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Unformatted text preview: R =0. (f) Show that the projection operator onto the spin in the direction can be expressed as n;+><n;+ = n ˆ ) ˆ ( 2 1 r ⋅ + n I . Can an arbitrary pure state of the spin 1/2 particle be expressed this way? 2. Suppose we have an ensemble of spin 1/2 particles that consists of a statistical mixture with 1/3 of the ensemble in the state z;+> and 2/3 of the ensemble in the state z;>. (a) Find the matrix of the density operator in the basis {z;+>, z;>}, and in the basis of eigenstates of S x , {x;+>, x;>} (b) Find, <S x >, <S y >, <S z > for this state, and compare it to the result for the mixed state ) ; ;   ; ; ( 2 1 − >< − + + >< + = z z z z ρ . Please comment on your findings. (c) Find n S ˆ ⋅ r for an arbitrary axis , for the 1/32/3 mixture and for the completely incoherent state with a 1/21/2 mixture. Please comment n ˆ 3. Sakurai, Chap. 3. 25...
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 Spring '08
 JS
 mechanics, Energy, Photon, Polarization

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