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MIT5_74s09_lec03

# MIT5_74s09_lec03 - MIT OpenCourseWare http/ocw.mit.edu 5.74...

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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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3 - 1 Andrei Tokmakoff, MIT Department of Chemistry, 3/2/2007 3. IRREVERSIBLE RELAXATION 1 It may not seem clear how irreversible behavior arises from the deterministic TDSE, although this is a hallmark of all chemical systems. To show how this comes about, we will describe the relaxation of an initially prepared state as a result of coupling to a continuum. We will show that first-order perturbation theory for transfer to a continuum leads to irreversible transfer—an exponential decay—when you include the depletion of the initial state. The Golden Rule gives the probability of transfer to a continuum as: 2 ρ ( E k = A ) w k A = P k A = 2 π V k A t = P k A = w t t A ( − ) (3.1) k 0 P AA =− 1 P k A The probability of being observed in k varies linearly in time. This will clearly only work for short times, which is no surprise since we said for first-order P.T. ( ) ≈ k 0 k ( ) . bt b What long-time behavior do we expect? A time-independent rate is also expected for exponential relaxation. In fact, for exponential relaxation out of a state A , the short time behavior looks just like the first order result: P t P 1 wt " AA ( ) = AA ( 0exp ) ( w k A t ) (3.2) k A + So we might believe that w k A represents the tangent to the relaxation behavior at t = 0. P k A w k A = (3.3) t t 0 The problem we had previously was we don’t account for depletion of initial state. From an exact solution to the two-level problem, we saw that probability oscillates sinusoidally between the two states with a frequency given by the coupling:
3-2 2 2 Δ+ V Ω= k A R = But we don’t have a two-state system. Rather, we are relaxing to a continuum. We might imagine that coupling to a continuous distribution of states may in fact lead to exponential relaxation, if destructive interferences exist between oscillations at many frequencies representing exchange of amplitude between the intital state and continuum states.

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MIT5_74s09_lec03 - MIT OpenCourseWare http/ocw.mit.edu 5.74...

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