Chapter 2 – Wave functions and uncertainty

Uncertainty

We have an idea of what momentum (velocity) and position is for physical objects but what are these when particles are represented by a wave? We saw earlier that momentum roughly translates to the wavelength which gives this diagram for what we know about different types of waves. (Note the Δ symbol means uncertainty. Position is represented by x and momentum is usually represented by the letter p):

The top row shows a continuous wave. We know its momentum exactly but it doesn’t have a position.

The middle row has a wave packet that has a position but the position isn’t precise. There is also possible increasing error in the momentum.

The third row has a very precise position but not enough wave cycles to exactly know the momentum.

the Heisenberg Uncertainty Principle formalizes this by showing that the uncertainty in position multiplied by the uncertainty in momentum must be greater than a constant. In other words, we can’t know the exact position and momentum of a particle at the same time!

This uncertainty is a property of the wave nature of matter – even the most accurate measurements have this uncertainty!

What happens if we measure both in sequence? Accurately measuring the position collapses the wave function for momentum to having great uncertainty. Similarly measuring the momentum collapses the wave function for position to having great uncertainty.

Measurement causing changes also means the order of measurement matters. The result from measuring position then momentum will not be the same as from measuring momentum then position.

Uncertain energy

There are other pairs to complementary attributes that have linked uncertainty. One pair is energy and time. That means we can measure a particle’s energy exactly but not be sure exactly when it had that energy. Short-lived quantum states have an uncertainty in their energy