# Zeeman effect and lande g factor

1. Nov 3, 2008

### KFC

1. The problem statement, all variables and given/known data
Deduce the expression of Lande g factors from Zeeman effect

2. Relevant equations
For Zeeman effect, we treat the energy perturbation as $$\vec{S}\cdot\vec{L}$$, where $$S$$ is the spin and $$L$$ is the orbital momentum. Let magnetic field along z direction so the energy correction would be

$$\langle jm|J_z + S_z|jm\rangle$$

3. The attempt at a solution
I try to write

$$\vec{S}\cdot\vec{L} = \dfrac{1}{2}(\vec{J}^2 + \vec{S}^2 - \vec{L}^2)$$

The solution would be
$$g = 1 + \dfrac{j(j+1) + s(s+1) - l(l+1)}{2j(j+1)}$$

The above expression $$\vec{S}\cdot\vec{L}$$ gives the numerator of the second term of the solution, but how to get the denormenator? How to connect the $$\vec{S}\cdot\vec{L}$$ with $$J_z + S_z$$?