads 3 o’clock. Consequently, all of the clocks arriving at X point somewhere between 1 o’clock and 3 o’clock, which means they do not cancel each other out but instead add up to one big clock pointing to approximately 2 o’clock. All of this amounts to saying that there will be a reasonable chance of finding the particle at points close to, but outside of, the original cluster. By ‘close to’, we mean that there isn’t sufficient winding to get the clock hands around at least once. This is starting to have a whiff of the Uncertainty Principle about it, but it is still a little vague, so let’s explore exactly what we mean by a ‘large enough’ initial cluster and a point ‘far enough away’.
Our initial ansatz, following Dirac and Feynman, was that the amount the hands wind around when a particle of mass m hops a distance x in a time t is proportional to the action, i.e. the amount of winding is proportional to mx2/t. Saying it is ‘proportional to’ isn’t good enough if we want to calculate real numbers. We need to know precisely what the amount of winding is equal to. In Chapter 2 we discussed Newton’s law of gravitation, and in order to make quantitative predictions we introduced Newton’s gravitational constant, which determines the strength of the gravitational force. With the addition of Newton’s constant, numbers can be put into the equation and real things can be calculated, such as the orbital period of the Moon or the path taken by the Voyager 2 spacecraft on its journey across the solar system. We now need something similar for quantum mechanics – a constant of Nature that ‘sets the scale’ and allows us t
o take the action and produce a precise statement about the amount by which we should wind clocks as we move them a specified distance away from their initial position in a particular time. That constant is Planck’s constant.
A Brief History of Planck’s Constant
In a flight of imaginative genius during the evening of 7 October 1900, Max Planck managed to explain the way that hot objects radiate energy. Throughout the second half of the nineteenth century, the exact relationship between the distribution of the wavelengths of light emitted by hot objects and their temperature was one of the great puzzles in physics. Every hot object emits light and, as the temperature is increased, the character of the light changes. We are familiar with light in the visible region, corresponding to the colours of the rainbow, but light can also occur with wavelengths that are either too long or too short to be seen by the human eye. Light with a longer wavelength than red light is called ‘infra-red’ and it can be seen using night-vision goggles. Still longer wavelengths correspond to radio waves. Likewise, light with a wavelength just shorter than blue is called ultra-violet, and the shortest wavelength light is generically referred to as ‘gamma radiation’. An unlit lump of coal at room temperature will emit light in the infra-red part of the spectrum. But if we throw it on to a burning fire, it will begin to glow red. This is because, as the temperature of the coal rises, the average wavelength of the radiation it emits decreases, eventually entering the range that our eyes can see. The rule is that the hotter the object, the shorter the wavelength of the light it emits. As the precision of the experimental measurements improved in the nineteenth century, it became clear that nobody had the correct mathematical formula to describe this observation. This problem is often termed the ‘black body problem’, because physicists refer to idealized objects that perfectly absorb and then re-emit radiation as ‘black bodies’. The problem was a serious one, because it revealed an inability to understand the character of light emitted by anything and everything.
Planck had been thinking hard about this and related matters in the fields of thermodynamics and electromagnetism for many years before he was appointed Professor of Theoretical Physics in Berlin. The post had been offered to both Boltzmann and Hertz before Planck was approached, but both declined. This proved to be fortuitous, because Berlin was the centre of the experimental investigations into black body radiation, and Planck’s immersion at the heart of the experimental work proved key to his subsequent theoretical tour de force. Physicists often work best when they are able to have wide-ranging and unplanned conversations with colleagues.
We know the date and time of Planck’s revelation so well because he and his family had spent the afternoon of Sunday 7 October 1900 with his colleague Heinrich Rubens. Over lunch, they discussed the failure of the theoretical models of the day to explain the details of black body radiation. By the evening, Planck had scribbled a formula on to a postcard and sent it to Rubens. It turned out to be the correct formula, but it was very strange indeed. Planck later described it as ‘an act of desperation’, having tried everything else he could think of. It is genuinely unclear how Planck came up with his formula. In his superb biography of Albert Einstein, Subtle is the Lord …, Abraham Pais writes: ‘His reasoning was mad, but his madness has that divine quality that only the greatest transitional figures can bring to science.’ Planck’s proposal was both inexplicable and revolutionary. He found that he could explain the black body spectrum, but only if he assumed that the energy of the emitted light was made up of a large number of smaller ‘packets’ of energy. In other words the total energy is quantized in units of a new fundamental constant of Nature, which Planck called ‘the quantum of action’. Today, we call it Planck’s constant.
What Planck’s formula actually implies, although he didn’t appreciate it at the time, is that light is always emitted and absorbed in packets, or quanta. In modern notation, those packets have energy E = hc/λ, where λ is the wavelength of the light (pronounced ‘lambda’), c is the speed of light and h is Planck’s constant. The role of Planck’s constant in this equation is as the conversion factor between the wavelength of light and the energy of its associated quantum. The realization that the quantization of the energy of emitted light, as identified by Planck, arises because the light itself is made up of particles was proposed, tentatively at first, by Albert Einstein. He made the proposition during his great burst of creativity in 1905 – the annus mirabilis which also produced the Special Theory of Relativity and the most famous equation in scientific history, E = mc2. Einstein received the 1921 Nobel Prize for physics (which due to a rather arcane piece of Nobelian bureaucracy he received in 1922) for this work on the photoelectric effect, and not for his better-known theories of relativity. Einstein proposed that light can be regarded as a stream of particles (he did not at that time use the word ‘photons’) and he correctly recognized that the energy of each photon is inversely proportional to its wavelength. This conjecture by Einstein is the origin of one of the most famous paradoxes in quantum theory – that particles behave as waves, and vice versa.
Planck removed the first bricks from the foundations of Maxwell’s picture of light by showing that the energy of the light emitted from a hot object can only be described if it is emitted in quanta. It was Einstein who pulled out the bricks that brought down the whole edifice of classical physics. His interpretation of the photoelectric effect demanded not only that light is emitted in little packets, but that it also interacts with matter in the form of localized packets. In other words, light really does behave as a stream of particles.
The idea that light is made from particles – that is to say that ‘the electromagnetic field is quantized’ – was deeply controversial and not accepted for decades after Einstein first proposed it. The reluctance of Einstein’s peers to embrace the idea of the photon can be seen in the proposal, co-written by Planck himself, for Einstein’s membership of the prestigious Prussian Academy in 1913, a full eight years after Einstein’s introduction of the photon:
In sum, one can say that there is hardly one among the great problems in which modern physics is so rich to which Einstein has not made a remarkable contribution. That he may sometimes have missed the target in his speculations, as, for example, in his hypothesis of light quanta, cannot really be held too much against him, for it is not possible to introduce really new ideas even in the most exact sciences without sometimes taking a risk.
In other words, nobody really believed that photons were real. The widely held belief was that Planck was on safe ground because his proposal was more to do with the properties of matter – the little oscillators that emitted the light – rather than the light itself. It was simply too strange to believe that Maxwell’s beautiful wave equations needed replacing with a theory of particles.
We mention this history partly to reassure you of the genuine difficulties that must be faced in accepting quantum theory. It is impossible to visualize a thing, such as an electron or a photon, that behaves a little bit like a particle, a little bit like a wave, and a little bit like neither. Einstein remained concerned about these issues for the rest of his life. In 1951, just four years before his death, he wrote: ‘All these fifty years of pondering have not brought me any closer to answering the question, what are light quanta?’
Sixty years later, what is unarguable is that the theory we are in the process of developing using our arrays of little clocks describes, with unerring precision, the results of every experiment that has ever been devised to test it.
Back to Heisenberg’s Uncertainty Principle
This, then, is the history behind the introduction of Planck’s constant. But for our purposes, the most important thing to notice is that Planck’s constant is a unit of ‘action’, which is to say that it is the same type of quantity as the thing which tells us how far to wind the clocks. Its modern value is 6.6260695729 × 10−34 kg m2/s, which is very tiny by everyday standards. This will turn out to be the reason why we don’t notice its al
l-pervasive effects in everyday life.
Recall that we wrote of the action corresponding to a particle hopping from one place to another as the mass of the particle multiplied by the distance of the hop squared divided by the time interval over which the hop occurs. This is measured in kg m2/s, as is Planck’s constant, and so if we simply divide the action by Planck’s constant, we’ll cancel all the units out and end up with a pure number. According to Feynman, this pure number is the amount we should wind the clock associated with a particle hopping from one place to another. For example, if the number is 1, that means 1 full wind and if it’s ½, it means ½ a wind, and so on. In symbols, the precise amount by which we should turn the clock hand to account for the possibility that a particle hops a distance x in a time t is mx2/(2ht). Notice that a factor ½ has appeared in the formula. You can either take that as being what is needed to agree with experiment or you can note that this arises from the definition of the action.6 Either is fine. Now that we know the value of Planck’s constant, we can really quantify the amount of winding and address the point we deferred a little earlier. Namely, what does jumping a distance of ‘10’ actually mean?
Let’s see what our theory has to say about something small by everyday standards: a grain of sand. The theory of quantum mechanics we’ve developed suggests that if we place the grain down somewhere then at a later time it could be anywhere in the Universe. But this is obviously not what happens to real grains of sand. We have already glimpsed a way out of this potential problem because if there is sufficient interference between the clocks, corresponding to the sand grain hopping from a variety of initial locations, then they will all cancel out to leave the grain sitting still. The first question we need to answer is: how many times will the clocks get wound if we transport a particle with the mass of a grain of sand a distance of, say, 0.001 millimetres, in a time of one second? We wouldn’t be able to see such a tiny distance with our eyes, but it is still quite large on the scale of atoms. You can do the calculation quite easily yourself by substituting the numbers into Feynman’s winding rule.7 The answer is something like a hundred million years’ worth of clock winding. Imagine how much interference that allows for. The upshot is that the sand grain stays where it is and there is almost no probability that it will jump a discernible distance, even though we really have to consider the possibility that it secretly hopped everywhere in the Universe in order to reach that conclusion.