# Homework Help: Quantum Angular momentum Question

1. Feb 9, 2008

### malawi_glenn

[SOLVED] Quantum Angular momentum Question

1. The problem statement, all variables and given/known data

Show that $$\dfrac{{\vec{p}} ^2 }{2m} = \dfrac{{\vec{l} }^2}{2mr^2} - \dfrac{\hbar ^2}{2mr^2}\dfrac{\partial}{\partial r} (r^2 \dfrac{\partial}{\partial r})$$ is rotational invariant under the rotation generated by: $$\vec{j} = \vec{l} + \vec{s}$$ , s is intrinic spin.

2. Relevant equations

[H,J] = 0 and/or [H,J^2] = 0

3. The attempt at a solution

I think that the second part, the radial operator $$\dfrac{\hbar ^2}{2mr^2}\dfrac{\partial}{\partial r} (r^2 \dfrac{\partial}{\partial r})$$ commutes with the angular momentas, since it is just a function of radial coordinate, whereas angular momenta depends on the angles (direction) Is that correct?

Last edited: Feb 9, 2008
2. Feb 9, 2008

### kdv

Yes, any function of r i strivially invariant under rotation since $$[f(r), L_i] = 0$$ because the $$L_i$$ depend only on the angles, as you say.

3. Feb 9, 2008

### malawi_glenn

Just as I thought then, Didn't wanna search or work out the differential form of J before I was sure :) Thanx!

4. Feb 9, 2008

### kdv

You're welcome.

They are given in spherical coordinates toward the middle of the page at xbeams.chem.yale.edu/~batista/vvv/node16.html

5. Feb 9, 2008

### malawi_glenn

Thanx!

Was wondering if you know if spin also have differential form? and j (total angular mom.). Or if you must work em out using group theory?

6. Feb 9, 2008

### kdv

You have to work using group theory. The differential form approach only produces the integer angular momentum representations. It's only by working with the abstract formalism of commutator and operators that one can generate all the spin representation including the half integer ones. For the spatial angular momentum calculations, one has the choice of working with explicit spatial wavefunctions or with matrices and column vectors, etc. For spin, one must work with the matrix representations.

7. Feb 9, 2008

### malawi_glenn

Ok I got it :)

So how can I argue that f(r) and s commutes? same as with L, that s only depends on direction?

8. Feb 9, 2008

### kdv

They commute but the reason is that s acts on a totally different space, so they commute trivially. The total Hilbert space is a direct product of the spin space and the Hilbert space of spatial wavefunctions. Any operator acting in one of the space commutes with any operator acting in the other space.

9. Feb 9, 2008

### malawi_glenn

yeah, of course.. I have done angular momentum in QM now in 6h.. maybe shall go and cook some food ;) this one was so obviuos, I should be ashamed..