• Choose a category
• All
• Classic-Fiction

By Root 804 0

should multiply the length of the hand, h, by the cosine of the angle the hand makes with the 12 o’clock direction. For example, the angle that a 3 o’clock makes with 12 o’clock is 90 degrees and the cosine of 90 degrees is zero, which means the wave height is zero. Similarly, a time of half-past-one corresponds to an angle of 45 degrees with the 12 o’clock direction and the cosine of 45 degrees is approximately 0.707, so the height of the wave is 0.707 times the length of the hand (notice that 0.707 is 1/√2). If your trigonometry is not up to those last few sentences then you can safely ignore the details. It’s the principle that matters, which is that, given the length of a clock hand and its direction you can go ahead and calculate the wave height – and even if you don’t understand trigonometry you could make a good stab at it by carefully drawing the clock hands and projecting on to the 12 o’clock direction using a ruler. (We would like to make it very clear to any students reading this book that we do not recommend this course of action: sines and cosines are useful things to understand.)

That’s the rule for adding clocks, and it works a treat, as illustrated in the bottom of the three pictures in Figure 3.4, where we have repeatedly applied the rule for various points along the waves.

Figure 3.6. Three different clocks all with the same projection in the 12 o’clock direction.

In this description of water waves, all that ever matters is the projection of the ‘time’ in the 12 o’clock direction, corresponding to just one number: the wave height. That is why the use of clocks is not really necessary when it comes to describing water waves. Take a look at the three clocks in Figure 3.6: they all correspond to the same wave height and so they provide equivalent ways of representing the same height of water. But clearly they are different clocks and, as we shall see, these differences do matter when we come to use them to describe quantum particles because, for them, the length of the clock hand (or equivalently the size of the clock) has a very important interpretation.

At some points in this book and at this point especially, things are abstract. To keep ourselves from succumbing to dizzying confusion, we should remember the bigger picture. The experimental results of Davisson, Germer and Thomson, and their similarity with the behaviour of water waves, have inspired us to make an ansatz: we should represent a particle by a wave, and the wave itself can be represented by lots of clocks. We imagine that the electron wave propagates ‘like a water wave’, but we haven’t explained how that works in any detail. But then we never said how the water wave propagates either. All that matters for the moment is that we recognize the analogy with water waves, and the notion that the electron is described at any instant by a wave that propagates and interferes like water waves do. In the next chapter we will do better than this and be more precise about how an electron actually moves around as time unfolds. In doing that we will be led to a host of treasures, including Heisenberg’s famous Uncertainty Principle.

Before we move on to that, we want to spend a little time talking about the clocks that we are proposing to represent the electron wave. We emphasize that these clocks are not real in any sense, and their hour hand has absolutely nothing to do with what time of day it is. This idea of using an array of little clocks to describe a real physical phenomenon is not so bizarre a concept as it may seem. Physicists use similar techniques to describe many things in Nature, and we have already seen how they can be used to describe water waves.

Another example of this type of abstraction is the description of the temperature in a room, which can be represented using an array of numbers. The numbers do not exist as physical objects any more than our clocks do. Instead, the set of numbers and their association with points in the room is simply a convenient way of representing the temperature. Physicists call this mathematical structure