The spin of an electron is described by a vector [psi] = mat([psi_up],[psi_down]) and the spin operator S = Sxi+Syj+Szk with components Sx = (h/2)*mat([0,1],[1,0]), Sy = (h/2)*mat([0,-i],[i,0]), Sz = (h/2)*mat([1,0],[0,-1]).

a)i) State the normalisation condition for [psi].

ii) Give the general expressions for the probabilities to find Sz =+-(h/2) in a measurement of Sz.

iii) Give the general expression of the expectation value <Sz>.

b)i) Calculate the commutator [Sy,Sz]. State whether Sy and Sz are simultaneous observables. 

ii) Calculate the commutator [Sx,S2], where S= Sx2 + Sy2 + Sz2. State whether Sx and S2 are simultaneous observables. 

c)i) Show that state [phi] = (1/sqrt(2))*mat([1],[1]) is a normalised eigenstate of Sx and determine the associated eigenvalue.

ii) Calculate the probability to find this eigenvalue in a measurement of Sx, provided the system is in the state [phi] = (1/5)*mat([4],[3]).

iii) Calculate the expectation values <Sx>, <Sy>, <Sz> in the state [psi].

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