BOX 11 The Two-Slit Experiment
If a beam of photons or electrons, all with the same energy, encounters a slit in a barrier and then impinges on a screen behind it, individual localized ‘hits’ invariably occur (Figure 32). This is so even if the beam has a very low density, so that at most one particle at a time is passing through the system. This strongly suggests that individual particles leave the beam generator, pass through the slit, and strike the screen. The impacts have a characteristic distribution over a region.
Now introduce a second identical slit in the barrier (Figure 33). The interpretation of the first experiment in terms of individual particles yields an unambiguous prediction for what will happen. The argument is as follows. All particles travel towards the barrier at right angles to it, and can be assumed to be uniformly distributed in space. The pattern behind a single slit is presumably created by the interaction between the particles and the slit as they pass through it. Entering the slit at different positions, the particles will have different deflections and will thus strike the screen at different points. When two slits are open, each should have an effect identical to that of the single slit, so the combined pattern should be simply the sum of the effects of two single slits.
Figure 32. The distribution of hits behind one slit.
Figure 33. The expected distribution of hits behind two slits.
Figure 34. The actual distribution behind two slits.
Nothing remotely resembling this is observed. The hits are distributed in the bands or fringes (Figure 34) characteristic of the interference that led Young to the wave theory of light (Figure 22). When, in the nineteenth century, it was believed that these fringes are built up continuously, and not in individual ‘hits’, it seemed that only a wave field could produce them.
In the absence of a detailed theory, the pattern observed behind a single slit can be explained equally well by particles or waves. But the pattern behind two slits seems totally inexplicable on the assumption of particles. For surely a particle can pass through only one slit, and what it does then will depend solely on the properties of that slit. It cannot ‘know’ whether the other slit is open or closed and change its behaviour accordingly. Moreover, we can do similar experiments with many slits of different shapes and sizes. Invariably, wave theory correctly predicts the pattern produced on the screen. As far as the total intensity pattern is concerned, there is no way to explain it except by a wave theory.
Yet the patterns are always built up by individual ‘hits’. This is extraordinarily strong evidence for particles. But if particles are creating the patterns, they must somehow explore all the slits at once. They must do what the very concept of a particle denies – be everywhere at once. Moreover, this ability to be present at several places at once gives rise to self-interference. Dirac put it memorably: ‘Each photon ... interferes only with itself.’ It is an important observational fact that the possibility for interference to occur continues until something like the screen forces the particle ‘to reveal itself.
As long as the particle is not forced to make a choice, its behaviour in quantum mechanics is described by what Schrödinger called a wave function, which he denoted by the Greek letter psi, ψ, and this has become traditional. Sometimes the capital is used: ψ. I shall use this suitably grander capital in quantum cosmology, keeping ψ for the things that happen in laboratories. The wave function is like an intensity. If x is a point in space, ψ(x) is the value of ψ at x. In general ψ has a different value for each x. The wave function represents something completely new in physics. A further novelty is that the wave function is not an ordinary number, as it would be for a simple intensity, but a complex number (Non-mathematicians should not get alarmed: it will be quite sufficient to think of a complex number as a pair of ordinary numbers. ‘Complex’ in this context means ‘composite’, not ‘complicated’.)
The status of the wave function is contentious to say the least. Some claim it merely represents knowledge, while others want to make it as physical as Faraday’s magnetic field. As I see things, the wave function is incorporeal (not some physical thing like a field or particle) and establishes a ranking of things. The real things are the points of Platonia, the instants of time. Quantum cosmology – at least in one embryonic form – will associate a value of ψ (note the capital) with each point of Platonia. To emphasize how different the wave function is, I like to think of it as some ‘mist’ that hangs or hovers over Platonia, its intensity varying from point to point.
Actually, there are two mists because the wave function, being complex, contains two numbers, which are its two components. I shall call them the red mist and green mist, respectively. I shall also introduce a third number, calling it the blue mist. The intensity of this third mist is determined by the two primary components as the sum of the squares of the red and green intensities. This is the mist mentioned in the early chapters. Those in the know will recognize the three mists as the real and imaginary parts of the wave function and the square of its amplitude.
The prominence that I give to these mists could be regarded by most theoretical physicists (above all Dirac and Heisenberg, were they still alive) as a one-sided, if not to say distorted and naive picture of quantum mechanics. The mists (as opposed to things called operators) are not particularly appropriate for talking about most quantum experiments currently performed in laboratories. However, the experiment I have in mind is not done in a laboratory. It is what the universe does to the instants of time. For this experiment, the one that really counts, I think the language of mists is appropriate. Those who disagree might have second thoughts if they really started to think of how inertial frames and duration arise. I come back to these issues later.
I shall now give, in familiar space-time terms, a quantum-mechanical account of the two-slit experiment (Figure 35). At an initial time, the wave function associated with a particle is in a ‘cloud’ well to the left of the barrier. Inside the cloud, ψ is not zero. Outside, it is zero. As time passes, this cloud moves to the right and, in general, changes its shape. It evolves (in accordance with some definite rules). Typically, it ‘spreads’. At the barrier, some of the cloud is reflected back to the left but some passes through the two slits. Initially there are two separate clouds, but they spread rapidly if the slits are narrow, and soon overlap. Characteristic wave interference occurs. Thus, when the merged wave reaches the screen, ψ is not the same everywhere, and fringes can form. In fact, the best fringes are formed by a steady ‘stream’ of wave function, not a cloud.
The question now arises: where will the particle in Figure 35 be observed? The answer, given already by the German physicist Max Born in 1926, is that ψ determines, through the intensity of the blue mist, the probability of where the particle will be observed. The blue mist enables you to guess where the particle will ‘hit’ – twice the intensity means twice the probability.