Zeeman effect and lande g factor

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KFC
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Homework Statement


Deduce the expression of Lande g factors from Zeeman effect


Homework Equations


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

[tex]\langle jm|J_z + S_z|jm\rangle[/tex]


The Attempt at a Solution


I try to write

[tex]\vec{S}\cdot\vec{L} = \dfrac{1}{2}(\vec{J}^2 + \vec{S}^2 - \vec{L}^2)[/tex]

The solution would be
[tex]g = 1 + \dfrac{j(j+1) + s(s+1) - l(l+1)}{2j(j+1)}[/tex]

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

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