(6-9)  We know, from Question 6-8, that our normalised wavefunctions are:

Therefore, our probability density functions, ψ2, are:

And we further know that we can find the probability that the particle will  be found, if a measurement is made, between x1 and x2 by evaluating the definite integral

Recall that we already found a nifty trick to help with integrating sin2x,  in Question 6-7. So, have a quick look back at what we did there and then:

①  See if you can find a general expression for the probability that the particle would be found between x=0.09L and x=0.11L, for any value of n.

②  Now consider the cases of n = 1, 5 and 10. Calculate the probabilities for these three cases. Then sketch the forms of ψ2 in the box for these cases and hence convince yourself that the numbers you have calculated make sense.

③  What if the quantum number is very big?  Let’s say n= 5000, or whatever big number comes into your head. Work it out, then sit back and have a think about the significance of what you find.