# Calculating variance of momentum infinite square well

1. May 18, 2015

### Robsta

1. The problem statement, all variables and given/known data
Work out the variance of momentum in the infinite square well that sits between x=0 and x=a

2. Relevant equations
Var(p) = <p2> - <p>2

$$p = -i\hbar \frac{{\partial}}{\partial x}$$
3. 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?

2. May 18, 2015

### Orodruin

Staff Emeritus
You forgot to normalise your eigenstates.

3. May 18, 2015

### Robsta

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

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