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 ( ( [ δω k A ω ) k A + ) ] 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 = A k A k A A , km n = , 2 2 ˆ A δ ω ) + δ + ) = E 0 k A pk A ( k A ( k A = , 2 = A , 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 = E 0 p m n ( mn ) absorption mn n 2 = 2 ( nm + ) stimulated emission ˆ + p n m nm m
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p. 82 Note: δ ( ω nm + ) = ( mn + ) = ( mn ) since ( x ) = (− x ) Also: the matrix elements squared are the same. mn =− nm 2 2 ± ˆ E rad = π E 0 ( p n p m ) m µ n ω ( mn ) mn 2 = At equilibrium p = exp[ β E ]/ Z A A p p = p ( 1 e x p [ βω m n ]) = n m n Now, the energy incident on the sample per unit time is ± = E in 8 c E 0 2 ± () E rad So we can write the absorption coefficient, αω = ± E in 2 2 () = 4 π = ˆ ( 1 e ) p n m n ( m ) = c Now this is written for two discrete states, but in general we will want to sum over all possible initial and final states. We can now separate α into a product of factors that represent the field,
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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

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