11_lecnotes_rwf

11_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Robert Field 5.74 RWF Lecture #11 11 – 1 Complex Energy H eff : Nondegenerate Perturbation Theory Reading : Chapter 9.3, The Spectra and Dynamics of Diatomic Molecules , H. Lefebvre-Brion and R. Field, 2 nd Ed., Academic Press, 2004. pure rotation Last time: vibration - rotation spectra I () ω electronic - vibration - rotation genuinely localized plucks grand recurrences Supplements: 1. Stationary Phase for Vibration and Electronic Transitions (nature of pluck) 2.± Heller’s Fractionation Measure (inversely proportional to the fraction of state- or phase-space accessed.) Today: sharp bright state broad doorway state (broad because it couples to the quasi-continuum bath) reduce many-state interactions to a simple two-level problem. vs. bright doorway continuum bright continuum “Lumpy continuum” As the bright state is systematically tuned (perhaps by changing rotational level), it can mix either indirectly or directly with the continuous, dark bath. Our goal is to find a shortcut for dealing with the interaction between a bright state and a bath, especially via a doorway state.
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5.74 RWF Lecture #11 11 – 2 We can develop an intuitive understanding for a two- or three-level problem much more easily than for a large-dimension random matrix problem. It is also computationally much quicker. The basis for the idea is, if H is time-independent, j Q Ψ j ( Q , t ) = ψ j () e iE t / h H ψ j = E j ψ j Suppose we allow E j to be complex? E j = ε j i γ j /2 i ε j i j 2 ) t / h iE t / h i t / h ( γ j e = e e ε j −γ j t /2 h it / h = e e exponential decay = e j t / h ! = Pt t t Ψ j () Ψ j j (an exponentially decaying population! Should come in handy.) γ τ γ τ = or γ has units of energy h h = Width of ± t j j t Ψ j in energy or cm –1 ? ;, ∆ν ) = 1 ( ν 0 ) Lorentzian L ( νν ( ∆ν ) 0 π ( ∆ν ) 2 + 2 * ∆ν is FWHM FWHM of a Lorentzian line νν 0 dL ( ;, ∆ν ) = 1 normalized to 1 associated with exp. decay of 1 ] = 1 probability with lifetime τ * ∆ν [ cm 2 π c τ
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5.74 RWF Lecture #11 11 – 3 () = e −γ t / h h Pt τ = γ for our case: h units of E E FWHM [] == γ τ We can think of decay rate as a property of a “quasi-eigenstate”. We can just tack the decay rate onto the normal Ψ j ( Q , t ) without changing anything. Seems to be too good to be true!
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11_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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