# Lande g-factor and total angular momentum conservation

## Main Question or Discussion Point

I'm reading about the derivation of the lande' g-factor which comes about when one considers an electron moving about a nucleus which is put in an external magnetic field. This gives rise to a perturbative hamiltonian

$$H = - (\vec \mu_s + \vec \mu_s) \cdot \vec B_{ext} = \frac{e}{2m} \vec{B}_{ext} \cdot (\vec J+ g \vec S)$$[/tex]

and to find the associated energy (expectation value of H) one encounters the problem that the spin S is not conserved in this situation but and one then states that the toatal spin J is conserved and that S will be precessing about J. Therefore the average value of S which is interessting for the expectation value can be expressed as

$$\vec S_{av} = \frac{(\vec S \cdot \vec J)}{J^2} \vec J$$

and we can express

$$2 \vec S \cdot \vec J = J^2 + S^2 - L^2$$

and the problem is essentially solved from there. Now what I wonder about is how one really figures out that the total angular momentum _is_ conserved. I would like a classical (and QM) proof of this statement. Could anyone lead me in the right direction?