5.7 How localised can you get?
The smaller the region of space to which a wave packet is restricted, the bigger the range of wavelengths of the waveforms that must be superimposed.
We have been looking at the generation of a localised wave packet by constructing superpositions of sinusoidal waveforms with the aid of integration:
Let's investigate what happens when we change the range of k over which we integrate. First we'll try keeping <k>, the mean value of k, the same, but look at what happens when we integrate over a wider range. Some results can be seen in figure V-v, below:
Fig. V-v Scaled superpositions of waveforms y = cos(kx), spanning the range of k shown. These curves were created using equation V-8.
This shows that, for a given <k>, the wave packet narrows as the range of angular wavenumbers (k) of the superposed waveforms increases. When the range is big enough, the wave packet is dominated just by a single central peak. The interesting thing, though, is that the width of this central peak is always essentially the same. This is our first hint of something of immense importance in quantum mechanics: there seems to be a fundamental lower limit on the width of a wave packet.
In the example above, <k> (the mean value of k) is the same (1.0) in each case; only the spread to either side of the mean was varied. What if we now try allowing the mean, <k>, to change? This is explored in figure V-vi, below:
The message is clear: we can narrow down our wave packet, but only by increasing <k>, the mean value of the angular wavenumber. It's easy to visualise why that is so: increasing k means decreasing the wavelength, so that individual peak in the waveform at x=0 where the constructive interference occurs will be narrower. What does this result mean, though? Time to remind ourselves of why we are doing all this.
First, remember what Max Born has taught us: if we detect the position of a particle whose normalised waveform is ψ, the probability density of a particle being found at a position x is given by ψ2. This means that we can take one of our superposition waveforms and convert it into a probability density distribution, just by squaring and then normalising (which means scaling so that the area under the curve, which equates to the probability of particle being somewhere is equal to 1). Here are probability density distributions for matter waves corresponding to the superpositions we constructed in figure V-vi:
This picture shows explicitly what we’ve already hinted at: superposition of waveforms spanning a bigger range of k – and therefore with a bigger value of <k> - focusses the probability density in a narrower range of x. In other words it localises the particle in a smaller region of space. This comes at a price, however, as we shall see.
