Non-degenerate Perturbation Theory
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1 No-degeerate Perturbato Theory Proble : H E ca't solve exactly. But wth H H H' H" L H E Uperturbed egevalue proble. Ca solve exactly. E Therefore, kow ad. H ' H" called perturbatos Copyrght Mchael D. Fayer, 17
2 Solutos of H E coplete, orthooral set of ket vectors,, 1 wth egevalues ad E, E, E, 1 Kroecker delta 1 Copyrght Mchael D. Fayer, 17
3 Expad wavefucto ad E E E E also have H H H' H" Have seres for H E Substtute these seres to the orgal egevalue equato H E Copyrght Mchael D. Fayer, 17
4 Su of fte uber of ters for all powers of equals. ' H ' E H H E E H H H E E E ' " Coeffcets of the dvdual powers of ust equal. zeroth order - H E frst order - 1 secod order - H H' E E H H' H" E E E Copyrght Mchael D. Fayer, 17
5 Frst order correcto H E E H' Wat to fd E ad. Expad The c H c H c E also substtutg Substtutg ths result. After substtuto ' c E E E H Copyrght Mchael D. Fayer, 17
6 After substtuto ' c E E E H Left ultply by ' c E E E H ' c E E E H uless =, but the E E Therefore, the left sde s. Copyrght Mchael D. Fayer, 17
7 We have E H' The E E E E E H' E uber, kets oralzed, ad trasposg, H' The frst order correcto to the eergy. (Expectato value of H zeroth order state ) Absorbg to E E E H ad E E H' H The frst order correcto to the eergy s the expectato value of H '. Copyrght Mchael D. Fayer, 17
8 Frst order correcto to the wavefucto Aga usg the equato obtaed after substtutg seres expasos ' c E E E H Left ultply by ' c E E E H ' c E E E H c E E H' Equals zero uless =. H ' c E E Coeffcets expaso of ket ters of the zeroth order kets. Copyrght Mchael D. Fayer, 17
9 H ' c E E c E H ' E H ' s the bracket of H ' wth ad. Therefore ' H ( E E) correcto to zeroth order ket The pre o the su ea. zeroth order ket eergy deoator Copyrght Mchael D. Fayer, 17
10 Frst order correctos E E H' H' H ' H H H ( E E) Copyrght Mchael D. Fayer, 17
11 Secod Order Correctos Usg coeffcet Expadg Substtutg ad followg sae type of procedures yelds E ' E E H H H coeffcets have bee absorbed. H H H H Secod order correcto due to frst order pece of H. Secod order correcto due to a addtoal secod order pece of H. H H H H ' ' k k ' k k k k k ( ) k k E E k E E E E E E H Secod order correcto due to frst order pece of H. Secod order correcto due to a addtoal secod order pece of H. Copyrght Mchael D. Fayer, 17
12 Eergy ad Ket Corrected to Frst ad Secod Order E E H' ' E E H H H ' H ( E E) ' ' H kh H H k E E E E E E k k k k ' H E E k k ( k) k Copyrght Mchael D. Fayer, 17
13 Exaple: x 3 ad x 4 perturbato of the Haroc Oscllator E x Vbratoal potetal of olecules ot haroc. Approxately haroc ear potetal u. Expad potetal power seres. Frst addtoal ters potetal after x ter are x 3 ad x 4. Copyrght Mchael D. Fayer, 17
14 p 1 H kx cx qx H p 1 1 kx H aa a a 3 4 quartc force costat cubc force costat haroc oscllator kow solutos 1 E zeroth order egevalues zeroth order egekets 3 4 perturbato H' cx qx c ad q are expaso coeffcets lke. Whe c ad q, H H Copyrght Mchael D. Fayer, 17
15 H H' 3 4 cx qx 3 4 cx qx I Drac represetato 1 x a a k 3 3 x a a Frst cosder cubc ter. Multply out. May ters. 3 3 a, a a, aa a, a. Noe of the ters have the sae uber of rasg ad lowerg operators. 3 x (At secod order wll ot be zero.) Copyrght Mchael D. Fayer, 17
16 4 4 x aa 4k a a 4 has ters wth sae uber of rasg ad lowerg operators. Therefore, 4 x Usg 1/ 1/ a a 1 ad ( 1) 1 1 aaa a a a aa aa aa a aa a aa a a a aaa 1 Oly ters wth the sae uber of rasg ad lowerg operators are o-zero. There are sx ters. Copyrght Mchael D. Fayer, 17
17 Su of the sx ters 4 aa 6( 1 ) Therefore q 3 1 k H Wth k k E q Eergy levels ot equally spaced. Real olecules, levels get closer together q s egatve. Correcto grows wth faster tha zeroth order ter decrease level spacg. Copyrght Mchael D. Fayer, 17
18 Perturbato Theory for Degeerate States H H E 1 1 E 1 ad oralze ad orthogoal 1 ad Degeerate, sae egevalue, E. If wth c c cc cc H E Ay superposto of degeerate egestates s also a egestate wth the sae egevalue. Copyrght Mchael D. Fayer, 17
19 learly depedet states wth sae egevalue syste -fold degeerate Ca for orthooral fro the degerate. Ca for a fte uber of sets of. Nothg uque about ay oe set of degeerate egekets. Copyrght Mchael D. Fayer, 17
20 Wat approxate soluto to H H E ' zeroth order Haltoa perturbato H E zeroth order egeket zeroth order eergy But E s -fold degeerate. Call these egekets belogg to the -fold degeerate E 1 1, orthooral Wth E E E E 1 1 Copyrght Mchael D. Fayer, 17
21 Here s the dffculty perturbed ket zeroth order ket havg egevalue, E 1 But, s a lear cobato of the. c c c 1 1 We do t kow whch partcular lear cobato t s. s the correct zeroth order ket, but we do t kow the c. The correct zero order ket depeds o the ature of the perturbato. p states of the H ato exteral agetc feld p 1, p, p -1 electrc feld p x, p z, p y Copyrght Mchael D. Fayer, 17
22 To solve proble Expad E ad 1 E E E c 1 Soe superposto, but we do t kow the c. Do t kow correct zeroth order fucto. Substtutg the expasos for E ad to H H E ' ad obtag the coeffcets of powers of, gves zeroth order frst order H c E c 1 H E 1 c EH' wat these Copyrght Mchael D. Fayer, 17
23 1 H E 1 c EH' substtute A k k k To solve Need H ' Use proecto operator k k H' H' k k k The proecto operator gves the pece of H ' that s. The the su over all k gves the expaso of H ' ters of the. k Defg H H' ' Hk k k H' k k Kow kow perturbato pece of the Haltoa ad the zeroth order kets. Copyrght Mchael D. Fayer, 17
24 1 H E 1 c EH' H ' Hk k k ch ch k k 1 1 k ths pece becoes Substtutg ths ad Ak k gves Ek E1 Ak k Ec ch k k k 1 k 1 k Result of operatg H o the zeroth order kets. Left ultplyg by Ek E1 Ak k Ec ch k k k 1 k 1 Copyrght Mchael D. Fayer, 17
25 Ek E1 Ak k Ec ch k k k 1 k 1 Correcto to the Eerges Two cases: (the degeerate states) ad >. Left had sde su over k equals zero uless k =. But wth, E E E E The left had sde of the equato =. 1 Therefore, 1 Rght had sde, frst ter o-zero whe =. Bracket = 1, oralzato. Secod ter o-zero whe k =. Bracket = 1, oralzato. The result s 1 Hc Ec We do t kow the c s ad the Es. Copyrght Mchael D. Fayer, 17
26 1 Hc Ec s a syste of of equatos for the c s. H E c H c H c H c H E c H c 1 1 Oe equato for each dex of c. H c H c H E c Besdes trval soluto of c c c oly get soluto f the deterat of the coeffcets vash. H E H H H E H H H H E 1 We kow the H H k k Have th degree equato for the E s. Copyrght Mchael D. Fayer, 17
27 Solve th degree equato get the E s. Now have the correctos to eerges. To fd the correct zeroth order egevectors, oe for each E, substtute E (oe at a te) to syste of equatos. Get syste of equatos for the coeffcets, c s. H E c H c H c H c H E c H c 1 1 Kow the H. H c H c H E c 1 1 * * * 1 1,, 1 cc cc c c There are oly 1 codtos because ca ultply everythg by costat. Use oralzato for th codto. Now we have the correct zeroth order fuctos. Copyrght Mchael D. Fayer, 17
28 The solutos to the th degree equato (expadg deterat) are E E E 1,, Therefore, to frst order, the eerges of the perturbed tally degeerate states are E E E 1 1 Have dfferet E s (uless soe stll degeerate). Wth E E as 1 Copyrght Mchael D. Fayer, 17
29 Correcto to wavefuctos Aga usg equato foud substtutg the expasos to the frst order equato Ek E1 Ak k Ec ch k k k 1 k 1 Left ultply by k Orthogoalty akes other ters zero. Noralzato gves 1 for o-zero brackets. k 1 k k 1 E E A c H Therefore k ch A k k 1 E1 Ek gves 1 gves Noralzato gves A =for. Already have part of wavefucto for Copyrght Mchael D. Fayer, 17
30 Frst order degeerate perturbato theory results E E E 1 k 1 k k E1 Ek ch Correct zeroth order fucto. Coeffcets c k detered fro syste of equatos. Correcto to zeroth order fucto. Copyrght Mchael D. Fayer, 17
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