notes19 - 5.73 Lecture #19 19 - 1 Density Matrices I (See...

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19 - 1 5.73 Lecture #19 modified 10/9/02 10:20 AM Density Matrices I Last time: Variational Method Linear variation: 0= H ε S ⇒0= ˜ ˜ H 1 ψ = c n φ n n d dc n =0 [Variation vs. pert. theory TODAY phase ambiguity – but for all observables each state always appears as a bra and a ket. what is needed to encode motion in the probability density? A superposition of eigenstates belonging to several different values of E. coherent superposition vs. statistical mixture: think about polarized light ρ no phase ambiguity coherences ” in off-diagonal position populations ” along diagonal A = Tr ρ A () = Tr A Quantum Beats prepared state ρ detection D (detect or destroy coherences) ρ Α Α Η Α Α ρ Η ρ , , t d dt i t ih d dt t t = [] + = h * state: ρ * evolution: H * detection: D (See CTDL pp. 252-263, 295-307**, 153-163, 199-202, 290-294)
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19 - 2 5.73 Lecture #19 modified 10/9/02 10:20 AM These 2 cases seem to be identical if you make 2 measurements with analyzer polarizers along ê x then ê y . But they are different with respect to 2 measurements with analyzer polarizers along then Let us define a quantity called “Density Matrix” ρ≡ ψ a pure state can be any sort of QM wavefunction * eigenstate of H * coherent superposition of several eigenstates of H but cannot represent a statistical (i.e. incoherent) mixture of several different ’s However, ρ can represent a statistical (i.e. equilibrium) mixture of states p k k k k = p k ρ k k p k =1 Example * one beam of linarly polarized light, with the polarization axis ( ε -field) * two beams of linearly polarized light, 50% along ê x , 50% along ê y . x y ê 45° ˆe =2 −1 / 2 x + y () 2 −1 / 2 x + y 2 −1 / 2 x y . In the statistical mixture, it does not matter how the analyzer is oriented.
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19 - 3 5.73 Lecture #19 modified 10/9/02 10:20 AM ρ nm = n ψ m nm = c n c m * What are the properties of ρ ?
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notes19 - 5.73 Lecture #19 19 - 1 Density Matrices I (See...

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