solution5 - 16. In the spin space of a particle of spin...

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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 z 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 . We have [ S u ]= u x [ S x ]+ u y [ S y u z [ S z ] i n θ [ S x ]+cos θ [ S z ] [ S u μ 1 2 cos θ 1 2 sin θ 1 2 sin θ 1 2 cos θ . (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. The only values that can be obtained are the eigenvalues m u of S u . These are found by solving the characteristic equation det ( S u m u )=0 , which gives m u = ± 1 2 , the same as it is for any of the spin components. To f nd the probability we must solve for the eigenvectors. Putting m u = ± 1 / 2intothe eigenvalue equation gives, after normalization, the eigenvectors | 1 2 , 1 2 i u = r 1+cos θ 2 | 1 2 , 1 2 i z sin θ θ | 1 2 , 1 2 i z ¸ | 1 2 , 1 2 i u = r 1 cos θ 2 | 1 2 , 1 2 i z + sin θ 1 cos θ | 1 2 , 1 2 i z ¸ The probability that a measurement of S u on | ψ i = | 1 2 , 1 2 i z will yield m u = 1 2 is P μ m u = 1 2 = ¯ ¯ ¯ ¯ u h 1 2 , 1 2 | 1 2 , 1 2 i z ¯ ¯ ¯ ¯ 2 = θ 2 and P μ m u = 1 2 = ¯ ¯ ¯ ¯ u h 1 2 , 1 2 | 1 2 , 1 2 i z ¯ ¯ ¯ ¯ 2 = 1 cos θ 2 . (c) What is the mean value h S u i for this state? We evaluate h ψ | S u | ψ i = ¡ 10 ¢ μ 1 2 cos θ 1 2 sin θ 1 2 sin θ 1 2 cos θ ¶μ 1 0 = 1 2 cos θ . 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 the associated eigenvalues.
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solution5 - 16. In the spin space of a particle of spin...

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