The Lande' g-factor is a geometric factor which arises in the evaluation of the magnetic interaction which gives the Zeeman effect. The magnetic interaction energy
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which is continuous in the classical case takes on the quantum form
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which is like a vector operation based on the vector model of angular momentum
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The problem with evaluating this scalar product is that L and S continually change in direction as shown in the vector model. The strategy for dealing with this problem is to use the direction of the total angular momentum J as a coordinate axis and obtain the projection of each of the vectors in that direction. This is done by taking the scalar product of each vector with a unit vector in the J direction.
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These vector relationships must be evaluated and expressed in terms of quantum numbers in order to evaluate the energy shifts. Carrying out the scalar products above leads to
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