University:
Widener University
Course:
PHYS-121 | Fundamentals of Physics I
Academic year:

2023
Views:

251

Pages:

5
Author:

Sarvadonre5

Page 1 of 5 pages Lecture #34: Wavepacket Dynamics Read Bernath, Chapter 6, especially pages 159-168, 181-196 Do problems 2, 5, 6, 8 in Bernath Absorption Spectrum is FT of autocorrelation function. I(ω) = ∫ ∞ −∞ e−iωt ψ(t) ψ(0) dt * Derive this by manipulating familiar stuff * Interpret I(ω) = ∑ I jδ (ω − ω j ) j spectrum is collection of lines of intensity Ij located at ω j = (E v′j − E v′′ ) h ≡ E j h (define E = 0 at Ev″ ) I j = M q v′j v′′ 2 = M * v′′ v′j v′j v′′ M FC factor times M M= eµg 2 ψ(0) ≡ v′′ M = ∑ v′j v′j v′′ M j ψ(0) = M * v′′ = M * ∑ v′′ v′j v′j j ψ(t) = M * ∑ v′′j v′ v′j e j iE j t h Page 2 of 5 pages return to this ψ(t ) ψ (0) = M * ∑ v′′ v′j v′j e iE j t vk′ vk′ v′′ M j ,k = (M *M )∑e iE j t v′′ v′j v′j vk′ vk′ v′′ j ,k =| M |2 ∑ qv′j v′′ e δ kj iE j t j ∫ ∞ −iωt 2 e ψ(t) ψ(0) dt =| M | −∞ =| M |2 ∫∑ q v′j ∑ v′j v′j v′′ e i(ωt −ω)t dt q v′j v′′δ (ω j − ω) = ∑ I jδ (ω j − ω) j (E v′j − E v′′ ) So we have shown that absorption spectrum I(ω) is FT of autocorrelation function. But what is this autocorrelation function? Page 3 of 5 pages ψ (t ) ψ (0) = M * ∑ + iω t v′′ v′j v′j e j v′′ M j + iω t = [ M * v′′ ] ∑ v′j v′j e j [ v′′ M ] j ′ e + iH t / propagation of ψ(0) on upper potential surface Now we have a new picture → spectrum is generated by a sequence of events occurring for a perfectly defined initial state propagating on unknown potential surface. 1 ² E/h 1 = ωe c * Recurrences at T = classical period = * Initial decay — gradient of potential is Force Page 4 of 5 pages dV = mR dR R(0) = R< F =− R(0) = 0 R (t ) = − R(t ) = − 1 dV m dR t1/ 2 related to 1 dV m dR t R< t 2 + R< R< dV dR R (t1/ 2 ) − R(0) = ² ⎡⎛ −m ⎞ ⎢⎜ dV ⎟ ² ⎢⎜ ⎟ ⎣⎝ dR ⎠ ⎤ ⎥ ⎥ ⎦ 1/ 2 = t1/ 2 What is t1/ 2 ? ⎡ ⎤ 1 ⎢∝ 1/ 2 ⎥ ⎣ ω (R − R e ) ⎦ Page 5 of 5 pages dV = k (R − R e ) dR = µω2 (R − R e ) Questions 1. 1 2 k (R − R e ) 2 Two Harmonic oscillators of identical k but different Re. V(R) = vs. small displacement large displacement which ψ(t) ψ(0) decays faster? What does this imply about I(ω)? [use time dependent approach instead of old FC envelope ideas to answer.] Does this give correct limiting behavior when ∆Re = 0? 2. For 2 modes of same R – Re, which will give wider FC region from v = 0? large ω′ or small ω′? fast decay slow decay large width small width 3. 2 modes of same ω, but different R – Re, which will give longer FC progression? 4. pictures of OODR.