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I am trying to find the time dependent expectation value for J ($\langle J(t) \rangle$) for a spin 3/2 particle in a uniform magnetic field (in the z direction). My method is as follows: $$H=-\frac{\mu B}{\hbar}J_z$$ Then to find the equaions of motion for $J_i$: $$i\hbar \dot J_i=[J_i,H]$$ $$\dot J_x=\mu B/\hbar \space \space J_y$$ $$\dot J_y=-\mu B/\hbar \space \space J_x$$ $$\dot J_z=0$$ Now I am given tha t the particle is initially in an eigenstate of $J_x$, with eigenvalue +3/2$\hbar$. So I know that: $$J^2|j,m\rangle=\frac{15\hbar^2}{4}|j,m\rangle$$ $$J_x|j,m\rangle=3/2 \hbar|j,m\rangle$$ So to calculate $\langle J(t) \rangle$, I need to know the equations, which I have solved. But then I need to know what $J_y,J_z$ acting on $|j,m\rangle$ is. For the z component of J. $$\langle J_z(t)\rangle=\langle j,m|J_z(0)|j,m\rangle$$ In order to find what $J_z(0)$ is, do I have to create a new raising operator, which would be $Jy\pm iJ_z$ and then find what I initially have in terms of J squared and the x component?

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