The Generating Equations of Rayleigh—Schrödinger Perturbation Theory.pdf

8/13/2019 The Generating Equations of RayleighSchrdinger Perturbation Theory.pdf

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Vlume 9, nmber HMA Y LR Augst 1982

and o retcion are poed on e degeneracieo Ea The emande V dened byV=H- h 5and operaor are ten o be hermia a corepond to a coed yem Therefore wavefunconmay be ohonormazed


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Volume 91, nber 1 CHEMICAL PHYSC TR 27 Agst 1982

reg the orhonormity onditon eq.

6whih

in ers of U nd Rk reds

(4a4b>= Eu}kb ab,k tht s

U Uh : k;]-1 whh together wh eq 9 ompeey determnesU pt from rreevn phse ftors

An essent hypohess ws he non-sngurhrter of wch orresponds o syng th hentions

4ja ' 'jb bja , 1, d (23)re inery ndependent Asa s he projeton ofa over the subspe spnned by the eigenfunionsofh wth egenvue he ssumpion mposes restrition on how dfferentH ndh my be I snownh n soe ses, notby in superondutvity the ondition is not fuled bu tis s the exepion rther hn the ue No presrpion sknownat presen onerning the wys of pritionngH, eq5 h fu e ner ndependene ondion sone is fored here to te prt pproh ndssume the ondtion o be rue uness the n resusprve the ontrry

is neresng to pon out h one n gve oreonvenient propertes to he mtrx W by mng hnge of bsis o soe iner obnion over thefuntons


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lme 9 nmbr CHEMICAL PYSCS LTTERS 27 Ag 1982

sates xcpt h gond one are obly egeneat; he egneracy of the mh exct sa is rovonly he 2mh or o prtrbation theoy e hexac soons ae known.

Lt I b e oao's oent of ineria, L s oal angar oent an p s ipol oenwhic es on he pan noa o L ony the coponn of E noral o conrbtes o h enegy,one ay tae whout oss of gneraiy h x axslong and the z as along Ten te toal haonian ay be wtn

H=Li2 - ps=-(2 2d2 - p cos (3)whre s te azal ange. e Scding qaon s hee exacly solbl [910] as eq 1 eads o

(2fu2 + a 2 cos 2=O (33)

a!Ejh2 =-Ipef2, 2 . (39)

Becase peaton v conncs only adacent1evels loorde rs in A ogh o vans ess q. (3) concde w hos obtand y

Mc can [1 1 n a coplely dfferent fashon Iay be sn ta e owes or in whic e egeneracy of te and _ sas s eove is hat inwch the at prouc 10 01 appars for e rsi n h expanson (3) Ths appens n h 2jhorder of prrbaon ory an reslng spngs ns o o be

( )S= 2 '

2 2 2(- )S (2j )(a 1 2 C21) A 1 r

an xpresson wic has no been gven bef nss clt o oan n any oer way ne ayalso wt wtout cty the coecents kan fo h k or hs prpos one ayeihe noaze exacty te k coecens bysng (6) or eng ore consisen wth the perubation schee noaz h o he esid oery an expanson of qs. 22 and () in a annrsiiar to tha of qs. (2)30)

Becase the coefcents k, e cease veyrapy wn incass a sp appoxaton solve qs (5) assng a ty vanish fo a cran val of onwar As an ilstraion w proxa gonstae negy and cofcnsangRO =O for ;2 erg hat y ntonR,an ang us o qs (5) 20 ()an 16 is oaned


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Volume 9 uber 1 CHEMICLPYC RS 27 Aut 982

R+0=[1(18X2)/]j3/2A,

fSh=1/2;Rl+,O '

T - (1 2 )-1/2v +Rl+,O (41)

where accordng o the tng condion 1)onesud take inR10 he lowe s n he squae rooFoX= (q= ),a vaue wch s we outsde herang of pertubaon heory one otns

f/S 2 u0 = 077 , U+ o = -0.64,' (4)

o be compared with the ounded exact vues [11 ,pp 31320/Sh = -27 , u0=01 U1+0=-0.67,' {4)

a remarkae agreemen or such a crude approxmaon Th pcson ncreases vey apdy one owsie coecens o be non-vasng Notc ha

om eq. 39one woud on 0/Sh = 4In concuson, t has been shown ha Ryegh

Schrdnger peruraon heoy s jus a pcarway (eraion mehod) o soving he generaigeuatons u by no means he ony one

Reeeces

] Dagao, in: Qnu heory V ed D.RBaes (cademc e New ok96 pp 1-09

[] . Boch Nuc ys 6 1958 329[ oveez, m J. Pys 35 1967 62 C. ovee J Phy C2 1969 61

5 F Jogese, Mo Py 29 1975 137.6] ovee Pys. ettes 79A 190 ] oivee Pys Re A2 19 [8] Dud, uio iqes des uatios

gbues Vo 2 Mo Pa 1961 59 MP. vemn Py R. A2 1981 339

MP lvermn Am Phy 9 1981 8711] NW Mccn Teory d ppito ofMtheu

cton Cedo e Oxod 195

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The Generating Equations of Rayleigh—Schrödinger Perturbation Theory.pdf