Time Evolution of Spin in a Magnetic Field

[JoseGG]

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.

 

[JoseGG]

I ofcourse had to use the commutation relations, for S got it figured out. $$\dot{S}$$ is then just a vector.