How Does the Quantum Operator \(\hat{p}^2\) Derive from \(\hat{p}\)?

[N00813]

Homework Statement

Given that $\hat{p} = -i\hbar (\frac{\partial}{\partial r} + \frac{1}{r})$, show that $\hat{p}^2 = -\frac{\hbar^2}{r^2} \frac{\partial}{\partial r}(r^2 \frac{\partial}{\partial r})$

Homework Equations

Above

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.

 

[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.

 

[N00813]

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.

Thanks!

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