(5-11)   ①   In the examples we used in figures V-vi and V-vii we integrated cos (kx) over a range of k from 0 to kmax. This means that for any value of k, the probability density function F(k) equals zero if k<0 or k>kmax. If k is in the range of 0 to kmax, F(k) equals a constant, c, (since, in effect, we weighted all these waveforms equally in our superpositions).

Now, recall that since the momentum, were we to measure it, must have some value, we can state that:                          

Use this requirement to deduce the value of c. You can either do this by formal integration or by using a graphical argument. 

②   What is the mean value of the momentum, <p>, in these examples?

③   Use integration to deduce an expression for <p2>.

④    Use equation V-11 to find an expression for the standard deviation of the momentum in these examples.