Calculating variance of momentum infinite square well

[Robsta]

Homework Statement

Work out the variance of momentum in the infinite square well that sits between x=0 and x=a

Homework Equations

Var(p) = <p²> - <p>²

$$ p = -i\hbar \frac{{\partial}}{\partial x} $$

The Attempt at a Solution

I've calculated (and understand physically) why <p> = 0

Now I'm calculating $$<p^2> = \int_{0}^{a} sin(\frac{nπx}a)(-\hbar^2)\frac{{\partial}^2}{\partial x^2}sin(\frac{nπx}a) dx$$

$$<p^2> = ({\frac{n\pi\hbar}{a}})^2 \int_{0}^{a} sin(\frac{nπx}a)sin(\frac{nπx}a) dx$$

$$ <p^2> = ({\frac{n\pi\hbar}{a}})^2 * \frac{a}2 $$

I'm out here by a factor of a/2 because of the integral and I'm not sure why, does anybody have any suggestions?

 

[Orodruin]

You forgot to normalise your eigenstates.

 

[Robsta]

Oh yes, that's exactly right, thanks very much. Was staring at this for ages, much appreciated :)