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As we saw in lecture, a spin exposed to a magnetic field experiences Larmor precession where the spin polarization vector rotates around the magnetic field vector. This is because the time- translation operator takes the form of a spin rotation operator. Let's close out the semester by looking at that more closely! The spin operator S2 is the generator of spin rotations about the z-axis. That is, to rotate a state about the z-axis by angle 4, we apply the operator R(+z,4)=ettips, Consider an observable A that is compatible with S2. That is, an observable with associated operator  such that = 0. (d) (4 points] Show that this implies [Â, Â(z,)] = 0. From this, show that  remains invariant under an active transformation by R(z,4). Finally, consider a particle of spin S = 1. This is a three-state system. We will use the S2-eigenbasis {|1,-1),|1,0),|1,-1)}. (e) [5 points] Find the matrix representation of the operator St with respect to the S2-eigenbasis and find its eigenvalues. solution.] [Note: It is not enough to give an answer. You need to show all details and explanations in your

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