Relate Magnetic Field, Rotations, and Length of Area in Spin 1/2 Systems?

[Cosmossos]

Here is the problem:

How can relate the magnetic field and the rotations and then the length of the area? I only know that 2pi rotation give the initial state a pi phase but that's all.
where the hell I get the length from?

thanks

 

[nickjer]

This is just a time evolution problem. You know the initial wavevector will evolve in time according to:

$$|\psi(t)\rangle = \exp(\frac{-iHt}{\hbar})|\psi(0)\rangle$$

Use the hamiltonian for a spin in a magnetic field. It will look very similar to the rotation generator. Also you know the time the particle is in the magnetic field knowing its velocity.