There are many mysteries in quantum mechanics, and the first is the probabilities. We can send identical clouds through the slits many times. The fringe patterns are always exactly reproduced, but the hits are distributed randomly. Only after many ‘runs’ does a pattern of hits build up. The blue mist gives that pattern. Where its intensity is high, many hits occur; where it is low, few; where it is zero, none. Quantum mechanics determines these probabilities perfectly, but says nothing about where the individual hits will occur.
Figure 35. A ‘cloud’ of wave function ψ approaching two slits (at t = 0 and t = 1), passing through them, dividing into two (at t = 2), spreading and overlapping (t = 3) and impinging on a screen (t = 4).
Einstein found this decidedly disturbing. He could not believe that God reaches for a die every time physicists set up such an experiment and force the particle to show up somewhere. For that is what standard quantum mechanics implies – brute chance determines outcomes. But there are even more puzzling things. It is worth saying that quantum mechanics has a remarkably beautiful and self-contained structure. Examined mathematically, it is a very harmonious whole. It is hard to see how its structure could be modified naturally to make it determine where individual hits occur, especially when relativity is taken into account.
The next mystery is the collapse of the wave function. Just before the particle hits the screen, its ψ can be spread out over a large region. What happens to ψ when the particle is suddenly found somewhere? The standard answer is that the wave is instantaneously annihilated everywhere except where the particle is now known to be.
If we want to determine what now happens, we have to start afresh from a small, reduced cloud. The large cloud has been ‘collapsed’ and has no more relevance. This too provokes much puzzling, especially for those (like Schrödinger in 1926) who would wish to think of ψ as something real, a density of charge, say. How can something real disappear instantaneously? Nothing in the equations describes the collapse – it is simply postulated. Lawful evolution, in accordance with the rules (equations) of quantum mechanics, continues until an observation is made, but then the rules are simply set aside. Quite different rules apply in measurements, as they are called. (In quantum mechanics, the term ‘measurement’ is used a very precise way. It means that some definite arrangement of instruments is used to establish the value of some physical property – say the speed or position of a particle.) The abrupt and schizophrenic change of the rules when measurements are made is a major part of the notorious measurement problem. There are rules for evolution and rules for measurement – and they are even more different than chalk and cheese. Nevertheless, both are excellently confirmed, though we have to be careful when saying that the collapse is instantaneous, and even when it occurs.
Just as mysterious as the rule change when measurements are made is a certain mutual exclusivity about the kinds of measurement that can be made. So far, I have talked only about particle positions. However, we can also measure other quantities – for example, a particle’s energy, momentum or angular momentum. It is particularly fascinating that information about them all is coded at once in ψ. This is another big difference from classical mechanics.
Imagine a perfect sinusoidal wave that extends with constant wavelength from infinity to infinity. For the moment, suppose that it is ‘frozen’, like the wave patterns you see in damp sand at low tide. Let me call this the red wave, because it represents the red mist. Now imagine another identical though green wave, shifted forward by a quarter of a wavelength relative to the red wave (Figure 36). Then the red peaks lie exactly at the green wave’s nodes, where the green wave has zero intensity. As time passes, the red and green waves move to the right, maintaining always their special relative positioning. A wave function in this special form represents a particle that has a definite momentum: if it hit something, it would transmit a definite impulse to it. A particle with the opposite momentum is represented similarly, but travels in the opposite direction and has the green peaks a quarter of the wavelength behind the red peaks. According to the quantum rules, the particle has a definite momentum because its ψ has a definite wavelength and is perfectly sinusoidal. Such wave functions give the best interference effects in two-slit experiments. They are called momentum eigenstates. (The German word eigen means ‘proper’ or ‘characteristic’.)
Figure 36. The wave function of a particle with a definite momentum.
The striking thing about this situation is that the probability for the position of the particle, given by the sum of the squares of the red and green intensities, is completely uniform in space. The reason is that for two sinusoidal waves displaced by a quarter of a wavelength, this sum is always 1 if the wave’s amplitude (its height at the peaks) is 1. This is a consequence of the well-known trigonometric relation sin2A + cos2A = 1, which itself is just another expression of Pythagoras’ theorem. Thus, for a particle in this state, we have absolutely no information about its position, but we do know that it has a definite momentum.
So far we have considered waves of only one wavelength. However, we can add waves of different wavelengths. Whenever waves are added, they interfere, enhancing each other here and cancelling out there. By playing around with waves of different wavelengths we can make a huge variety of patterns (Figure 37 is an example). In fact the French mathematician Joseph Fourier (one of Napoleon’s generals) showed that more or less any pattern can be made by adding, or superposing, sinusoidal waves appropriately. Any wave pattern created in this way and concentrated in a relatively small ‘cloud’ is called a wave packet. The same pattern can be made by superpositions of quite different kinds. The primary meaning of ψ is that its value at x determines, through the squares of its two intensities, the probability that the particle will be ‘found’ at x. Now, a ‘cloud’ could be so narrow that it becomes a ‘spike’ at some value of x. The particle can then be at only one place – at the spike. Such a wave function is called a position eigenstate.
Figure 37. Superposition of the two waves at the top gives rise to the very different wave pattern at the bottom.
Thus, the same wave pattern can be regarded either as a superposition of plane waves or as a superposition of many such spikes added together with different coefficients (Figure 38). Any wave function is a superposition of either position or momentum eigenstates. There is a duality at the heart of the mathematics. What is remarkable – and constitutes the essential core of quantum mechanics in the standard form it was given by Dirac – is that it perfectly reflects a similar duality found in nature. This is where the measurement problem becomes even more puzzling. We need to consider the ‘official line’, known as the Copenhagen interpretation because it was established by Heisenberg and Bohr at the latter’s institute in Copenhagen shortly after the creation of quantum mechanics.