(6-9)   ① The probability density function is, for a given n, 

So we can find the probability that the particle would be found between x=0.09L and x=0.11L by evaluating the definite integral:

Following the tricks we figured out in Question 6-7, we can rewrite this integral as:

And, doing the same integration by substitution we did in Question 6-7, we get:

②    Plugging in the numbers, this gives us:       

For n = 1,   Pr  =  0.02 – 0.0162  =  0.0038

For n = 5,   Pr  =  0.02 + 0.0197  =  0.0397

For n = 10,   Pr  =  0.02 – 0.0187  =  0.0013

The graphs of ψ2 are shown below, with the range of the integration highlighted:

Recall, from Question 6-8, that if the probability were constant along the whole length of the box (and, because of the normalisation, therefore equal to the mean value of ψ2 for the actual wavefunctions) the probability of finding the particle in this range of x would be 0.02.

• For the ground state, n=1, the probability of finding the particle is maximal near the middle of the box. Therefore it seems right that near the edge of the box, where x is around 0.1L, the probability is considerably less than it would be if the probability density were constant across the box.

• For n=5, 0.1L coincides with a maximum in ψ2  so the calculated probability of finding the particle there is, as expected, bigger than it would be if the probability density were constant across the box.

• For n=10, 0.1L coincides with a node in the wavefunction, so the probability density is close to zero, hence the very small calculated probability in this case.

④  Following the same method, this time we get (for n = 5000):

So, to a very good approximation, the probability is the same as the classical (i.e. it has an equal chance of being anywhere in the box) prediction. You can see that this will be the case for any large value of n by considering the expression that we evaluate to find the probability:

As n becomes very large, the denominator (2nπ) gets so big that the second term becomes negligible.

The reason for this is that, as we noted earlier there are n-1 nodes in the box of length L. So when n is large, there will be many, many nodes and maxima even in a narrow range such as 0.02L. The mean probability density in this range is therefore going to be very close to the overall mean value.