36_580ln_fa08

36_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
± ± Lectures # Wavepacket Dynamics II and III An eigenstate does not move and the population in an eigenstate does not decay. Quantum Beats: polarization vs. population. Concept of “bright” and “dark” states Nonradiative decay into dense manifold of dark states Fermi’s Golden Rule Bixon-Jortner into continuum: Complex H eff (SDDM, pages 671-683) Dynamical Quantities some are in real coordinate/momentum space others are in state space key is to reduce the quantity of information from what is contained in Ψ( t ) yet retain visibility that * reveals mechanism * suggests experiments examples * probability density * density matrix * autocorrelation function * survival probability * transfer probability * Q i ± ( t ), P i ± ( t ) Ehrenfest’s Theorem * E i ± = ω i a i a i ( t ) OR anharmonic v i ± = a i a i ( t ) * expectation values of resonance and transfer rate operators (SDDM, pages 694-701) . It is also possible to go from a Quantum Mechanical H eff to a Classical Mechanical H eff and view structure and dynamics in an action-angle represention of phase space. Regular (quasi-periodic trajectories) and Chaotic regions of classical phase space. 1 36 : 35
Background image of page 2
5.80 Lectures # Fall, 2008 Page 2 An eigenstate ψ j , Ψ j ( t ) = ψ j e iE j t/ , from a harmonic oscillator potential surface Ψ j ( t ) | Q | Ψ j ( t ) ± = ψ j | Q | ψ j ± = 0 because Q is Δ v = ² 1 off-diagonal. Suppose ψ j = αφ 0 + (1 α 2 ) 1 / 2 φ 0 (a vibrationally mixed state) v v +1 ψ j | Q | ψ j ± = 2 α (1 α 2 ) 1 / 2 ² v + 1 ³´ | Q | v ± µ = 0 ³ . ( v +1) 1 / 2 Still no motion. Also P j = Ψ j ( t ) | Ψ j ( t ) ± = ψ j | ψ j ± = 1 does not decay. Suppose, using a short pulse of radiation, we excite a coherent superposition of two eigenstates, ψ j and ψ k . Then Ψ(0) = αψ j + (1 α 2 ) 1 / 2 ψ k φ 0 B Ψ( t ) = αψ j e iE j t/ + (1 α 2 ) 1 / 2 ψ k e iE k t/ Ψ ( t )Ψ( t ) = | α | 2 | ψ j | 2 + | 1 α 2 || ψ k | 2 + α ψ (1 α 2 ) 1 / 2 ψ k e i ( E k E j ) t/ + c.c. j = | α | 2 | ψ j | 2 + | 1 α 2 || ψ k | 2 + α (1 α 2 ) 1 / 2 ψ j ψ k e kj t + c.c. = | α | 2 | ψ j | 2 + | 1 α 2 || ψ k | 2 + 2Re α (1 α 2 ) 1 / 2 ψ j ψ k cos ω kj t + 2Im[ α (1 α 2 ) 1 / 2 ψ ψ k ] sin ω kj t. j We have a constant term and two oscillating terms. More insight is obtained if we ask for the time dependent probability of Fnding the system in Ψ(0) φ 0 B P B ( t ) = ¶· Ψ( t ) | φ 0 ¸ 2 = | Ψ( t ) | Ψ(0) ±| 2 B = Ψ( t ) | Ψ(0) ± Ψ(0) | Ψ( t ) ± ± ± = | α | 2 e iE j t/ + | 1 + α 2 | e iE k t/ | α | 2 e iE j t/ + | 1 α 2 | e iE k t/ = | α | 4 + | 1 α 2 | 2 + | α | 2 | 1 α 2 | ( e kj t + e kj t ) = (1 + 2 | α | 4 2 | α | 2 ) + 2 | α | 2 | 1 α 2 | cos ω kj t. The
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 11

36_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online