# Time average value of Spin operator

• B
• Muthumanimaran
In summary: The expectation value is given by $$\label{eq:ExpectationValue}E(\vec{S})=\int_{-\infty}^\infty \left( \frac{\vec S}{\vec J} \right)^{-1} J^2\vec S d\vec J$$
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?

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 $$A_B=A\cos\theta=\frac{(\vec A \cdot \vec B)}{AB}A=\frac{(\vec A \cdot \vec B)}{B^2}B.$$

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?

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.

BvU

## 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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