The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics (Oxford Landmark Science)

by Roger Penrose

Here, θ is the angle which the pair of points z and w subtend at the origin in the Argand plane (see Fig. 6.9). (Recall that the cosine of an angle is the ratio ‘adjacent/hypotenuse’ for a right-angled triangle. The keen reader who is unfamiliar with the above formula may care to derive it directly, using the geometry introduced in Chapter 3. In fact, this formula is none other than the familiar ‘cosine rule’, slightly disguised!) It is this correction term 2|w||z| |cos θ that provides the quantum interference between quantum-mechanical alternatives. The value of cos θ can range between –1 and 1. When θ = 0° we have cos θ = 1 and the two alternatives reinforce one another so that the total probability is greater than the sum of the individual probabilities. When θ = 180° we have cos θ = –1 and the two alternatives tend to cancel one another out, giving a total probability less than the sum of the individual ones (destructive interference). When θ = 90° we have cos θ = 0 and we get an intermediate situation where the two probabilities do add. For large or complicated systems, the correction terms generally ‘average out’ – because the ‘average’ value of cos θ is zero – and we are left with the ordinary rules of classical probability! But at the quantum level these terms give important interference effects.

This is fucking nuts. Pythagoras how are you so prominent