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Spin dynamics and Larmor precession

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Homework Statement



Consider an electron with spin, which should be in a homogenous magnetic field B=B0ez. This situation is described by the Hamiltonian of the shape [itex]\hat{H}=g\frac{\mu_B}{\hbar}\textbf{BS}[/itex].

Consider now the time dependent state [itex]|\psi(t)>[/itex] of the electron in spin space. The spatial structure of [itex]|\psi(t)>[/itex] is not considered here at all!

(a) Show that the expectation values of the spin operators satisfy the equations of motion:

[itex]\frac{d}{dt}<S_x>=-\omega_0<S_y>[/itex]
[itex]\frac{d}{dt}<S_y>=\omega_0<S_x>[/itex]
[itex]\frac{d}{dt}<S_z>=0[/itex].

What is the value of [itex]\omega_0[/itex]?

(b) Show that because of the equations of motion, the expectation value of the spin rotates in the xy-Plane (~Larmor precession)

Homework Equations



When looking up the Larmor precession I found this: [itex]\omega=\gamma\cdot|B|=\frac{gq}{2m}\cdot|B|[/itex].

The Attempt at a Solution


Is the formula I gave in (2) already the solution for [itex]\omega_0[/itex]? Being naive I would think that the rotation cannot happen in the z-Direction, as the temporal derivative of the z-Component of the spin operator is 0. Can that be?
 

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