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03_580ln_fa08

03_580ln_fa08 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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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 .

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5.80 Lecture #3 Fall, 2008 Page 1 Lecture #3: Building an Effective Hamiltonian Last time: diatomic molecule as anharmonic non-rigid rotor Q = R R e V(Q) = 1 kQ 2 + 1 aQ 3 + 1 bQ 4 +… B(Q)J(J + 1) 2 6 24  really KE 2 1 B(R) = hc 2µR 2 B(Q) = B e 1 1 2 R Q e + 1 3 R Q e 3 +… 1 [ k µ ] 1/2 ω e = 2 π c E vJ /hc = ω e ( v + 1/2 ) − ω e x e (v + 1 / 2) 2 + ω e y e (v + 1 / 2) 3 + J(J + 1) B e − α e (v + 1/ 2) + γ e (v + 1 / 2) 2 J(J + 1) 2 [ D e + β e (v + 1/ 2) +… ] Problem: find ω e , ω e x e , ω e y e , α e , D e in terms of k, a, b, R e , µ using non-degenerate perturbation theory (over-tilde implies additional corrections). H (0) ψ (0) v (0) ψ v = E v (0) defines basis states H (0) 1 p 2 (0) hc = ω e (v + 1/ 2) + B e J(J + 1) = kQ 2 + + B e J(J + 1) E vJ hc 2 v,J 0 = v HO JM J H is everything not in H (0) . Some tools: 1/2 Q = 2 π ω e Q P = [ 2 π ω e ] 1/2 P Q = 2 1/2 ( a + a ) P = 2 1/2 i( a a ) v| a |v 1 = v 1/2 = v 1/2 v 1| a |v
5.80 Lecture #3 Fall, 2008 Page 2 N = a a N | v = v | v [ a , a ] = a a aa [ a , a ] | v = [v – (v + 1)]|v = –|v [ a , a ] = –1 OR [ a , a ] = + 1 [ N , a ] = [ a aa a a a ] = a ( aa a a ) = a [ N , a ] = [ a aa aa a ] = ( a a aa ) a = – a Q 2 = 2 1 ( a + a ) 2 = 2 1 ( a 2 + a †2 + aa + a a ) 1 a 2 + a †2 + (2 N + 1) = 2 P 2 = 2 1 ( a a ) 2 = 1 a 2 + a †2 (2 N + 1) 2 Q 2 + P 2 = (2 N + 1) ( off-diagonal elements cance ) Q 3 = 2 3/2

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03_580ln_fa08 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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