hw4a9 - PHYS 455 Answers to Homework IV 1 Consider an...

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Unformatted text preview: PHYS 455 Answers to Homework IV 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. 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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hw4a9 - PHYS 455 Answers to Homework IV 1 Consider an...

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