Question

A spin-1 particle interacts with an external magnetic field B = B. The interaction Hamiltonian for the system is H = gB-S, where S-Si + Sỳ + SE is the spin operator. (Ignore all degrees of freedom other than spin.) (a) Find the spin matrices in the basis of the S. S eigenstates, |s, m)) . (Hint: Use the ladder operators, S -S, iS, and S_-S-iS,, and show first that s_ | 1,0-ћ /2 | 1.-1)) . Then use these to find the representations in terms of 3 matrices.) 3 (b) From the matrix representation of S, show that its eigenvectors Clarification on notation: The notation we used in class for spin-1/2 particle was +z for spin-up in the z-direction, -z) for spin-down in the z-direction, etc. In the case of spin-1 (as in this problem) there is also the possibility of spin zero in a given direction, so we could perhaps use a notation like +z then read z), 0z). The expressions above would It is better, though, to use the notation | s. m) in this problem, which makes it easier to display the action of the ladder operators. (There are many other alternative notations adopted by textbooks. It is good to get used to it.) (c) If the particle is in the state 11,1〉at t =0,show that the state at a later time , is (d) Show that the probability of finding the particle in the state ,-1) is sin' [gBt/2 (Hint: Transform back to the S, eigenstates.)

Answer 1

A) Accodinto n A Spin-1 pan Hcle 、ntenactSedith an ex4ennaf magnetic eld the intenacion amiltonian or the Sygm 8 2

さち'120 34

四凸。 2 2 firally 2

o pennsS on this Soes 2 to 0

984 gBt 在. 2 2 Bt 4 Bt in () 2 : probability pYO