6.7 Where it's at
Finding the Probability Distribution for Where the Particle Might Be
We can now calculate the square of this function, as a function of x, which will give us the probability density distribution for the particle in each state. Here (Figure VI-iii) what we get for the first three wavefunctions:
Whatever state it is in, the particle does not have a defined position in the box. It has a wavefunction whose square tells us the probability that - if we force it to make a decision by observing it in some way - it will be found in the vicinity of a given position. Until we make that observation it just has the potential to be at a position, if we make a position measurement.
Suppose you want to find the probability that an observation will find a particle near the left hand side of the box, in the region of x = 0.1L. To calculate a probability, we have to specify a finite range, so let’s go for 0.09L to 0.11L. What are our chances of finding the particle there?
Before we get on to the predictions of the Schrödinger equation, let's initially consider what we would expect from classical physics, which is that the probability density is constant across the whole length of the box.
We know this is not physically possible because it violates Erwin's smoothness requirement, which demands that this function must go to zero at the edges of the box, but this "classical" value will provide an interesting comparison with what we'll now go on to find for the actual allowed wave functions:
This Question should convince you that the wave functions we obtain by solving the Schrödinger equation suggest that the probability of finding the particle in a particular narrow window of x varies in rollercoaster fashion as we slide the window across the box. In the ground state (n=1), for example, the particle is much more likely to be close to the middle of the box than it is the near the edges. When n=2, on the other hand, the introduction of a node in the wavefunction at the exact centre of the box means that the probability of finding the particle close to the centre is now very small.
What also emerges clearly, both from the maths and from considering what the wavefunction looks like, is that at high quantum numbers, the effect of quantisation starts to become negligible and it starts to behave like a classical system. This is a point we will explore further later.
