How to Derive the Lande g Factor from the Zeeman Effect?

[KFC]

Homework Statement

Deduce the expression of Lande g factors from Zeeman effect

Homework 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

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

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 $$J_z + S_z$$?

 

[Jhero]

Thank you for your question. The expression for the Lande g factor can be derived from the perturbation theory of the Zeeman effect. In perturbation theory, the energy correction due to an external magnetic field is given by the expectation value of the perturbing Hamiltonian, in this case \vec{S}\cdot\vec{L}.

To connect this to the expression for the Lande g factor, we can use the fact that the total angular momentum J is given by the sum of the orbital momentum L and spin S. Therefore, we can rewrite the perturbing Hamiltonian as \vec{J}\cdot\vec{J} = \vec{L}\cdot\vec{L} + \vec{S}\cdot\vec{S} + 2\vec{L}\cdot\vec{S}.

Using this, we can see that the expectation value of the perturbing Hamiltonian can be written as \langle jm|\vec{J}\cdot\vec{J}|jm\rangle = \langle jm|\vec{L}\cdot\vec{L}|jm\rangle + \langle jm|\vec{S}\cdot\vec{S}|jm\rangle + 2\langle jm|\vec{L}\cdot\vec{S}|jm\rangle.

The first two terms can be calculated using the standard angular momentum operators, and the third term can be related to the expectation value of J_z + S_z using the commutation relations of angular momentum operators.

I hope this helps in understanding the connection between the perturbing Hamiltonian and the Lande g factor. Please let me know if you have any further questions.