5.5 What it all really means, probably
Introduces the idea that the wave associated with a particle of matter describes a probability distribution and, specifically, the Born interpretation of the wave function.
Let’s go back to the interference experiment we thought about in Chapter 1. Electrons passing through a pair of slits are detected, at some distance beyond the slits by hitting a screen. Each electron is registered at the position where it hits the screen. As we saw before, they do not arrive as we would expect if they sailed as localised projectiles through one or other slit. Instead, a pattern of alternating bands of many electrons and few electrons builds up, closely resembling the pattern of light and dark fringes that results from interference of diffracted waves in Young’s double slit experiment with light.
The intensity of the electron beam can be turned down so low that it becomes statistically highly unlikely that two electrons would pass through the slits simultaneously. Remarkably, the result is then still the same: even though only one electron at a time is passing through the double slit arrangement, you still get this pattern of alternating high and low probability of an electron arriving at a specific place on the detector. This seems to show not only that an electron has a wave-like ability to undergo diffraction and interference – but that a single electron can, as it were, interfere with itself. In a sense this means that the electron has passed through both slits: this clearly shows the inadequacy of the simple “blob of stuff” model in situations like this.
The crucial observation here is that each individual electron arrives at a specific position on the detector and it is impossible to predict what this position will be. It is only when a large number of electron impacts have been recorded that the non-random nature of the distribution becomes clear. This leads to the idea that it’s not possible to predict with any certainty the point in space where any individual electron will be detected. The best we can do is to find the probability that it might be at any particular point. This probability is not uniform: the similarity of the distribution of electron impacts to the pattern observed for interfering waves suggests that it is related to the function that described the electron’s wave form. Specifically, the bigger the magnitude (positive or negative) of this function at any point, the higher the probability that the electron will be found there if it is detected. Consequently, in the diffraction experiment, where constructive interference has occurred, so that the magnitude of the wave function is large, there is a high probability of the electron being detected and where the amplitude is small, because of destructive interference, there is a reduced probability of the electron being detected there.
This is all very unsettling. You grow up thinking that at any instant a thing is where it is and that’s it. But this experiment suggests (at least according to a widely accepted interpretation – not everyone agrees, as we'll see later) that a particle does not have a specific position until we force it to do so, using some kind of detection mechanism. Until we do that, there is only a probability that the particle will be detected at a particular place, and this probability is related to the value of the wave function at that point. You are thinking fast now: this is why my keys are not where I know I left them. Sadly, however, this fuzziness is a crucial aspect of the behaviour of small particles such as electrons but it’s entirely negligible on the scale of you and your elusive personal belongings.
Let’s get back to the maths: say that a particle’s matter-wave is described by a function Ψ = f(x,t). Notice that we’ve switched Y for its Greek cousin Ψ here (pronounced – for the benefit of those of us who didn’t pay attention in our Greek classes - psi): this is the convention for a quantum mechanical wave function, so we’ll use it from now on. Notice too that, just to keep it (relatively) simple, we’re keeping space one-dimensional here: in our 3D reality, Ψ would depend on x, y and z, as well as time.
In the last chapter, we developed the idea that it if it isn’t moving, a particle can be described in terms of a standing wave, in which the waveform does not move in space but oscillates in amplitude. We found that for such a wave, we could separate the time and space-dependent terms in the wavefunction, writing the equation as Ψ =ψ(x).f(t). Here we are (confusingly, Cormorant thinks and please feel free to agree) following tradition and using the upper case Ψ to represent the total wave function and lower case ψ to represent the spatial part, which we will refer to as the wave form. Since localisation in space affects only the spatial part of the wave function, this in turn means we can continue to focus just on this spatial part, ψ, as we continue our investigation of this phenomenon.
It was Max Born – a man who achieved the unique double distinction of being both one of the movers and shakers in those heady, early days of the quantum theory and of being a grandparent to the female lead in ‘Grease’ – who was first to develop the probabilistic interpretation of the wave function. Specifically, Max proposed that the probability density (F(x)) of a particle being detected at a position x is given by:
F(x) = ψ2 (V-3)
The idea of probability density is that you can only talk about the actual probability that the particle will be detected in a given region of space if you define the extent of that region. Probability density is defined by the wavefunction at each point and if you integrate it over a region of space between x1 and x2, you find the actual probability (Pr) that the particle will be detected in that region:
In order to make this probabilistic interpretation work, we have to scale the waveform ψ(x) such that if this integration is performed over all space, the probability is equal to 1 (because if the particle is detected, it has to be somewhere). This is called normalising the wave function.