Figure 38. Two ‘spiky’ wave patterns (thin curves) are superposed to make a much smoother pattern (heavy curve).
The wave function of a particle is assumed to be a maximal representation of its physical state. It codes everything that can ever be deduced about the particle at an instant. Using it, we can predict the outcomes of experiments performed on the particle. There are two cardinal facts about these predictions. First, they are probabilistic. Only if, for example, the particle is in a momentum eigenstate (represented by the two special plane waves described above) will measurement of the momentum confirm that the particle has the corresponding momentum. If it is in a superposition of momentum eigenstates, then any one of the momenta in the superposition may be found as a result of the measurement. The probabilities for them are determined by the strengths with which the corresponding momentum eigenstates are represented in the superposition.
It is a basic Copenhagen tenet that the probabilistic statements reflect a fundamental property of nature, not simply our ignorance. It is not that before the measurement the particle does have a definite momentum and we simply do not know it. Instead, all momenta in the superposition are present as potentialities, and measurement forces one of them to be actualized. This is justified by a simple and persuasive fact. If we do not perform measurement but instead allow ψ to evolve, and only later make some measurement, then the things observed later (like the two-slit fringes) are impossible to explain unless all states were present initially and throughout the subsequent evolution. Outcomes in quantum mechanics are determined by chance at the most fundamental level. This is the scenario of the dice-playing God that so disturbed Einstein.
If anything, the second cardinal fact disturbed him even more. There seems to be a thoroughgoing indefiniteness of nature even more radical than the probabilistic uncertainties. As we have seen, one and the same state can be regarded as a superposition of either momentum or position eigenstates. It is the way this mathematics translates into physics that is startling. The experimentalist has complete freedom to choose what is to be measured: position or momentum. Both are present simultaneously as potentialities in the wave function. The experimentalist merely has to choose between set-ups designed to measure position or momentum. Once the choice is made, outcomes can then be predicted – and one outcome is actualized when the measurement is made. In fact, the indefiniteness is even greater since other quantities, or observables as they are called, such as energy and angular momentum, are also present as potentialities in ψ.
Only one experiment can be made – for position or momentum, say, but not both. Every measurement ‘collapses’ the wave function. After the collapse, the wave function, which could have been used to predict outcomes of alternative measurements, has been changed irrevocably: there is no going back to the experiment we opted not to perform. It is a very singular business. Whatever observable we decide to measure, we get a definite result. But the observable that is made definite depends on our whim. The many people who, like Einstein, believe in a real and definite world find this immensely disconcerting. What is out there in the world seems to depend on mere thoughts that come into our mind. Most commentators believe that this radical indefiniteness – the possibility to actualize either position or momentum but not both – is the most characteristic difference between classical and quantum physics. In classical physics, position and momentum are equally real, and they are also perfectly definite.
The fact that in quantum mechanics one can choose to measure one but not both of two quantities was called complementarity by Bohr. Pairs of quantities for which it holds are said to be complementary.
Heisenberg’s famous uncertainty relation gives quantitative expression to complementarity for position and momentum. De Broglie’s relation λ=h/mv=h/p determines the wavelength of a particle of momentum p=mv, where m is its mass and v its velocity. Now, it follows from Fourier’s work on the superposing of waves that a wave packet restricted to a small spatial region contains many waves in a broad spread of wavelengths. To narrow down the spatial positions q, it is necessary to broaden the range of momenta p. Conversely, to get a nearly definite p, we must accept a wide range of positions q.
Mathematically, we can in fact construct wave packets in which the positions are restricted to a small range, from q to q + Δq, and the momenta to a correspondingly small range, from p to p + Δp. Any attempt to make Δq smaller necessarily makes Δp larger, and vice versa. Heisenberg’s great insight – his uncertainty relation – was the physics counterpart of this mathematics. There is always a minimum uncertainty: the product ΔqΔp is always greater than or, at best, equal to Planck’s constant h divided by 4π. If you try to pin down the position, the momentum becomes more uncertain, and vice versa. This is the uncertainty relation. Moreover, a wave packet of minimum dimensions will in general spread: the uncertainty in the position will increase. This is what in quantum mechanics is known as the ‘spreading of wave packets’.
Since Planck’s constant h is so small, an object like a pea or even a grain of sand can effectively have both a definite position and definite momentum, and the spreading of its wave packet takes place extremely slowly. This explains why all the macroscopic objects we see around us can seem to have definite positions. But though the quantum laws allow objects to be localized in space and to have effectively definite velocities, there is no apparent reason in the equations why this should habitually be so. They also allow – encourage, one might even say – a pea’s wave packet to be localized in two or more places at once. Nothing forces ψ to ‘localize’ around a single point. Einstein used to look at the Moon and ask why we do not see two. It is a real problem. Quantum measurements on microscopic systems are actually designed to create situations in which a macroscopic instrument pointer is, according to the equations, in many places at once. Yet we always see it at only one.
We shall come back to this mystery, which is one aspect of another: Hilbert space and transformation theory. If you find this section a bit abstract, don’t worry; it is helpful at least to mention these things. In quantum mechanics, position and momentum (and other observables) play a role rather like coordinates – ‘grid lines’ – on a map. Just as in relativity the coordinates on space-time can be ‘painted’ in different ways, so too in quantum mechanics there are many mathematically equivalent ways of arranging the coordinates. This was one of Dirac’s first great insights, and it led to his transformation theory.
According to this, the state of a quantum system is some definite but abstract thing in an equally abstract Hilbert space. The one state can, so to speak, be looked at from different points of view. A Cubist painting might give you a flavour of the idea. In relativity, different coordinate systems on space-time correspond to different decompositions into space and time. In quantum mechanics, the different coordinate systems, or bases, are equally startling in their physical significance. They determine what will happen if different kinds of measurement, say of position or of momentum, are made on the system by instruments that are external to the system. The state in Hilbert space is an enigmatic gem that presents a different aspect on all the innumerable sides from which it can be examined. As Leibniz would say, it is a city multiplied in perspective. Dirac was entranced, and spoke of the ‘darling transformation theory’. He knew he had seen into the structure of things. What he saw was some real but abstract thing not at all amenable to easy visualization. But the multiplication of viewpoints and the mathematical freedom it furnished delighted him.