6sup - MIT Department of Chemistry 5.74, Spring 2004:...

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p. Supp. 6-1 Slowly Applied (Adiabatic) Perturbation All of our perturbations so far have been applied suddenly at t > t 0 (step function) Vt ( ) = θ t t 0 ( ) ( ) This leads to unphysical consequences—you generally can’t turn on a perturbation fast enough to appear instantaneous. Since first-order P.T. says that the transition amplitude is related to the Fourier Transform of the perturbation, this leads to additional Fourier components in the spectral dependence of the perturbation—even for a monochromatic perturbation! So, let’s apply a perturbation slowly . . . () = V e η t : small and positive 0 t V(t) t V The system is prepared in state A at t = −∞ . Find P k t ( ) . b k = kU I A = i = d τ e i ω k A −∞ t kV A e ητ b k = iV k A = exp t + i k A t [] + i k A = V k A exp t + iE k E A () t / = E k E A + i = P k = b k 2 = V k A 2 = 2 exp 2 t 2 + k A 2 = V k A 2 exp 2 t E k E A 2 + = 2 MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Andrei Tokmakoff
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p. Supp. 6-2 This is a Lorentzian lineshape in ω k
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6sup - MIT Department of Chemistry 5.74, Spring 2004:...

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