# 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:

## Answers and Replies

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..

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