WHAT IS QUANTUM THEORY ABOUT?

The Uncertainty Principle

Zeno thought about an arrow in flight. It moves continuously through space. But suppose we want to think about exactly where the arrow head is at some instant. An arrow can only occupy a region of space exactly equal to its size, it can't occupy a larger space or a smaller, and it can't exist in two different places at the same instant. Now the next instant follows immediately - there is no time between one instant and the next - so how can the arrow have moved to a new position? It looks as though if the arrow is at some particular position at one instant of time there is no way for it to arrive at a new position at the next instant!

It is quite difficult to explain today that this is a paradox because we have all been brought up on the infinitesimal calculus of Newton and Leibnitz, even if only subconsciously. The separation between one point and the next and between one instant and the next is NOT zero - it is treated as an infinitesimally small amount which we can make as small is we like as long as we don't let it actually equal zero. Without this sophisticated concept which allows us to deal with continuous changes in position and time, it seems that the only way an arrow can move is by jumping discontinuously from one position to the next.

In quantum theory, however, not all changes are continuous. That is, not all changes can be made up of an infinite number of infinitesimally small parts. For example we saw in the previous section that the energy of an atom changes discontinuously. So Zeno's paradox becomes relevant again. It tells us that we have two options: Either we look at the motion of the arrow through space, and concentrate on the rate at which it passes a point (measured by its momentum), or we think of its position at some instant during its flight. We can't simultaneously take both points of view.

This is the essence of the uncertainty principle. If we measure the position of a particle precisely, we can have no information about its momentum. If on the other hand we measure its momentum precisely, then we have no information about its position. But in the case of quantum mechanics other possibilities are allowed as well. We may measure the position to within some limit x (which we call the uncertainty in position). Then the uncertainty principle sets limits on the accuracy to which the momentum is defined. The uncertainty in momentum cannot be less than p, which is determined by the equation: