How Does Spin Behavior Change in a Rotating Magnetic Field?

[guyafe]

Homework Statement

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.

Homework Equations

Convention given in the description

The Attempt at a Solution

Already solved:
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: $\theta=\arctan\frac{\omega}{\Omega_0}$

Homework Statement

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 $\omega \to 0$?

Homework Equations

None

The Attempt at a Solution

Already solved:
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: $-\hat{x}$.

Homework Statement

(3) What is the maximal angular error $\Delta \theta$ from the previous section if $\omega$ is finite?

Homework Equations

None

The Attempt at a Solution

Sorry, had no idea how to approach this.

Homework Statement

(4) Which values of $\omega$ give zero angular error?

Homework Equations

None

The Attempt at a Solution

Sorry, had no idea ho to solve this

Homework Statement

(5) What is the final direction of the spin in the limit $\omega \to \infty$?

Homework Equations

None

The Attempt at a Solution

Already solved:
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: $+\hat{x}$.

 

[mark726]

Hi there,

It seems like you have already made some progress on this problem, so I will try to provide some additional guidance for the remaining questions.

For (3), we can use the fact that the spin will follow the magnetic field as long as the field rotates slowly enough. This means that there will be some maximum angular error that the spin can have in following the field. Can you think of a way to calculate this maximum error using the given information?

For (4), we are looking for values of \omega that will give zero angular error. In other words, we want to find the values of \omega that will make the spin follow the magnetic field perfectly. Can you think of a way to determine these values by using the given information?

For (5), you have already made an educated guess that the spin will stay in its original position (+\hat{x}) in the limit of \omega \to \infty. Can you think of a way to prove this using the given information and possibly some mathematical reasoning?