Average value in a one-dimensional well

[kreil]

Homework Statement

Show that the average value of x² in the one-dimensional well is

$$(x^2)_{av}=L^2(\frac{1}{3}-\frac{1}{2n^2 \pi^2})$$

Homework Equations

wave fuction in 1-dim well:
$$\psi_n(x)=\sqrt{\frac{2}{L}}sin(\frac{n \pi x}{L})$$

$$x^2_{av}=\int_{0}^{L}|\psi(x)|^2 x^2 dx$$

The Attempt at a Solution

Im having trouble evaluating the integral:

$$x^2_{av}=\frac{2}{L} \int_0^Lsin^2(\frac{n \pi x}{L})x^2dx$$

i think this needs to be integrated by parts, but could it be in a table somewhere?

 

[ziad1985]

There should be a table, when I took Quantum mechanics, usually the professor gave a table with solution of some integral(like this one for example), even in exams he did that.
But try to do it manually, more experience..

 

[joob]

yah, simplest way to solve it is integration by parts.
dont forget to use double angles to get rid of
$$sin^2$$