4.8 Adding Up
Superposition of waves with the same velocity always produces something which is also a wave.
Superposition of wave functions is of central importance in quantum mechanics. We have already seen examples of superpositions which are themselves still waves and we’re now going to generalise this and show that any solutions to the wave equations have this capacity to superpose and create a new wave:
We can extend this to say that any number of waves can be added together to produce a superposition that is itself a wave. Cormorant exclaims excitedly at the prospect of being able to generate waves with all sorts of weird and wonderful forms by adding up simple sinusoidal waves with different amplitudes, wavelengths and phases. “You are not wrong, mon cher oiseau” chimes in a ghostly Joseph Fourier, who is sipping his absinthe nearby. “In fact, I have proposed that any waveform can be deconstructed into a sum of sinusoidal waveforms”. Joseph went further and developed a tool – the Fourier transform – which can, in effect, calculate the amplitudes, wavelengths and phases of the contributory waves. This is fantastically useful in, for example, spectroscopy: but that is a story for another day.
Our interest is in how the properties of waves may help us to understand the quantum properties of matter and, indeed, that will be the focus of much of the rest of our story. Superposition will prove to play a crucial role in this. We have already learned about constructive and destructive interference and about the creation of stationary waves, both of which are examples of superposition effects, and both of which we will come back to haunt us.
