Problem 1 We consider a particle with spin 1 in magnetic field. We already derived in class that the matrices correspond

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Problem 1 We Consider A Particle With Spin 1 In Magnetic Field We Already Derived In Class That The Matrices Correspond 1 (61.41 KiB) Viewed 24 times

Problem 1 We consider a particle with spin 1 in magnetic field. We already derived in class that the matrices corresponding to all spin operators are the following in the basis of eigenstates of Sx): SX 0 1 0 1 0 1 0 1 0 0 -1 0 1 0 0 1 -1 0 10 Sz=100 0 0 0 0 -1 2 2i And the Hamiltonian is still H=-yS.B 1. Determine the eigenstates of Sx and Sy 2. A particle is initially in the Sx=ħ eigenstate and the magnetic field is equal to B in the z direction. Determine the time evolution of the wavefunction. 3. Determine <S>>(t), <Sy(t) and <S>>(t). Explain what is happening physically. 4. We now assume that the particle starts in the same state, but that the magnetic field is now pointing in the y direction. Write the system of differential equations for all three components of the spinor, which we will call ci(t), cz(t) and c3(t). 5. Get a second order decoupled D.E. for cz(t). Solve it. (Hint: since your original system of D.E. is first order, if you know ci, c, and cz at t=0, you also know their derivatives at t=0, which is a trick that was used in class) 6. Use your result to find ci(t) and c(t) as well. 7. Determine <Sx>(t), <Sy>(t) and <S>(t). Explain what is happening physically now.