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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 to 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