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Lecture 24

# Lecture 24 - for a perturbation of the form h t = AF t the...

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( 29 ( 29 = - for a perturbation of the form t AF t h ( 29 ( 29 ( 29 ( 29 = Φ - + ' 2 0 the response function is given by t ' ' t B t t F t O F ( 29 ( 29 ( 29 ϖ ϖ ϖ χ ϖ χ ϖ - = = Φ 0 t B F where e t i ( 29 ( 29 β Φ = g in the classical limit A t B t

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( 29 ε χ ϖ π - = 1 4 ( 29 ϖτ ε ϖ βμ π ϖ τ = + 2 0 2 2 " 4 1 ϖ βμ ϖ τ ϖ τ = - + + 2 0 2 2 2 2 1 1 1 t i ( 29 ( 29 ( 29 ( 29 ϖ ε ϖ ϖ ε ϖ ϖ ε ϖ ε τ ϖ ε ϖ = = ↑↑ = max ~ 0 " =small as ' 1 for , " " in the limit of , " 0 ( 29 ( 29 ( 29 ϖ ε ϖ ε ϖ ϖ ε ϖ = ↑↑ = 0 ' 0 =conts. as ' and in the limit of , ' 1 ( 29 ε ϖ βμ π ϖ τ = + + 2 0 2 2 1 ' 4 1 1 ( 29 ( 29 { ε ϖ α ϖ ϖ ϖ τ = Re( ) since " all absorption occurs for in the range of the dipole correlation function ! ABSORPTION COEFFICIENT c n
( 29 ( 29 ( 29 ( 29 ϖ χ ϖ ε ϖ πχ ϖ - = Φ 0 and =1+4 t t e dt i ( 29 ( 29 ( 29 ( 29 ( 29 ε ϖ ε ϖ π χ ϖ χ ϖ = + - ' - " 1 4 ' " i i ( 29 ( 29 ( 29 ϖ π ε ϖ - Φ = Φ 0 and considering that t is an odd funtion of t, 4 " 2 t t e dt i i ( 29 ( 29 πχ ϖ πχ ϖ = + - 1 4 ' 4 " i ( 29 ( 29 { } ρ μ μ Φ = h 1 we also recall that t , z z Tr t i ( 29 ( 29 ( 29 { } ρμ μ ρμ μ Φ = - h 1 t z z z z Tr t t i ( 29 ( 29 ( 29 { } ϖ ε ϖ π ρμ μ μ μ ρ - = - h 0 1 " 2 t z z z z Tr t t e dt i i i

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( 29 ( 29 ϖ β ϖ ϖ ρ ρ - - = h from the properties of commutation of operators and traces (see the book for details on an unneccessary derivation) t t e Tr AB t e dt Tr B t A e dt i i β ϖ β ϖ h h within the fourier transform, A and B do not commute.
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Lecture 24 - for a perturbation of the form h t = AF t the...

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