852_2010hw1 - PHYS852 Quantum Mechanics II, Fall 2008...

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Unformatted text preview: PHYS852 Quantum Mechanics II, Fall 2008 HOMEWORK ASSIGNMENT 1: Density Operator 1. [10pts] The trace of an operator is defined as T r {A} = m m|A|m , where {|m } is an arbitrary basis set. Introduce a second arbitrary basis set, and use it to 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 0 ρ2 = 5 7 ρ4 = 1 5 √ 2 5 √ 2 5 4 5 1 4√ −i 3 4 i √ 3 4 3 4 ρ5 = 00 01 ρ3 = 1 9 2 9 2 9 2 9 4 9 4 9 2 9 4 9 4 9 d 5. [10 pts] Derive the equation of motion, dt ρ(t) = − i [H, ρ(t)], using Schr¨dinger’s equation and o the most general form of the density operator, ρ = j Pj |ψj ψj |. ...
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This note was uploaded on 12/21/2011 for the course PHYS 852 taught by Professor Moore during the Spring '11 term at Michigan State University.

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