16. In the spin space of a particle of spin 1 / 2 , suppose the particle is in a spin state which is spin-up along the z direction: | ψ s i = | 1 2 , 1 2 i z , so that S z | ψ s i = 1 2 | ψ s i . Consider the component S u = ~ S · ˆ u, of the spin operator ~ S along a direction ˆ u =s in θ ˆ x +cos θ ˆ y in the xz plane, making an angle θ with respect to the z axis. (a) In the standard representation | 1 2 , ± 1 2 i z of eigenstates of S 2 and S z , construct the 2 × 2matr ix[ S u ] representing the operator S u . (b) If S u is measured on the state | ψ s i , what values can be obtained, and with what probability will those values be obtained. (c) What is the mean value h S u i for this state? 17. A quantum system is in an eigenstate | j,m i of J 2 and J z . (a) Showthatisitalsoinaneigenstateof J 2 z and of J 2 x + J 2 y , (but not, generally of J x or J y ) and determine
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