Setting down some foundations

For tonight’s blog post, I thought I’d begin to elaborate on the main idea that I introduced in my last post: the idea that something can exhibit both particle and wave-like properties, known more commonly as “wave-particle” duality. Wave-particle duality effects all particles to a certain extent, and is most easily observed in particles of very small (or no) mass. Whereas in the first post, we considered the wave-particle duality of light, this time we’re going to talk about electrons.

Imagine we have the set up shown in figure 1, where you have an electron fired from a source, through one of two slits in a middle screen, and detected at some position on the screen on the far right. As a particle, you'd think that, say, if a the electron were to pass through slit one, it'd be more likely to land somewhere on the top part of the screen, and if it were to go through slit two, the opposite would happen. This result is easily confirmed by experiment: if you stick a measuring device, say, some light source by each of the slits, you can tell which of the slits an electron has gone through by watching to see at which slit some of the light is scattered. When you do this, you get the graphs below:

As you can see, the probabilities match up with the predictions: most of the electrons land in line with the slit it went through in each case, with the chance of a few electrons appearing further away from the slit.

However, things start to get more interesting if you take the measuring device away, which leaves you in the dark as to which slit the electron passed through. As you no longer care which hole the electron goes through, it can take either path, meaning the probabilities compile, and you add them up. This, generally, would be : P(x) = P1 + P2, where P1 and P2 are the probabilities from the graph on the left and right respectively.

Therefore logically, you'd expect the shape of the graph to be some sort of average between the two graphs above, shaped like a quadratic curve, symmetrical around the horizontal axis.

As you've probably guessed, this is far too simple for quantum mechanics! The graph looks something more like the graph on the right hand side:

In this graph, you've got a much more complex curve, which, if you've done a bit of AS physics (if not, read more here), should actually look pretty familiar to you: it's exactly the same shape as the interference patterns you get when you pass light through the same set up! The electrons are acting as waves, constructively and destructively interfering with each other, causing a variety of maxima and minima of concentrations of electrons. Because of the more complicated nature of the maths involved in such a wave, the probabilities are no longer as simple as they first appeared. In this case, we say that P(x) is the square of the "probability amplitude". The probability amplitude is calculated by doing the sum of the solutions of wave equations representing the wave of the electron spreading from each of the slits to the detectors. This is why, when describing a particle from a quantum perspective , we consider its "wavefunction". 

But hold on! All we did was take away our measuring device, and suddenly we have a completely different set of results! How can this be? To really understand what's going on, we have to consider what we're actually doing when we measure what slit the electron is going through. To "see" the electron, they have to scatter the photons from the measuring device, and it's this process that causes the interference pattern to break down and a regular pattern to reemerge. However, you could argue that, well, you're using photons that have a momentum of Planck's constant divided by their wavelength, and we could minimise the disruption the photons causes by increasing its wavelength. But then we meet another problem here too: if the wavelength is too long it's impossible to tell which slit the electron went through, and the interference pattern returns! 

This completely counter-intuitive effect, that an observer can destroy the interference between two events, was first observed by Heisenberg. Heisenberg went on to state a number of things, including the idea that there is a limitation to the subtlety to which experiments could be performed, and these ideas went on to become Heisenberg's uncertainty principle. This principle gives rise to all sorts of weird things, and forced people to consider even the actual limits of reality! However, this blog post is definitely long enough as it is, and I'll get on to those sort of things next time xD

  - So, that's it for this post. Although the material may not seem like the most exciting stuff in the world, it leads on nicely to all sorts of things, such as quantum tunneling, which you can expect in the posts to come ^^ My contribution to this blog is mostly going to be quantum type things, as I'm planning on doing an extended project on the subject, and I'm using this as a place to have a good go at writing about it.

Thanks for reading,

GM