Time Evolution of Spin in a Magnetic Field

AI Thread Summary
The discussion focuses on the time evolution of a spin 1/2 particle in a magnetic field, described by the Hamiltonian H = g S · B. Participants are tasked with finding the time derivative of the spin operator, dot S, using the Heisenberg equation of motion. The specific case of a magnetic field aligned along the z-axis is considered, requiring the calculation of eigenstates and eigenvalues of dot S_y. One participant expresses difficulty in progressing without first determining dot S and seeks assistance with the calculations. The conversation emphasizes the importance of understanding commutation relations to solve the problem effectively.
JoseGG
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Homework Statement


The Hamiltonian of a spin 1/2 particle is given by:
$$H=g\overrightarrow { S }\cdot \overrightarrow { B } $$
where ##\overrightarrow { S }=\hbar \overrightarrow{\sigma }/2## is the spin operator and ##\overrightarrow { B }## is an external magnetic field.
1. Determine ##\dot { \overrightarrow{ S } } ## as a function of S⃗ and B⃗ .
2. Consider now the particular case in which B⃗ = ##\hat{z}##B is oriented along ##\hat{z}##. Calculate the eigenstates and eigenvalues of ##\dot{S_y}##.

3. For t = 0 the system is in one of the eigenstates of ##\dot{S_y}## . Calculate the time evolution of the spin state and of the expectation value of the energy.


2. Homework Equations

1. So we are working with Heisenberg, equaiton of motion. They ask what is the rate of change of Spin operator with time. We are dealing with,
$$ \frac { dS }{ dt } =\frac { 1 }{ i\hbar } \left[ \overrightarrow { S } ,H \right] $$

2. ?

3. I think the timeevolution operator on the state

$$e^{-iHt/\hbar}\left |s \right> $$

The Attempt at a Solution


1. I interpet the ##\overrightarrow { S}## in the Heisenberg equation of motion as, a vector of [Sx,Sy,Sz], I don't know how to work with the Heisenberg equation to find the dirrevative.

2. Use the found ##\dot{\overrightarrow {S}}## vector and pick ##\dot{S_y}##, solve it as an eigenvalue problem with, spin up, with an eigen value of one.

I am not able to move forward without the first task. Any help would be nice. It is possible my attempt are not correct.
 
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I ofcourse had to use the commutation relations, for S got it figured out. $$\dot{S}$$ is then just a vector.
 
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