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12 - MIT Department of Chemistry 5.74 Spring 2004...

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MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II� Instructor: Prof. Andrei Tokmakoff p. 81 Time-Correlation Function Description of Absorption Lineshape Let’s express the absorption of radiation by dipoles as a dipole correlation function. Start with the rate of absorption and stimulated emission between an initial state k induced by monochromatic field: A and final state π E 0 2 2 ˆ w k A = k ∈⋅ µ A ( ( [ δ ω k A ω ) δ ω k A + ω ) ] 2 = 2 m E Let’s consider a two-level system and n with E > E . m m n m w w nm mn The rate of energy absorption is proportional to the absorption rate and the transition energy: E ± rad = w = ω . But more generally we E n n nn nm need to consider the difference between the rates of absorption and stimulated emission, so the rates of transitions between these two states is ± E rad = p w = ω A k A k A A , k m n = , 2 2 ˆ ∈⋅ µ A δ ω ω ) + δ ω + ω ) = π E 0 ω k A p k A ( k A ( k A = , 2 = A , k m n Here we have to sum over the rates of absorption from n m and the rates of stimulated emission from n m . 2 2 ± ˆ E rad = π E 0 ω p m ∈⋅ µ n δ ω ( mn ω ) absorption mn n 2 = 2 ( nm + ω ) stimulated emission ˆ + ω p n ∈⋅ µ m δ ω nm m
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p. 82 Note: δ ( ω nm + ω ) = δ ( ω mn + ω ) = δ ( ω mn ω ) since δ ( x ) = δ ( x ) Also: the matrix elements squared are the same. ω mn = − ω nm 2 2 ± ˆ E rad = π E 0 ω ( p n p m ) m ∈⋅ µ n δ ω ( mn ω ) mn 2 = At equilibrium p = exp[ β E ]/ Z A A p p = p ( 1 exp [ β ω mn ] ) = n m n Now, the energy incident on the sample per unit time is ± = E in 8 c π E 0 2 ± ( ) E rad So we can write the absorption coefficient, α ω = ± E in 2 2 ( ) = 4 π β ω = ˆ α ω ω ( 1 e ) p n m ∈⋅ µ n δ ω ( mn ω ) = c
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