Perturbatio Theory I (See CTDL 1095-1107, 1110-1119) 14-1 Last time: derivatio of all matrix elemets for Harmoic-Oscillator: x, p, H selectio rules scalig ij x i j i steps of 2 e.g. x : = ± 3, ± 1 xii i / 2 ( 3 ) dimesioless quatities x ~ = mω 1/2 x h p = ( hmω) 1/2 p ~ H ~ = 1 hω H aihilatio a = 2 12 12 / x+ ip a = 1 / ~ ~ creatio a = 2 12 / x ip a 12 = ( + ) / + ~ ~ umber aaot aa aa commutator [a,a ] = +1 = 1 1
14-2 a little more: a 01 = 1 1/2 0 1 0 0 0 0 0 2 0 0 a = 0 0 0 3 0 0 0 0 0 O 0 0 0 0 0 0 L 12 /! 0 0 12 / ( + 1)! 0 0 L 0 1! a = L L L L L ( + q)! 0 L L L q! selectio rule for a ij j-i= selectio rule for a ij j-i= 12 / = [! ] ( a ) m [( a ) ( a) ] = δ j, + m j 12 4 34 selectio rule 0!! j! j m! 12 / (oe step to right of mai diagoal) 12 / ( steps to right) Selectio rules are obtaied simply by coutig the umbers of a ad a ad taig the differece. The actual value of the matrix elemet depeds o the order i which idividual a ad a factors are arraged, but the selectio rule does ot. Lots of ice trics ad shortcuts usig a, a ad a a
14-3 Oe of the places where these trics come i hady is perturbatio theory. We already have: 1. WKB: local solutio, local (x), statioary phase 2. Numerov Cooley: exact solutio - o restrictios 3. Discrete Variable Represetatio: exact solutio, ψ as liear combiatio of H-O. Why perturbatio theory? replace exact H which is usually of dimesio by H eff which is of fiite dimesio. Trucate ifiite matrix so that ay eigevalue ad eigefuctio ca be computed with error < some preset tolerace. Fit model that is physical (because it maes localizatio ad couplig mechaisms explicit) yet parametrically parsimoious derive explicit fuctioal relatioship betwee the -depedet observable ad e.g. E hc = ω e( + 1/2) ω e x e ( + 1/2) 2 + ω e y e ( + 1/2) 3 establish relatioship betwee a molecular costat (ω e, ω e x e, ) ad the parameters that defie V(x) e.g. ω e x e ax 3 There are 2 ids of garde variety perturbatio theory: 1. Nodegeerate (Rayleigh-Schrödiger) P.T. simple formulas 2. Quasi-Degeerate P.T. matrix H eff fiite H eff is corrected for out-of-bloc perturbers by va Vlec or cotact trasformatio ~4 Lectures Derive Perturbatio Theory Formulas * correct E ad ψ directly for eglected terms i exact H * correct all other observables idirectly through corrected ψ
Formal treatmet E = λ 0 E + λ 1 E + λ 2 E (2) usually stop at λ 2 14-4 ψ = λ 0 ψ + λ 1 ψ H = λ 0 H + λ 1 H usually stop at λ 1 (because all observables ivolve ψ ψ ) order-sortig is MURKY λ is a order-sortig parameter with o physical sigificace. Set λ = 1 after all is doe. λ = 0 1 is lie turig o the effect of H. Equatios must be valid for 0 λ 1. Plug 3 equatios ito Schr. Equatio Hψ = E ψ ad collect terms accordig to order of λ. λ 0 terms H ψ = E ψ left multiply by ψ m H m = E δ m requires that H be diagoal i ψ eigevalues E { } ad eigefuctios { ψ ( 0) } of H CALLED BASIS FUNCTIONS CALLED ZERO ORDER MODEL
14-5 So we choose H to be the part of H for which: * it is easy to write a complete set of eigefuctios ad eigevalues * it is easy to evaluate matrix elemets of commo perturbatio terms i this basis set * sometimes choice of basis set is based o coveiece rather tha goodess does t matter as log as the basis is complete. examples: Harmoic Oscillator Morse Oscillator Quartic Oscillatr -fold hidered rotor V(x) = 1 2 x2 [ ] 2 V(x) = D1 e ax V(x) = bx 4 0 V (φ) = ( V 2 )1 cosφ Now retur to the Schr. Eq. ad examie the λ 1 ad λ 2 terms. λ 1 terms H ψ + H ψ = E ψ + E ψ multiply by ψ from H operatig to left H +E ψ ψ =E +E ψ ψ same get rid of them could also require ψ ψ = 0 we do require this later
14-6 H = 1st-order correctio to E is just E expectatio value of perturbatio term i H: H. retur to λ 1 equatio ad this time multiply by ψ m H m + Em ψ m ψ H m = ψm ψ ψ m ψ = = 0 + E ψ m ψ E ( Em ) H m E Em completeess of { ψ }: ψ ψ ψ = ψ ψ ψ but we ow this ψ = ψ H E E * idex of ψ matches 1st idex i deomiator * = is problematic. Isist Σ exclude =. * we could have demaded ψ ψ 1 = 0
14-7 λ 2 terms most importat i real problems although excluded from may text boos. H ψ = E ψ + E (2) ψ multiply by ψ ψ ψ = 0 ψ H ψ =0+E (2) completeess ψ H ψ ψ ψ = E ( 0 ) ( 1 ) ( 0 ) ( 0 ) ( 1 ) ( 2 ) H, Σ H, E E (2) E = Σ H 2, E E always first matrix elemet squared over eergy differece i eergy deomiator we have derived all eeded formulas E,E,E (2) ;ψ,ψ Examples V(x) = 1 2 x2 + ax 3 (a < 0) V(x) H = 1 2 x2 + p2 2m H = ax 3 (actually ax 3 term with a < 0 maes all potetials uboud. How ca we preted that this catastrophe does ot affect the results from perturbatio theory?) x
14-8 eed matrix elemets of x 3 two ways to do this * matrix multiplicatio * a,a trics 32 / x 3 il = xij x j x l j, 32 / / [ ] 3 3 12 x = x 2 a a h = h + mω ~ mω 32 = / a a aa aa a aaa aa a a aa a a a a h + + + 2 +( + + )+ mω 3 3 3 [ ] each group i has their ow v selectio rule (see pages 13-8 ad 9): simplify usig [a,a ] = 1 Goal is to maipulate each mixed a,a term so that the umber operator appears at the far right ad exploit a a = Oly ozero elemets: 3 a 3 3 a +3 [ ( 2) ] 1/2 ( +2) ( +1) = 1 [ ] 1/2 = +3 square root of larger q.. ( a aa + aa a + aaa )= 3aa a because aaa= aaa+[ a, a] a= aa a a aaa = aa a + a[ a, a ]= aa a + a ( aa a) 32 / = 1
14-9 ( aa a + a aa + a a a)=3a a a +3a 3a a a +3a [ ] +1 =3+1 1/2 +3( +1) 1/2 =3 ( +1) 3/2 So we have wored out all x 3 matrix elemets leave the rest to P.S. #5. Property other tha E? Use ψ = ψ + ψ e.g. trasitio probability (electric dipole allowed vibratioal trasitios) 2 P x for H - O 2 h x = 2(m) 1/2 > δ >, < +1 mω (oly = ±1 trasitios) for a perturbed H O, e.g. H = ax 3 ( 0) H ψ = ψ + Σ ψ ( 0) ( 0) E E ψ () 1 () 1 ( 0) () 1 ( 0) H + 3 ψ ω ψ ( 0) H + 1 ω ψ ( 0) H 1 ω ψ ( 0) H 3 = + + + + ω ψ + 3 + 1 + 1 3h h h + 3h () 1 () 1 ( 0) 3
1st idex Allowed 2d Idices + 3 + 4, + 2 + 1 + 2, X + 1, 1 1, 2 3 2, 4 14-10 For matrix elemets of X. cubic aharmoicity of V(x) ca give rise to = ±7, ±5, ±4, ±3, ±2, ±1, 0 trasitio h x + 7 = 2m 1/2 mω 2 a 4 x +7 m 7 11 7 ω 7/2 a 2 1/2 ( + 7)! ( 3hω ) 2! 7/2 other less extreme trasitios go as lower powers of 1 ω ad