- #1
Garlic
Gold Member
- 181
- 72
Dear PF,
As an excercise I am to find out how the expectation value of the spin operator evolves over time.
There was a hint, stating that it is enough to show that
$$
e^{i \frac{\phi ( \hat{n} \cdot \sigma )}{2}} \sigma_i e^{- i \frac{\phi ( \hat{n} \cdot \sigma )}{2}} = [R_{ \hat{n} }]_{ij} \sigma_j
$$
Where ## R_{ \hat{n} } ## is a 3x3 matrix that describes a rotation with the angle Φ around the n axis.
In the last exercise we showed that putting a vector between these exponantial terms, we get a rotated vector.
In the given equation the free parameter is the angle, where the spin expectation value would evolve over time.
-So I don't quite get how showing that equation holds would tell us how the spin expectation value evolves over time.
Also: In the exercise it was written S-vector, I was surprised because usually it is asked to find the expectation value of S_z.
The expectation value of S_z would be ±hbar/2.
And when we search for the expectation value of S-vector, I think of a "vector inside a unit sphere" pointing in a specific direction, and this spin vector would somehow rotate over time.
-Am I thinking in the right way?
-In Griffiths I could not find much information about the time evolution of spin over time. Do you know a source where I can find more explanations about this subject?
Thank you very much for your time,
Garli
As an excercise I am to find out how the expectation value of the spin operator evolves over time.
There was a hint, stating that it is enough to show that
$$
e^{i \frac{\phi ( \hat{n} \cdot \sigma )}{2}} \sigma_i e^{- i \frac{\phi ( \hat{n} \cdot \sigma )}{2}} = [R_{ \hat{n} }]_{ij} \sigma_j
$$
Where ## R_{ \hat{n} } ## is a 3x3 matrix that describes a rotation with the angle Φ around the n axis.
In the last exercise we showed that putting a vector between these exponantial terms, we get a rotated vector.
In the given equation the free parameter is the angle, where the spin expectation value would evolve over time.
-So I don't quite get how showing that equation holds would tell us how the spin expectation value evolves over time.
Also: In the exercise it was written S-vector, I was surprised because usually it is asked to find the expectation value of S_z.
The expectation value of S_z would be ±hbar/2.
And when we search for the expectation value of S-vector, I think of a "vector inside a unit sphere" pointing in a specific direction, and this spin vector would somehow rotate over time.
-Am I thinking in the right way?
-In Griffiths I could not find much information about the time evolution of spin over time. Do you know a source where I can find more explanations about this subject?
Thank you very much for your time,
Garli