Chemistry 3730 assignment 2 solutions

First, define the wavefunction:

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

Calculate the probability for n=4:

> evalf(int(psi(4,L,x)^2,x=L/10..L/5));

For n=8:

> evalf(int(psi(8,L,x)^2,x=L/10..L/5));

For n=32:

> evalf(int(psi(32,L,x)^2,x=L/10..L/5));}{%

For n=128:

> evalf(int(psi(128,L,x)^2,x=L/10..L/5));

For n=512:

> evalf(int(psi(512,L,x)^2,x=L/10..L/5));

The limit seems to be 1/10. We can confirm this with Maple:

> limit(int(psi(n,L,x)^2,x=L/10..L/5),n=infinity);

This makes sense because for large n, the probability density has more and more increasingly closely packed bumps and starts to look pretty flat over large distances. Thus an interval of 10% of the length of the box should contain 10% of the probability.