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Unformatted text preview: PHYS 455 Answers to Homework IV 1. Consider an ensemble for a spin1/2 particle where the particle is in “spinup along z ” state with probability p 1 = 1 / 6, “spindown along z ” state with probability p 2 = 1 / 3, and “spinup along x ” state with probability p 3 = 1 / 2. (a) Compute the density matrix ρ of this ensemble. ρ = X p i  ψ i ih ψ i  = 1 6 • 1 0 0 0 ‚ + 1 3 • 0 0 0 1 ‚ + 1 2 1 2 • 1 1 1 1 ‚ = • 5 / 12 1 / 4 1 / 4 7 / 12 ‚ (b) As an example, compute the expectation value of σ z , by using (i) the density matrix and (ii) the ensemble states and probabilities. ( i ) h σ z i = tr σ z ρ = tr • 5 / 12 1 / 4 1 / 4 7 / 12 ‚ = 1 6 , ( ii ) h σ z i = X i p i h σ z i ψ i = 1 6 h σ z i z ↑ + 1 3 h σ z i z ↓ + 1 2 h σ z i x ↑ = 1 6 (+1) + 1 3 ( 1) + 1 2 (0) = 1 6 . 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?) * tr ρ = 1 , OK....
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 Spring '11
 starg
 Probability, Hilbert space, Sylvester, density matrix

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