MIT5_74s09_lec06

MIT5_74s09_lec06 - MIT OpenCourseWare http:/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu 5.74 Introductory Quantum Mechanics II Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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6-1 Andrei Tokmakoff, MIT Department of Chemistry, 2/25/2009 6.1. Time-Correlation Function Description of Absorption Lineshape The interaction of light and matter as we have described from Fermi’s Golden Rule gives the rates of transitions between discrete eigenstates of the material Hamiltonian H 0 . The frequency dependence to the transition rate is proportional to an absorption spectrum. We also know that interaction with the light field prepares superpositions of the eigenstates of H 0 , and this leads to the periodic oscillation of amplitude between the states. Nonetheless, the transition rate expression really seems to hide any time-dependent description of motions in the system. An alternative approach to spectroscopy is to recognize that the features in a spectrum are just a frequency domain representation of the underlying molecular dynamics of molecules. For absorption, the spectrum encodes the time-dependent changes of the molecular dipole moment for the system, which in turn depends on the position of electrons and nuclei. A time-correlation function for the dipole operator can be used to describe the dynamics of an equilibrium ensemble that dictate an absorption spectrum. We will make use of the transition rate expressions from first-order perturbation theory that we derived in the previous section to express the absorption of radiation by dipoles as a correlation function in the dipole operator. Let’s start with the rate of absorption and stimulated emission between an initial state and final state l k induced by a monochromatic field 2 π E 2 w 0 = k ε ˆ μ l δ ω + + (6.1) k l 2 ( k l ) ( k l ) 2 h We would like to use this to calculate the experimentally observable absorption coefficient (cross- section) which describes the transmission through the sample T = αω L exp ⎡−Δ N ( ) . (6.2) The absorption cross section describes the rate of energy absorption per unit time relative to the intensity of light incident on the sample α = E & rad . (6.3) I The incident intensity is I = c E 0 2 . (6.4) 8
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6-2 If we have two discrete states m and n with E m > E n , the rate of energy absorption is proportional to the absorption rate and the transition energy E & rad = w nn h ω nm . (6.5) For an ensemble this rate must be scaled by the probability of occupying the initial state. More generally, we want to consider the rate of energy loss from the field as a result of the difference in rates of absorption and stimulated emission between states populated with a thermal distribution. So, summing all possible initial and final states l and k over all possible upper and lower states m and n with E > E m n E & = p w h rad l k l k l , = , l kmn π E 2 . (6.6) 2 ˆ = 0 k l pk εμ l ( k δω ) ( l + l + k ) h l , , l 2 = The cross section including absorption n m and stimulated emission m n terms is: 2 2 ˆ ˆ α
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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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MIT5_74s09_lec06 - MIT OpenCourseWare http:/ocw.mit.edu...

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