# Lande g-factor and total angular momentum conservation

• center o bass
In summary, the derivation of the lande' g-factor involves considering an electron moving around a nucleus in an external magnetic field. The resulting perturbative Hamiltonian leads to the problem of non-conservation of spin, but it is found that the total spin is conserved and the average value of spin can be expressed in terms of the total angular momentum. The Zeeman and Paschen-Back effects are classified based on the strength of the magnetic energy compared to the spin-orbit energy, with the former resulting in diagonal shifts of energy for each state. A classical and quantum mechanical proof of the conservation of total angular momentum is desired.

#### center o bass

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?

"The effect can be classified as weak or strong according to whether the magnetic energy is small or large in comparison with the spin-orbit energy. The Zeeman Effect refers to the weak-field case, while Paschen-Back Effect refers to the strong-field case. In the weak-field case, the magnetic energy has matrix elements between states of different j for but not between states of the same j and different m. We neglect the former, because of the relatively large energy separation between states of different j. Thus the magnetic energy is diagonal with respect to m for each j and shifts the energy of each of the states by its expectation value for that state."