The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [46]
The ability ‘not to ask too many questions’ is a necessary skill in physics because we have to draw the line somewhere in order to answer any questions at all; no system of objects is perfectly isolated. It seems reasonable that if we want to understand how a microwave oven works, we don’t need to worry about any traffic passing by outside. The traffic will have a tiny influence on the operation of the oven. It will induce vibrations in the air and ground which will shake the oven a little bit. There may also be stray magnetic fields that influence the internal electronics of the oven, no matter how well they are shielded. It is possible to make mistakes in ignoring things because there might be some crucial detail that we miss. If this is the case, we’ll simply get the wrong answer and have to reconsider our assumptions. This is very important, and goes to the heart of the success of science; all assumptions are ultimately validated or negated by experiment. Nature is the arbiter, not human intuition. Our strategy here is to ignore the details of the mechanism that traps the electron and model it by something called a potential. The word ‘potential’ really just means ‘an effect on the particle due to some physics or other that I will not bother to explain in detail’. We will bother to describe in detail how particles interact later on, but for now we’ll talk in the language of potentials. If this sounds a bit cavalier, let us give an example to illustrate how potentials are used in physics.
Figure 6.4. A ball sitting on a valley floor. The height of the ground above sea-level is directly proportional to the potential that the particle experiences when it rolls around.
Figure 6.4 illustrates a ball trapped in a valley. If we give the ball a kick then it can roll up the valley, but only so far, and then it will roll back down again. This is an excellent example of a particle trapped by a potential. In this case, the Earth’s gravitational field generates the potential and a steep hill makes a steep potential. It should be clear that we could calculate the details of how a ball rolls around in a valley without knowing the precise details of how the valley floor interacts with the ball – for this we’d have to know about the theory of quantum electrodynamics. If it turned out that the details of the inter-atomic interactions between the atoms in the ball and the atoms in the valley floor affected the motion of the ball too much, then the predictions we make would be wrong. In fact, the inter-atomic interactions are important because they give rise to friction, but we can also model this without getting into Feynman diagrams. But we digress.
This example is very tangible because we can literally see the shape of the potential1. However, the idea is more general and works for potentials other than those created by gravity and valleys. An example is the electron trapped in a square well. Unlike the case of the ball in a valley, the height of the walls is not the actual height of anything; rather it represents how fast the electron needs to be moving before it can escape from the well. For the case of a valley, this would be analogous to rolling the ball so fast that it climbed up the walls and out of the valley. If the electron is moving slowly enough then the actual height of the potential won’t matter much, and we can safely assume that the electron is confined to the interior of the well.
Let us now focus on the electron trapped inside a box described by a square well potential. Since it cannot escape from the box, the quantum waves must fall to zero at the edges of the box. The three possible quantum waves with the largest wavelengths are then entirely analogous to the guitar-string waves illustrated in Figure 6.2: the longest