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# HW4 - PHYS851 Quantum Mechanics I Fall 2008 HOMEWORK...

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Unformatted text preview: PHYS851 Quantum Mechanics I, Fall 2008 HOMEWORK ASSIGNMENT 4: Schr¨dingers Equation, Tensor Product, Density Operator o 1. [10pts] The trace of an operator is deﬁned as T r {A} = m m|A|m , where {|m } is a suitable basis set. Prove that the trace is independent of the choice of basis. 2. [10pts] Prove the linearity of the trace operation by proving T r {aA + bB } = aT r {A} + bT r {B }. 3. [10pts] Prove the cyclic property of the trace by proving T r {ABC } = T r {BCA} = T r {CAB }. 4. [10pts] Which of the following density matrices correspond to a pure state? ρ1 = 2 7 0 5 7 0 ρ2 = −i 1 4√ i √ 3 4 3 4 3 4 1 9 2 9 2 9 2 9 4 9 4 9 ρ3 = 2 9 4 9 4 9 00 01 ρ4 = 1 5 √ 2 5 √ 2 5 4 5 d o 5. [10 pts] Derive the equation dt ρ(t) = − i [H, ρ(t)], using Schr¨dinger’s equation and the most general form of the density operator. ρ5 = 6. [10pts each] Cohen Tannoudji, pp341-350: problems 3.6, 3.7, 3.11, 3.14, 3.17, and 3.18 ...
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