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15_580ln_fa08

15_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 #15 Fall, 2008 Page 1 of 6 pages Lecture #15: 2 and 2 Matrices Last Time: effect of A 2 , A i , A ± on | A α M A basis set case (a) basis set | n(L) Λ S | v | JM L-destroyed, but not Λ : L z , L ± , selection rules H ROT = B(R)R 2 matrix elements n ′Λ′ S Diagonal : v Ω JM H ROT n Λ S v Ω JM = δ n n δ Λ′Λ δ S S δ δ v v δ δ J J δ M M × B v [J(J + 1) – 2 + S(S + 1) – 2 + L 2 ] ∆Ω = ∆∑ = ±1 within Λ –S multiplet state ( S-uncoupling ): n Λ S ± 1 v Ω ± 1JM H ROT n Λ S v Ω JM = –B v [J(J + 1) – ( ± 1) ] 1/2 [S(S + 1) – ( ± 1) ] 1/2 **** In some of my handouts I call J + 1/2 = x. Here, I'll call it y ***** Here x = J(J + 1), y = J + 1/2 For example: Start by listing all relevant basis states. Λ S Σ 2 n n n n 1 1 / 2 1 / 2 1 1 / 2 1 / 2 1 1 / 2 1 / 2 1 1 / 2 1 / 2 2 3/2 2 1/2 2 3/2 2 1/2 H ROT 2 ( ) = 2 3/2 x 9 4 + 3 4 1 4 [x 3 4 ] 1/2 [ 3 4 + 1 4 ] 1/2 0 2 1/2 sym x 1 4 + 3 4 1 4 0 0 2 1/2 0 0 x 1 4 + 3 4 1 4 [x 3 4 ] 1/2 1 2 3/2 0 0 sym x 9 4 + 3 4 1 4 B v π × ∆Ω = ±1 Λ = ±2 B v π 0 x 4 7 ( x 4 3 ) 1/2 0 0 x + 1 4 0 0 1 4 3 4 sym ) 1/2 0 0 ( x x + 0 0 sym x 7 4 Two identical blocks for > 0 and < 0 - later we will consider parity basis. What about 2 + ? Class should do this.
5.80 Lecture #15 Fall, 2008 Page 2 of 6 pages ∆Ω = Λ = ±1 between Λ -S multiplet states ( L-uncoupling ) n Λ + 1S v Ω ± 1JM H ROT n Λ S v Ω JM = –B v v [J(J + 1) – ( ± 1) ] 1/2 × n ′Λ ± 1 L ± n Λ  β a perturbation parameter to be determined by a fit to the spectrum. effective operators Today: H SO , H SS , H SR matrix elements 2 effective H . Matrix elements of 2 , H

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