A4. The spin of an electron is described by a vector :01Suj + S₂k with components ST(18), Sy10ħ2=24&₁ħ21√2 1,and the spin operator Ŝ = Ŝ₂i+00(17)d), §. = 1/(1-9).i020(a) (i) State the normalisation condition for .(ii) Give the general expressions for the probabilities to find S₂ = ±ħ/2 in a measure-ment of $₂.(iii) Give the general expression of the expectation value (Ŝ₂).[6](b) (i) Calculate the commutator [Sy, S₂]. State whether S, and Ŝ, are simultaneous ob-servables.(ii) Calculate the commutator [S, $²], where S² = S²+²+S2. State whether S, and$² are simultaneous observables.[6]is a normalised eigenstate of S, and determine the(c) (i) Show that the stateassociated eigenvalue.(ii) Calculate the probability to find this eigenvalue in a measurement of Ŝ, provided1/4the system is in the state = -5 3(iii) Calculate the expectation values (Ŝz), (Sy) and (S₂) in the state .[8]

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