Chapter 38, Problem 57GP

To determine

The proof that the average value of square of position $x$ of a particle in state $n$ inside an infinite well of width $l$ is ${\overline{x}}^2={\displaystyle \int x^2{\left|{\psi }_n\right|}^{2}dx}=l^2\left[\frac{1}{3}-\frac{1}{2}{\left(n\pi \right)}^{-2}\right]$ and the graph for ${\overline{x}}^2/l^2$ versus $n$.