Assignment 2 solutions

Let's define the wavefunction:

> psi := (n,x) -> sqrt(2/L)*sin(n*Pi*x/L);

For n=1:

> evalf(int(psi(1,x)^2,x=0..L/10));

For n=2:

> evalf(int(psi(2,x)^2,x=0..L/10));

For n=3:

> evalf(int(psi(3,x)^2,x=0..L/10));

Bonus:

Define a function whose value is the probability as a function of n:

> pr_left_tenth := n -> int(psi(n,x)^2,x=0..L/10);

Let's test it:

> evalf(pr_left_tenth(1));

To plot this function, use the plotting trick given in the handout:

> plot(pr_left_tenth,0..100,style=POINT,adaptive=false,
> sample=[seq(i,i=1..100)]);

The probability approaches . As n gets larger, there are more and more peaks in the probability density and these peaks are more and more tightly spaced. Accordingly, every interval becomes more or less the same and the probability becomes uniform in the box. This agrees with the correspondence principle.