8.3 Perturbation theory
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1 8.3 Perturbatio theory Slides: Video Costructig erturbatio theory Text referece: Quatu Mechaics for Scietists ad gieers Sectio 6.3 (u to First order erturbatio theory )
2 Perturbatio theory Costructig erturbatio theory Quatu echaics for scietists ad egieers David Miller
3 Tie-ideedet erturbatio theory Presue soe uerturbed Hailtoia H that has kow oralized eige solutios i.e., H 0 We ca iagie that our erturbatio could be rogressively tured o at least i a atheatical sese For exale we could be rogressively icreasig alied field fro zero to soe secific value o
4 Tie-ideedet erturbatio theory We look successively for the chages i the solutios for exale, for the th eergy eigevalue roortioal first to electric field first-order correctios (0) a (0) 0 lectric field,
5 Tie-ideedet erturbatio theory We look successively for the chages i the solutios for exale, for the th eergy eigevalue roortioal first to electric field first-order correctios roortioal to 2 secod-order correctios ad so o (0) 2 a b 0 (0) lectric field,
6 Tie-ideedet erturbatio theory It is ore coveiet ad geeral if we iagie a secific fixed erturbatio (e.g., a field ) ad we atheatically icrease a house-keeig araeter fro 0 to 1 so our erturbatio is with fixed Now we exress chages as orders of rather tha of the field itself (0) (1) 2 (2) (0) 0 1
7 The house-keeig araeter (0) 2 So, istead of writig ab (0) (1) 2 (2) we are writig ad istead of workig out a ad b we are goig to work out araeters (1) (2) ad ad so o These have diesios of eergy ad reflect the first order ad secod order correctios to the eergy as a result of the secific erturbatio e.g., a secific field
8 The house-keeig araeter I geeral, the, we iagie that our erturbed syste has soe additioal ter i the Hailtoia the erturbig Hailtoia H I our exale case of a ifiitely dee otetial well with a alied field that erturbig Hailtoia would be H e zl /2 I the theory, we write the erturbig Hailtoia as H usig to kee track of the order of the correctios through the owers of i the exressios We ca set = 1 at the ed if we like z
9 The house-keeig araeter (0) 2 So, we could set u the theory usig ab i which case we would work out a ad b ad soe other araeters But, to ake it ore geeral we use (0) (1) 2 (2) ad work out the araeters (1) (2) ad ad soe other araeters If this is cofusig at first the just thik of as the stregth of the electric field i our secific roble
10 Costructio of the orders of erturbatio theory With this way of thikig about the roble atheatically we ca write the erturbed Schrödiger equatio as H o H We ow resue that we ca exress the resultig erturbed eigefuctio ad eigevalue as ower series i this araeter, i.e.,
11 Costructio of the orders of erturbatio theory We ow substitute these ower series ito the erturbed Schrödiger equatio H o H to get H 0 H
12 Costructio of the orders of erturbatio theory Now, at ay secific oit i sace, each fuctio ad each fuctio H 0 H is just soe uber So, at ay secific oit i sace, the left had side of H H is just a ower series i, e.g., ad so is the right had side, e.g., a a a a b b b b
13 Costructio of the orders of erturbatio theory Because a ower series exasio is uique the oly way the equality of two ower series ca work a a a a b b b b for every value of withi soe covergece rage e.g., 0 to 1 is if the ters are equal, oe by oe, i.e., a0 b0 a1 b a 1 2 b2 a3 b3 ad so o
14 Costructio of the orders of erturbatio theory Hece, i H 0 H we ca equate each ter with a secific ower of ad hece obtai a rogressive set of equatios which we ca solve to evaluate correctios to whatever order we wish
15 Progressive set of erturbatio theory equatios I H 0 H equatig ters i 0, i.e., ters without gives the zeroth order equatio H o i.e., the uerturbed Hailtoia equatio with eigefuctios ad eigevalues So if we ow resue we start i a secific eigestate we write ad 0 istead of ad 0
16 Progressive set of erturbatio theory equatios So, with H 0 H we get a rogressive set of equatios each equatig a differet ower of H ad so o o H H o H H o
17 Progressive set of erturbatio theory equatios We ca rewrite these equatios as o H H H o H H o ad so o H 0 o 1 1 H H o H H o
18
19 8.3 Perturbatio theory Slides: Video First ad secod order theories Text referece: Quatu Mechaics for Scietists ad gieers Sectio 6.3 (startig at First order erturbatio theory u to xale of well with field )
20 Perturbatio theory First ad secod order theories Quatu echaics for scietists ad egieers David Miller
21 First order erturbatio theory Now we ca calculate the various erturbatio ters 1 1 Startig with H H o ad reultilyig by gives H H 0 o o 1 H 1 H i.e., 1 H a forula for the first-order eergy correctio i the resece of our erturbatio H 1
22 First order erturbatio theory For the first order correctio to the wavefuctio we exad that correctio i the basis set 1 1 a Substitutig this is i 1 1 H H o i ad reultilyig by gives 1 1 H a 1 i o i i 1 i H i i 1 1 i i H
23 First order erturbatio theory So we have i ai i i H We resue the eergy eigevalue is ot degeerate i.e., oly oe eigefuctio for this eigevalue With o degeeracy, we still eed to distiguish two cases First, for, fro above 1 i i ai i H 1 i H So ai i Secod, for i 1 1 a 0a H (1) a which gives o costraits o 1 1 0
24 First order erturbatio theory (1) We are therefore free to choose The choice that akes the algebra silest 1 is to set a 0 which is the sae as sayig 1 we choose to ake orthogoal to The sae haes for the higher order equatios Hece, quite geerally we ake the coveiet choice j 0 a
25 First order erturbatio theory 1 i H 1 Hece with ai ad a 0 i the first order correctio to the wavefuctio is 1 H ad we have the first order correctio to the eergy 1 H
26 This iage caot curretly be dislayed. Secod order erturbatio theory We cotiue siilarly to fid the higher order ters Preultilyig H 2 1 H 1 2 o both sides by so o H gives 0 H H j Sice we chose orthogoal to H 2 1
27 Secod order erturbatio theory Usig our result for the first-order wavefuctio correctio 1 H the fro we obtai quivaletly H H H H 2
28 Secod order erturbatio theory For the secod order wavefuctio correctio 2 we exad 2 otig ow that is chose orthogoal to a 2 2 We reultily H 2 1 H 1 2 by i to obtai H 2 a 2 o i o i i H i i i i a a H
29 Secod order erturbatio theory So, we have a a a H i i i i Note this suatio excludes the ter = 1 because we chose to be orthogoal to 1 i.e., we have chose a 0 Hece, for i we have a i H ai ai i i Note that the secod order wavefuctio deeds oly o the first order eergy ad wavefuctio
30 First ad secod order erturbatio results 1 H First order a 1 1 a 1 i H i i, a 1 0 H 2 1 Secod order a 2 2 a 2 i a i H ai i i, a 2 0
31
32 8.3 Perturbatio theory Slides: Video Alyig erturbatio theory Text referece: Quatu Mechaics for Scietists ad gieers Sectio 6.3 (startig at xale of well with field )
33 Perturbatio theory Alyig erturbatio theory Quatu echaics for scietists ad egieers David Miller
34 xale of a well with field We write the Hailtoia as the su of the uerturbed Hailtoia which is, i the well, i our diesioless uits 2 1 d H o 2 2 d ad the erturbig Hailtoia H f 1/2 where agai we take f 3 for a exlicit calculatio
35 First order eergy correctio I first order, the eergy shift with alied field is 2si 1/2 2si 1 H f 0 2 2f 1/2 si d 0 d The itegrals here are zero for all because the sie squared fuctio is eve with resect to the ceter of the well whereas 1/2 is odd Hece, for this articular roble there is o first order eergy correctio
36 First order eergy correctio There o first order eergy correctio because of syetry
37 First order eergy correctio There o first order eergy correctio because of syetry If the eergy chaged roortioately with alied field?
38 First order eergy correctio There o first order eergy correctio because of syetry If the eergy chaged roortioately with alied field chagig field directio (or sig) would chage the eergy correctio sig But, by syetry here the eergy chage caot deed field directio?
39 Matrix eleets for erturbatio calculatios The geeral atrix eleets that we will eed for further erturbatio calculatios are 1 0 H H 2si u 1/2 2si v d uv u v f I geeral we eed u ad v to have oosite arity i.e., if oe is odd, the other ust be eve for these atrix eleets to be o-zero sice otherwise the overall itegrad is odd about 1/2
40 First order correctio to the wavefuctio We calculate the first order wavefuctio correctio for the first state, i.e., for = 1 q i H 1 a ai 2 o 1 oi 2 where o are the eergies of the uerturbed states, ad q is a fiite uber we ust choose i ractice Here, we chose q = 6 though a saller uber would likely be quite accurate
41 First order correctio to the wavefuctio xlicitly, for the exasio coefficiets 1 a H / i i 1 o1 oi for 3 uits of field we have uerically 1 1 a a Here the value of for a 2 coares closely with the value of obtaied above i the fiite basis subset ethod 1 a
42 First order correctio to the wavefuctio We su the zero-order (uerturbed) wavefuctio si
43 First order correctio to the wavefuctio ad the first order correctio art fro the secod basis fuctio si 2
44 First order correctio to the wavefuctio To get our aroxiate wavefuctio solutio si si2
45 First order correctio to the wavefuctio Addig the ext correctio akes egligible differece si si si4
46 Secod order eergy correctio Sice the first order correctio to the eergy was zero to get a erturbatio correctio to the eergy we go to secod order 2 q 2 1 H 1 xlicitly, we have 1 H which uerically here gives or a total eergy of which coares with the result of fro the fiite basis subset ethod 2 2 1
47 Aroxiate aalytic forulas Note that is aalytically roortioal to the square of the field f q H /2 q f /2 q f
48 Aroxiate aalytic forulas Hece erturbatio theory gives a aroxiate aalytic result for the eergy which we ca ow use for ay field xlicitly, we ca write for the eergy of the first state i diesioless uits f This tyical kid of result fro erturbatio theory gives us a aroxiate aalytic forula valid for sall erturbatios
49 Aroxiate aalytic forulas Siilarly, for the wavefuctio the correctio is aroxiately roortioal to field for exale with exasio coefficiet 1 i H i 1/2 ai f i i So, keeig oly the doiat cotributio fro the secod-state wavefuctio i our exale we would have the aroxiate forula for sall f 2 si 0.06f 2 si 2 (This is ot quite oralized, though that could be doe)
50 Aroxiate aalytic results f Wavefuctio ergy Field
51
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