The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [49]
That last sentence is the big clue we need to establish the significance of ‘frequency’. We encountered the law of energy conservation earlier in the chapter, and it is one of the few non-negotiable laws of physics. Energy conservation dictates that if an electron inside a hydrogen atom (or a square well) has a particular energy, then that energy cannot change until ‘something happens’. In other words, an electron cannot spontaneously change its energy without a reason. This might sound uninteresting, but contrast this with the case of an electron that is known to be located at a point. As we know very well, the electron will leap off across the Universe in an instant, spawning an infinity of clocks. But the standing wave clock pattern is different. It keeps its shape, with all the clocks happily rotating away for ever unless something disturbs them. The unchanging nature of standing waves therefore makes them a clear candidate to describe an electron of definite energy.
Once we make the step of associating the frequency of a standing wave with the energy of a particle then we can exploit our knowledge of guitar strings to infer that higher frequencies must correspond to higher energies. That is because high frequency implies short wavelength (since short strings vibrate faster) and, from what we know of the special case of the square well potential, we can anticipate that a shorter wavelength corresponds to a higher-energy particle via de Broglie. The important conclusion, therefore, and all that really needs to be remembered for what follows, is that standing waves describe particles of definite energy and the higher the energy the faster the clocks whizz round.
In summary, we have deduced that when an electron is confined by a potential, its energy is quantized. In the physics jargon, we say that a trapped electron can only exist in certain ‘energy levels’. The lowest energy the electron can have corresponds to its being described by the ‘fundamental’ standing wave alone,6 and this energy level is usually referred to as the ‘ground state’. The energy levels corresponding to standing waves with higher frequencies are referred to as ‘excited states’.
Let us imagine an electron of a particular energy, trapped in a square well potential. We say that it is ‘sitting in a particular energy level’ and its quantum wave will be associated with a single value of n (see page 100). The language ‘sitting in a particular energy level’ reflects the fact that the electron doesn’t, in the absence of any external influence, do anything. More generally, the electron could be described by many standing waves at once, just as the sound of a guitar will be made up of many harmonics at once. This means that the electron will not in general have a unique energy.
Crucially, a measurement of the electron’s energy must always reveal a value equal to that associated with one of the contributing standing waves. In order to compute the probability of finding the electron with a particular energy, we should take the clocks associated with the specific contribution to the total wavefunction coming from the corresponding standing wave, square them all up and add them all together. The resulting number tells us the probability that the electron is in this particular energy state. The sum of all such probabilities (one for each contributing standing wave) must add up to one, which reflects the fact that we will always find that the particle has an energy that corresponds to a specific standing wave.
Let’s be very clear: an electron can have several different energies at the same time, and this is just as weird a statement as saying that it has a variety of positions. Of course, by this stage in the book this ought not to be such a shock, but it is shocking