1.4 String Theory (sort of)
standing waves ..... a link to quantisation?
Nefertiti has a guitar: when she plucks a string, right in the middle, it vibrates and - because this in turn sets up a vibration in the air – you hear a note. The actual vibration is a superposition of many vibrations but the one that dominates the sound is called the fundamental and it looks a bit like this:
If the guitar and the air were frictionless, the string would keep vibrating like this until the end of time. This is a kind of wave but what makes it different from most waves is that it’s not going anywhere: the peak stays at the same place, while the amplitude oscillates. This is called a standing wave.
Now Nefertiti asks Cormorant to touch the string lightly, in the exact centre, with a wing of his choice while she plucks it further along. You hear a note – an octave higher than the fundamental – and when Cormorant removes his wing pressure the note continues to ring out (if you have a guitar and a seabird handy, you can try it – or improvise without the bird – or, if you don’t have the equipment, listen to the opening of Buffalo Springfield’s “For What it’s Worth” and hear a pair of these notes, which are called harmonics, ringing over the top of the music).
The key point here is that since the string is fixed at either end, any standing wave set up on the string must have zero displacement at these positions, and therefore only certain wavelengths of standing wave can exist on a given string. Specifically, since an oscillation like these goes to zero every half a wavelength, the string length must be an integral multiple of one half wavelength.
You can imagine that if you can set up standing waves on a string, like this, you could do it with other kinds of waves, including light. Standing electromagnetic waves are, in fact, important in a range of devices - for example standing light waves set up between two mirrors are used to build up the light intensity in a laser cavity. The same rules will apply as with the guitar string – there must be a whole number of half-wavelengths between the two reflecting surfaces to establish a standing wave.
We also know that the wavelength of light is related to the energy of an individual photon, according to the equation E = hc/λ . What that means is that if a standing wave is constrained in some way so that only certain wavelengths are possible, it follows that only certain photon energies are possible. In terms of wave-particle duality, as it applies to light, this means that restrictions on the wavelength of the waveform are associated with limitations on the energy of the particle ……
Lightbulb moment! What if it works like this for matter too? What if a material particle such as an electron also exhibits wave-particle duality? Then we could begin to see a plausible explanation for the existence of distinct energy levels for such a particle: if there are physical constraints on the possible wavelength of the standing wave form of the particle then, just as for light, this could translate into restrictions on its possible energy.
This idea was first set down and developed by a French prince called Louis de Broglie. We’ll get back to him and his ideas in chapter 5. But this is the beginning of our journey – if we could establish some kind of evidence that matter can, indeed, behave with wave-like properties, then we could begin to believe that this is just not the pet fantasy of a French aristocrat but a real aspect of a particle’s personality. And because waves lend themselves to mathematical description, this will, in turn, give us our way into to the theory of matter that we crave.
