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p137bp11

# p137bp11 - eigenstates(the standard “Copenhagen...

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Physics 137B, Fall 2007 Problem set 11 : scattering and density matrices Assigned Friday, 30 November. Due in box Friday, 7 December. 1. Calculate the expectation values of the spin-half operators S z and S x , represented in the z basis by the matrices S z = ¯ h 2 1 0 0 - 1 , S x = ¯ h 2 0 1 1 0 (1) for the states represented by the following two density matrices: ˆ ρ 1 = 1 / 2 1 / 2 1 / 2 1 / 2 , ˆ ρ 2 = 1 / 2 0 0 1 / 2 . (2) You will want to use ˆ O = Tr (ˆ ρ ˆ O ) . (3) Next, argue that the effect of a measurement of S z on the pure state represented by ˆ ρ 1 if the mea- surement outcome is unknown is to transform it into the mixed state represented by ˆ ρ 2 . Remember from 137A that measuring an initial state that is not an eigenstate causes the system to jump into different eigenstates with probabilities determined by the projection of the initial state over
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Unformatted text preview: eigenstates (the standard “Copenhagen interpretation” of QM). 2, 3. Bransden 13.15 (counts as 2 problems) 4. In class we showed for pure states that ˆ ρ 2 = ˆ ρ , by working in the basis where ˆ ρ is diagonal. Show for yourself that if ˆ ρ represents a mixed state made up of orthogonal pure states α 1 ,α 2 ,...,α m with nonzero probabilities W 1 ,W 2 ,...,W m , m > 1, then ˆ ρ 2 6 = ˆ ρ by working in an appropriately chosen basis. (This result also holds if the constituent pure states are not orthogonal, but you need not treat this case.) 1...
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