The End of Time: The Next Revolution in Physics

Figure 39 The two-dimensional configuration space Q of two particles on one line. The line is shown in multiple copies on the left. Nine different configurations of the two particles on it are shown. The positions of particles 1 and 2 are indicated by the black and white triangles, respectively. The axes of Q on the right show the distances of particle 1 (horizontal axis) and particle 2 (vertical axis) from the left-hand end of the line. The points on the 45° diagonal in Q correspond to configurations for which the two particles coincide (points 1, 4 and 8). You might like to check how the nine configurations on the left are represented by the nine corresponding points on the right.

In Chapter 3 we imagined tipping triangles out of a bag. That exercise was presented because it mimics one of the ways in which we can interpret a quantum state. Imagine now that the blue mist has the distribution shown in Figure 40. To avoid problems with infinite numbers of configurations, we divide up Q by a grid of cells sufficiently fine that ψ hardly changes within any one of them (Figure 40, on the right). The intensity of the blue mist at the central point of each cell then gives the relative probability of the nearly identical configurations in that cell. On a piece of cardboard, let us depict one of these configurations (as shown on the left in Figures 39 and 40). This will serve as the representative of all the configurations of that cell. For the grid in the figure with 100 cells, there are 100 relative probabilities whose sum should be conveniently large, say a million. Then we shall not distort things seriously by replacing exact relative probabilities like 127.8543... by the rounded-up integer 128.

We now imagine putting into a bag the number of copies of each representative configuration equal to its rounded probability, 128 for example. In quantum mechanics, performing a measurement to determine the positions of both particles is like drawing at random one piece of cardboard from the bag. We get some definite configuration. In the process, we destroy the wave function and replace it by one entirely concentrated around the configuration we have found. If we recreate the original wave function, by repeating the operations that we used to set it up, and repeat the experiment millions of times, then the relative frequencies with which the various configurations are ‘drawn from the bag’ will match, statistically, the calculated relative probabilities.

Figure 40 Like Figure 39, this shows nine different configurations of two particles (black and white triangles) on a line and the points corresponding to them in the configuration space Q (on which a grid has been drawn). A possible distribution of the intensity of the blue probability mist is shown as the height of a surface over Q in the top part of the figure (you are seeing the surface in perspective from above, and rotated). In the state of the system shown here, the probabilities for configurations 4, 6 and 9 are high, while 5 has a very low probability.

Figure 41 (a) The effect of measuring, for the probability density of Figure 40, the position of the particle represented by the horizontal axis, and finding that it lies in the interval on which the vertical strip stands. All the wave function outside the strip is instantaneously collapsed.

This is only the start. We can select from a menu of different kinds of measurement. For example, we can opt to find the position of only one particle, which has remarkable implications for what we can say about the other one. Suppose first that we measure the position of just one of the particles. According to the quantum rules, this instantaneously collapses the wave function from its original two-dimensional ‘cloud’ to a one-dimensional profile (Figure 41). The point is that we now know the position of one particle to within some small error, so none of the wave function outside the narrow strip is relevant any longer. It is annihilated. If the particle whose position is measured is represented along the horizontal axis, only a vertical strip of ψ survives (Figure 41(a)); if the position of the other particle is measured, only a horizontal strip survives (Figure 41(b)).

Figure 41 (b) The same for a position measurement of the other particle.

Either profile then gives conditional information. If we know where one particle is, the possible positions of the other are restricted to a narrow strip. The relative probabilities for the position of the second particle are determined by the values of ψ within the strip. Provided we know the original wave function, acquired knowledge about one particle sharpens our knowledge – instantaneously – about the other. This is the place to explain entangled states, or quantum inseparability (Box 12).

BOX 12 Entangled States