Time average value of Spin operator

In summary: The expectation value is given by $$\label{eq:ExpectationValue}E(\vec{S})=\int_{-\infty}^\infty \left( \frac{\vec S}{\vec J} \right)^{-1} J^2\vec S d\vec J$$
  • #1
Muthumanimaran
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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?
 
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  • #2
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 $$A_B=A\cos\theta=\frac{(\vec A \cdot \vec B)}{AB}A=\frac{(\vec A \cdot \vec B)}{B^2}B.$$
 
  • #3
kuruman said:
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?
 
  • #4
Muthumanimaran said:
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.
 
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1. What is the time average value of the spin operator?

The time average value of the spin operator is a measure of the average spin of a particle over a period of time. It is calculated by taking the average of the spin operator over a specific time interval.

2. How is the time average value of the spin operator calculated?

The time average value of the spin operator is calculated by taking the integral of the spin operator over a specific time interval and dividing it by the total length of the time interval.

3. What does the time average value of the spin operator tell us?

The time average value of the spin operator tells us the average orientation of the spin of a particle over a period of time. It can provide information about the magnetic moment of the particle and its interaction with external magnetic fields.

4. Is the time average value of the spin operator a constant value?

No, the time average value of the spin operator is not a constant value. It can change depending on the specific time interval and the behavior of the particle's spin during that interval.

5. How is the time average value of the spin operator related to quantum mechanics?

The time average value of the spin operator is a concept in quantum mechanics that helps us understand the behavior of particles with spin. It is used to calculate the average spin of a particle over a period of time and is essential in describing the quantum state of a system.

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