Average value in a one-dimensional well

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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?

Last edited:

$$sin^2$$