Orbital angular momentum and kinetic energy

[bon]

Homework Statement

Please see attached.

I have two questions:

how does pr get to be -ihbar (D/Dr + 1/r) ? where does the extra factor of D/Dr come from? one comes from r.del, but what about the other? surely div (r/r) = 2/r?

Also, why does [r,pr]=-ih[r,D/Dr] = ih?

surely -ih[r,D/Dr] = -ih(rD/Dr - 1)?

am i missing something?

Homework Equations

The Attempt at a Solution

See my attempt above. thanks!

 

[vela]

To calculate [r, d/dr], you need to use the product rule in calculating the second term:

$$\left[r,\frac{\partial}{\partial r}\right]\psi = r\left(\frac{\partial}{\partial r}\psi\right) - \frac{\partial}{\partial r}(r\psi) = r\left(\frac{\partial}{\partial r}\psi\right) - \left[\left(\frac{\partial}{\partial r}r\right)\psi + r\left(\frac{\partial}{\partial r}\psi\right)\right] = -\psi$$

 

[bon]

Great thanks!

any ideas on the previous bit of my question?

Thanks!

 

[vela]

Probably a similar reason. Try working it out again.