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Zeeman effect and lande g factor

  1. Nov 3, 2008 #1


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    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 [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]

    3. 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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