The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [32]
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 all-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