Computing the net result of all possible alternatives is a monumental task, even in apparently simple cases. This is not a physics textbook, but it may be useful here to give some indication of how quantum motion is computed. This will show how far from everyday experience quantum behaviour is, and how unlike the classical motion we saw in Galileo's parabola and Huygens's spinning circle.

Motus inter corpora relativus tantum est, said Huygens: movement between objects is relative in all aspects. The absolute velocity of an object does not exist. Any dispute about the ‘true’ state of motion of a single object is meaningless. In particular, it makes no sense to distinguish between ‘rest’ and ‘motion’, provided that this motion proceeds with constant velocity.

A classical particle moves with constant velocity in the absence of accelerations, which are caused by forces. But what does a quantum do?

Picking just one path for the quantum to follow makes no sense in our Universe, because the windowpane experiment tells us that the same causes do not always have the same effects. In fact, we must do exactly the opposite: each path that is not explicitly forbidden, and that is not distinguished from other paths in any way, must be allowed. All force-free paths are to be considered collectively as equal-rights alternatives.

Now we are faced with the following problem. If all paths have equal rights, do they all have equal probability? It turns out that the answer is ‘no’. How must we compute the various probabilities? The answer to that is rather curious and totally contrary to everyday intuition. This is so bizarre, that it took scientists something like half a century to work it out.

A quantum on a path behaves like a rolling wheel. Imagine that we have a wheel with a given circumference. On the wheel, an arrow drawn from the axle to the air valve on the rim is called the amplitude of the quantum. When the wheel rolls along, that arrow rotates about the axle. The angle through which the wheel has rotated is called the phase of the quantum.