Time average value of Spin operator

[Muthumanimaran]

From the book Introduction to Quantum Mechanics by Griffiths,. In the section 6.4.1 (weak field zeeman effect) Griffiths tells that the time average value of S operator is just the projection of S onto J while finding the expectation value of J+S

$$S_{avg}=\frac{(S.J)J}{J^2}$$

How to prove this?

 

[kuruman]

Griffiths gives the standard argument in the vector model for the atom that when $S$ precesses rapidly about $J$, the tranverse components time-average to zero and the operator can be replaced with a time-averaged operator which is the projection of $S$ on $J$. Now if you have two regular old vectors, $A$ and $B$ with angle $\theta$ between them, you would write the projection of $A$ on $B$ as $$A_B=A\cos\theta=\frac{(\vec A \cdot \vec B)}{AB}A=\frac{(\vec A \cdot \vec B)}{B^2}B.$$

 

[Muthumanimaran]

Griffiths gives the standard argument in the vector model for the atom that when $S$ precesses rapidly about $J$, the tranverse components time-average to zero and the operator can be replaced with a time-averaged operator which is the projection of $S$ on $J$. Now if you have two regular old vectors, $A$ and $B$ with angle $\theta$ between them, you would write the projection of $A$ on $B$ as $$A_B=A\cos\theta=\frac{(\vec A \cdot \vec B)}{AB}A=\frac{(\vec A \cdot \vec B)}{B^2}B.$$

Im satisfied with the Griffith's explanation for the above expression, but out of curiosity I am looking for the mathematical proof of the same expression. While searching internet about this question, I saw "Wigner Eckart Theorem" could be used to find this expectation value, but I don't know how? Any idea how to do that?

 

[kuruman]

Im satisfied with the Griffith's explanation for the above expression, but out of curiosity I am looking for the mathematical proof of the same expression. While searching internet about this question, I saw "Wigner Eckart Theorem" could be used to find this expectation value, but I don't know how? Any idea how to do that?

I believe that $\vec S_{avg}$ is an operator, not an expectation value. If by "mathematical proof" you mean "Starting with an expression for the time-averaged spin operator, use the Wigner-Eckart theorem to show that $$
\vec{S}_{avg}=\frac{(\vec S \cdot \vec J)\vec J}{J^2}$$ in the weak field approximation", the answer is "no I don't have an idea how to do that."
However, you don't need the Wigner-Eckart theorem to find the expectation value $<\vec S_{avg}>.~$ Just follow Griffiths, equations 6-73 to 6.75.