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852_2010hw1_Solutions

# 852_2010hw1_Solutions - PHYS852 Quantum Mechanics II Fall...

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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. Let {|φn } be an arbitrary second basis. In the basis {|m }, the trace is Tr{A} = m|A|m (1) φn |A|φn (2) m In basis {|φm }, it is Tr{A} = n inserting the projector onto basis {|m } then gives φn |m m|A|φn (3) m|A|φn φn |m Tr{A} = (4) m|A|m (5) m,n re-ordering gives Tr{A} = m,n recognizing that n |φn φn | = I leads to Tr{A} = m 2. [10pts] Prove the linearity of the trace operation by proving T r {aA + bB } = aT r {A} + bT r {B }. Tr{aA + bB } = m|(aA + bB )|m m =a m|A|m + b m = aTr{A} + bTr{B } 1 m|B |m m (6) 3. [10pts] Prove the cyclic property of the trace by proving T r {ABC } = T r {BCA} = T r {CAB }. m1 |ABC |m1 Tr{ABC } = m1 m1 |A|m2 m2 |B |m3 m3 |C |m1 = m1 ,m2 ,m3 m3 |C |m1 m1 |A|m2 m2 |B |m3 = Tr{CAB } = m1 ,m2 ,m3 m2 |B |m3 m3 |C |m1 m1 |A|m2 = Tr{BCA} = m1 ,m2 ,m3 4. [10pts] Which of the following density matrices correspond to a pure state? ρ1 = 2 7 0 0 ρ2 = 5 7 ρ4 = 1 4√ 1 5 √ 2 5 −i 3 4 i √ 3 4 3 4 √ ρ5 = 2 5 4 5 00 01 ρ3 = 1 9 2 9 2 9 2 9 4 9 4 9 2 9 4 9 4 9 A pure state must satisfy ρmn ρnm = ρmm ρnn for every m and n. ρ1 : 10 49 = 0, so NO. ρ2 : 3 16 = 3 , 16 so YES. ρ3 : 0 = 0, so YES ρ4 : 4 25 = 2 , 25 so NO ρ5 : m, n=1, 2 → 4 81 = 4 ; 81 m, n=1, 3 → 4 81 = 4 ; 81 2 and m, n=2, 3 → 16 81 = 16 , 81 so YES. (7) 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 |. d d ρ= dt dt Pj |ψj ψj | j = d |ψj dt Pj j d ψj | dt ψj | + |ψj i i Pj − H |ψj ψj | + |ψj ψj |H = j i =−H Pj |ψj ψj | + i |ψj ψj | H j j i = − [H, ρ] (8) 3 ...
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