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

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p. 48 IRREVERSIBLE RELAXATION We want to study the relaxation of an initially prepared state. 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 (for a constant perturbation): +, 2 k kk k 0 k P2 wV E E t Pw t t P1 P wS U w 0 0 A AA A AA A = P t 0 1 t 0 k P A P 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. b k t | b k 0 . So !! w k A represents the tangent to the relaxation behavior at t 0. !! w k A w P k A w t t 0 The problem is we don’t account for depletion of initial state. What long-time behavior do we expect? 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: Cohen-Tannoudji, et al. p. 1344; Merzbacher, p. 510. MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II. Instructor: Prof. Andrei Tokmakoff
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p. 49 R / S: P t 1st order P.T. k P A P AA 0 1 0 !! : R ' 2 . V k A 2 = But we don’t have a two-state system. Rather, we are relaxing to a continuum. Fermi’s Golden Rule says we have a time-independent rate, !!
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7 - MIT Department of Chemistry 5.74, Spring 2004:...

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