6.6   It's Perfectly Normal

• Normalising a wavefunction to take account of the fact that the particle must be somewhere.

Something is bothering Nefertiti. We still have that multiplier, a, in our wavefunctions for the particle in a box:

We didn’t need this to find the energies, so do we care? Well, remember that the wavefunction is about more than that. It can give us the probability density distribution for our particle but for that, the wavefunction has to be normalised. This means, in this instance,  ensuring that the particle is definitely somewhere within the box. Recall from Chapter 5 that the probability density of finding the particle close to a position x is given by ψ2(x), and therefore the integral of ψ2(x) over all x must equal 1. So, we can write:

Substituting our expression for ψ(x), this becomes:

Let’s see if we can find the value for a which will make this work.

This process generates normalised wavefunctions for our particle in a box, which embodies the probability distribution for where the particle may be found, should we open the box and take a peek.