# Quantum Mechanics Operators

1. Mar 29, 2014

### N00813

1. The problem statement, all variables and given/known data
Given that $\hat{p} = -i\hbar (\frac{\partial}{\partial r} + \frac{1}{r})$, show that $\hat{p}^2 = -\frac{\hbar^2}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r})$

2. Relevant equations

Above

3. The attempt at a solution
I tried $\hat{p}\hat{p} = -\hbar^2((\frac{\partial}{\partial r})^2 + \frac{1}{r} \frac{\partial}{\partial r} + \frac{\partial}{\partial r}\frac{1}{r} +\frac{1}{r^2})$.

This gave me $-\hbar^2((\frac{\partial}{\partial r})^2 + \frac{1}{r} \frac{\partial}{\partial r} )$ instead of the 2 / r factor I needed.

Last edited: Mar 29, 2014
2. Mar 29, 2014

### BvU

Nope. ${\partial \over \partial r}{1\over r}$ gives ${1\over r}{\partial \over \partial r} -{1\over r^2}$
Remember p is an operator: you have to imagine there is something to the right of it to operate on.

3. Mar 29, 2014

### N00813

Thanks!

I suppose it makes it easier if I had used a test function, and then taken it away.