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# hw44 - PHYS 455 Homework IV Assigned Tuesday Due December 7...

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PHYS 455 Homework IV Assigned: November 29, 2011, Tuesday. Due: December 7, 2011, Wednesday. 1. Consider an ensemble for a spin-1/2 particle where the particle is in “spin-up along z ” state with probability p 1 = 1 / 6, “spin-down along z ” state with probability p 2 = 1 / 3, and “spin-up along x ” state with probability p 3 = 1 / 2. (a) Compute the density matrix ρ of this ensemble. (b) As an example, compute the expectation value of σ z , by using (i) the density matrix and (ii) the ensemble states and probabilities. 2. Consider a mixed state with the following matrix ρ = 1 5 2 1 + i 1 - i 3 . (a) Is this really a valid density matrix? (In other words, does ρ satisfy all of the condi- tions that a density matrix should satisfy?) (b) Compute h σ x i , h σ y i and h σ z i . Express your results compactly by expressing h i . (b 0 ) The vector h i expresses the position of the mixed state ρ inside the Bloch sphere. Verify that the length of this vector is less than or equal to 1.

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hw44 - PHYS 455 Homework IV Assigned Tuesday Due December 7...

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