Homework Help: Spin half in rotating field

1. Aug 6, 2011

guyafe

1. The problem statement, all variables and given/known data
A question given by my professor. Already solved three section and need help in other two. Please advice:
The Hamiltonian of a spin $1/2$ in a magnetic field is given by $\mathcal{H}=h(t)\cdot S$, where the magnetic field is of the size $|h(t)|=\Omega_0$ and it is rotating on $XY$ plan with angular velocity $\omega$, so the angle it is forming with $X$ axis after time $t$ is $\varphi = \omega t$.
In order to solve this question you may use the convention that the Hamiltonian of a spin in a rotating system is $\tilde{\mathcal{H}}=\mathcal{H}-\omega S_z$.

(1) Find the angle of inflection $\theta_0$ in which you have to prepare the spin in order for it to follow in a constant angle after the rotating field.

2. Relevant equations
Convention given in the description

3. The attempt at a solution
In the rotating system we can assume without any lost of generality that the magnetic field is always on the $X$ axis direction. Using this and the previous convention we get a new Hamiltonian: $\mathcal{H}=\Omega_0 S_x-\omega S_z$.
We can consider this as a Hamiltonian of a system with an effective magnetic field $\vec{B}=(\Omega_0,0,\omega)$.
If we want the magnetic field to stay constant in this system and not precssitate, we need to prepare it in that direction: $\theta=\arctan\frac{\omega}{\Omega_0}$

1. The problem statement, all variables and given/known data
At time zero you prepare the spin in $X$ direction, and let the spin rotate half a circle.
(2) What is the final direction of the spin in the limit $\omega \to 0$?

2. Relevant equations
None

3. The attempt at a solution
Using an educated guess, we can say that if the magnetic field is rotating "slowly enough", the spin will follow it and end the rotation in the same direction the field does: $-\hat{x}$.

1. The problem statement, all variables and given/known data
(3) What is the maximal angular error $\Delta \theta$ from the previous section if $\omega$ is finite?

2. Relevant equations
None

3. The attempt at a solution
Sorry, had no idea how to approach this.

1. The problem statement, all variables and given/known data
(4) Which values of $\omega$ give zero angular error?

2. Relevant equations
None

3. The attempt at a solution
Sorry, had no idea ho to solve this

1. The problem statement, all variables and given/known data
(5) What is the final direction of the spin in the limit $\omega \to \infty$?

2. Relevant equations
None

3. The attempt at a solution
Another educated guess: Now the field is rotating so fast so the spin won't have time to follow it and it will stay in its position: $+\hat{x}$.