The End of Time: The Next Revolution in Physics

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.

Figure 38 Two ‘spiky’ wave patterns (thin curves) are superposed to make a much smoother pattern (heavy curve).

THE COPENHAGEN INTERPRETATION

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.