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Lecture 15 Time-dependent perturbation theory (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies. Time-dependent perturbation theory The theory described here is essential for understanding the optical transitions from one state to another and thus spectroscopies. It gives the formula for intensities of spectral bands. The probability of transition (intensity of a spectral band for the transition n ← 0) is wn¬0 µ ò Y (0)* n 2 ˆzY0(0) dx E 2 Zeroth order Before the perturbation is turned on, the system is in time-independent ground state. (0) (0) (0) (0) ˆ H 0 E0 0 Let us remember that a “time-independent” wave function does have hidden time dependence of the form: Y (0) 0 (0) (0) -iE0 t/ 0 ®Y e Perturbation = light Next, we shine light on the system. Light (or a photon) is a time-oscillating electromagnetic field. It is very weak as compared to electrostatic interactions between electrons and nuclei in atoms and molecules. It can be treated as a time-dependent perturbation: ( (0) (1) (0) iw t -iw t ˆ ˆ ˆ ˆ H = H + H (t) = H + zˆE e + e ) Oscillating real electrostatic field in z direction with intensity E and angular frequency ω = 2πν. Perturbation = light ( (0) (1) (0) iw t -iw t ˆ ˆ ˆ ˆ H = H + H (t) = H + zˆE e + e ) Oscillating real electrostatic field in z direction with intensity E and angular frequency ω = 2πν. First order Once we turn on the time-dependent perturbation, the wave function will have time-dependent response. We consider the first-order response. We expand the firstorder correction to the zeroth-order wave function by eigenfunctions of the known (initial, time-independent) system. ¥ Y (t) = å ck (t)Y e (1) 0 k (0) (0) -iEk t/ k Light absorption First order Substituting into the time-dependent Schrödinger equation: ¶ ˆ HY = i YÛ ¶t ¥ (0) (0) æ ö (0) (1) (0) - iE0 t/ (0) - iEk t/ ˆ ˆ H + l H (t) ç Y 0 e + l å ck (t)Y k e +… ÷ è ø k ( ) ¥ (0) ö ¶ æ (0) - iE0( 0 )t/ (0) - iEk t/ =i Y0 e + l å ck (t)Y k e +… ÷ ç ¶t è ø k First order Collecting the terms in first order in λ Hˆ (1) (t)Y (0) e 0 -iE0( 0 )t/ ¥ + Hˆ (0) å ck (t)Y (0) e k k -iEk( 0 )t/ (0) ¶ ¥ (0) -iEk t/ =i ck (t)Y k e å ¶t k First order (0) Y Using that k are eigenfunctions of H(0) Hˆ (1) (t)Y (0) e 0 Hˆ (t)Y e (1) (0) (0) -iE0 t/ 0 -iE0( 0 )t/ ¥ + Hˆ (0) å ck (t)Y (0) e k k ¥ + å ck (t)E Y e k (0) k (0) (0) -iEk t/ k -iEk( 0 )t/ (0) ¶ ¥ (0) -iEk t/ =i ck (t)Y k e å ¶t k ¥ (0) ¶ck (t) (0) -iEk( 0 )t/ (0) (0) -iEk t/ =i å Yk e + å ck (t)Ek Y k e ¶t k k ¥ Cancel First order Multiplying Y (0)* from the left and integrating n Hˆ (1) (t)Y (0) e 0 -iE0( 0 )t/ + Hˆ (0) å ck (t)Y (0) e k k ¥ + å ck (t)E Y e Hˆ (t)Y e (1) ¥ (0) (0) -iE0 t/ 0 (0) k k (0) (0) -iEk t/ k -iEk( 0 )t/ (0) ¶ ¥ (0) -iEk t/ =i ck (t)Y k e å ¶t k ¥ (0) ¶ck (t) (0) -iEk( 0 )t/ (0) (0) -iEk t/ =i å Yk e + å ck (t)Ek Y k e ¶t k k ¥ Cancel òY ¶c (t) -i é Ek( 0 ) -E0( 0 ) ùt/ k û Hˆ (t)Y dx = i å e ë ¶t k ¥ (0)* n (1) (0) 0 (0)* (0) Y ò n Y k dx ( ) 1 only if n = k Hˆ (1) (t) = zˆE eiw t + e -iw t First order Simplifying òY é E ( 0 ) -E ( 0 ) ùt/ ¶c (t) -i Hˆ (t)Y dx = i å k e ë k 0 û ¶t k ¥ (0)* n (1) (0) 0 ( ) Hˆ (1) (t) = zˆE eiw t + e -iw t ( E e iw t +e -iw t (0)* (0) Y ò n Y k dx 1 only if n = k )òY (0)* n ¶cn (t) -iéë En( 0 ) -E0(0 ) ùût/ zˆY dx = i e ¶t (0) 0 Equal Energy conservation! E (0) n -E (0) 0 = w or - w First order E (0) n -E (0) 0 = w or - w Excitation or deexcitation First order Removing the time-oscillating factors Eò Y (0)* n 1 cn (t) = i ¶cn (t) zˆY dx = i ¶t (0) 0 t é E Y (0)* zˆY (0) dx ùdt ò0 ë ò n 0 û Fermi’s golden rule The probability of transition (intensity of a spectral band for the transition n ← 0) is wn¬0 µ cn (t) µ ò Y 2 (0)* n 2 zˆY dx E (0) 0 2 When the dipole moment interacts with the oscillating electric field (light), the transition probability is proportional to the square of the transition dipole moment (in the direction of polarization) and square of the intensity of light. Fermi’s golden rule wn¬0 µ ò Y (0)* n 2 zˆY dx E (0) 0 2 Summary Spectroscopic transitions can be explained by the time-dependent Schrödinger equation. One-photon transition can be understood by first-order time-dependent perturbation theory. Energy conservation Transition dipole moment Intensity of light