notes14 - 5.73 Lecture #14 14 - 1 Perturbation Theory I...

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14 - 1 5.73 Lecture #14 modified 9/30/02 10:13 AM Perturbation Theory I (See CTDL 1095-1107, 1110-1119) Last time: derivation of all matrix elements for Harmonic-Oscillator: x, p, H “scaling” xn n n n ij ii ij i −≤ ∆= ±± () in steps of 2 e.g. x n x 3 2 31 : , / dimensionless quantities x ~ = m ω h 1/2 x p ~ = h m p H ~ = 1 h H annihilation a =+ 2 12 / ~ ~ xi p a nn n =− 1 / creation a †/ ~ ~ 2 p a / n + 11 number aa †† not n = “commutator” [ a,a ] = + 1 “selection rules”
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14 - 2 5.73 Lecture #14 modified 9/30/02 10:13 AM a little more: a 01 = 1 1/2 a a = = () + + 01 0 00 00 20 0 00 0 30 00 0 0 00 0 0 0 0 1 1 0 0 12 O L L LL L L L LL L n n n nq q ! ! ! ! ! / / / selection rule for a ij n j-i = n selection rule for a ij =− n nn k kn j jm n m n jk jk n m = [] () [] = −+ ! ! ! ! ! / , / 0 a aa δ selection rule 43 4 (one step to right of main diagonal) (n steps to right) Selection rules are obtained simply by counting the numbers of a and a and taking the difference. The actual value of the matrix element depends on the order in which individual a and a factors are arranged, but the selection rule does not. Lots of nice tricks and shortcuts using a , a and a a
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14 - 3 5.73 Lecture #14 modified 9/30/02 10:13 AM One of the places where these tricks come in handy is perturbation theory. We already have: Why perturbation theory? 1. WKB: local solution, local k(x), stationary phase 2. Numerov–Cooley: exact solution - no restrictions 3. Discrete Variable Representation: exact solution, ψ as linear combination of H-O. replace exact H which is usually of dimension by H eff which is of finite dimension. Truncate infinite matrix so that any eigenvalue and eigenfunction can be computed with error < some preset tolerance.
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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notes14 - 5.73 Lecture #14 14 - 1 Perturbation Theory I...

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