# Commutation Rules for L and J

1. Apr 10, 2015

### fayled

So the total angular momentum operator J commutes with any scalar operator S. The argument for this is that J is the generator of 'turntable rotations' (by this I mean we rotate the whole object about an axis, along with its orientation) and the expectation value of any scalar operator has to be invariant under such a rotation. This tells us that S commutes with the rotation operator and thus its generator J.

My question is why doesn't a similar argument hold for the orbital angular momentum operator L? The difference is that L generates rotations that rotate only the object but not its orientation around an axis. However surely the expectation value of a scalar operator should still be invariant under this type of rotation, meaning L and S commute. However this is not the case, because I know that any component of L does not commute with J2.

2. Apr 10, 2015

### The_Duck

Scalar operators that involve the orientation of the object may not be invariant under the rotations generated by $\vec L$. For example consider the angle between an electron's spin axis and its momentum. This angle is a scalar quantity (related to the scalar operator $\vec S \cdot \vec P$) which is invariant under the full rotations generated by $\vec J$. But it is not invariant under the rotations generated by $\vec L$, which will rotate the momentum but not the spin. You can confirm that $[\vec J, \vec S \cdot \vec P] = 0$ while $[\vec L, \vec S \cdot \vec P] \neq 0$.

3. Apr 10, 2015

### fayled

Beautiful, thanks!