Rayleigh-Schrödinger Perturbation Theory

Transcription

1 Raylgh-Schrödngr Prturbaton Thory Introducton Consdr so physcal syst for whch w had alrady solvd th Schrödngr quaton copltly but thn wshd to prfor anothr calculaton on th sa physcal syst whch has bn slghtly odfd n so way. Could ths b don wthout solvng th Schrödngr quaton agan? Ths would b partcularly rso f for xapl all w wantd to do was to subct th syst whch w now fl s wll dscrbd by th wav functons and nrgs to a srs of sall changs such as posng a srs of lctrc or agntc flds of varous strngths. Suppos th orgnal syst wr dscrbd by th haltonan H, so that w had a coplt st of nrgy gnvalus and gnfunctons lablld by : H H H Th odfcaton of th syst s dscrbd by th addton of a tr to th orgnal haltonan so that th nw haltonan s H = H + and has solutons wth unsuprscrptd sybols H H, tc. Prturbaton thory s basd on th prncpl xprssd n McLaurn and Taylor srs that f a varabl x s altrd by a sall aount thn a functon fx + of th varabl can b xprssd as a powr srs n. Truncaton of th srs at th trs of dffrnt powrs,,,, dfns th ordr of th prturbaton as zroth, frst, scond ordr tc. So th haltonan H and th gnral soluton, for th prturbd syst ar wrttn as H H and th Schrödngr quaton for th prturbd syst s

2 H H 3 whr and 3,,, ar th st -ordr, nd -ordr, 3 rd -ordr corrctons to th th stat wav functons; stat nrgs. 3,,, ar th st -ordr, nd -ordr, 3 rd -ordr corrctons to th th Substtutng for and bac nto th Schrödngr quaton 3 w hav H H H H whch on quatng th sa powrs of gvs th quatons H H H H zroth ordr sa as qns. frst ordr frst ordr 5 scond ordr 6 4 and w could go on... Th zroth ordr quaton tlls us nothng nw t's ust. But 5 and 6 dfn th condtons of frst and scond ordr prturbaton thory, whch co nxt.. Frst ordr prturbaton a nrgs For ths w nd q. 5. W now th sts { } and { } but not th frstordr corrctons l { } so lt s xprss th lattr as a cobnaton of our bass st th

3 3 coplt zroth ordr st of functons { }: a Snc s a corrcton to { } th suaton xcluds =. Substtutng 7 n 5 w gt ' a H and snc, fro, H = th oprator H n th last quaton ay b rplacd by gvng ' a Multply by * and ntgrat: 7 Th ntgral on th LHS s th Kroncr dlta whch ans that t s unty f = noralsaton and zro othrws orthogonalty. Its ffct n th suaton s to ll all th trs for whch, lavng a whr < > s a concs way of wrtng th ntgral <> 8 * d. In 7 put =. Ths gvs our frst usful rsult: = < > 9 In othr words, th frst ordr corrcton to th th nrgy s th xpctaton valu * d of obtand usng th zroth ordr wav functon. Non of th othr functons ar nvolvd. So th nrgy corrctd to frst ordr s ust

4 4 b Wav functons Nxt, try puttng n 7. Ths provds th coffcnt: a But ths s what w nd n 6 to xprss th frst-ordr corrcton to th th stat wav functon. So th wav functon of th th stat, corrctd to frst ordr, s Unl th thory ladng to th frst-ordr nrgy n qn. 9, n ordr to xprss th wav functon to frst ordr th functons and nrgs and of all th othr zroth-ordr stats ar nvolvd. Th functon n qn. 7 s thn noralsd aftr calculatng all th coffcnts xprssd by qn... Scond ordr prturbaton To scond ordr th wav functon and th nrgy of th th stat ar = + + and = + + As was don to xprss th st ordr corrctons to th wav functons n 7, w also wrt th nd ordr corrctons to th functons as a lnar cobnaton of th zro th ordr st{ }: b 3 and substtut ths xprsson n 6: b H As bfor w us q. s H = to rplac H by, gttng Multply by * and ntgrat: b

5 5 Not th Kroncr dltas! Th on on th LHS lts th suaton to a sngl tr: b d d * * 4 Optng for = whch w ll call, not only dos th LHS n 4 bco zro but so also dos th frst tr on th RHS bcaus 7 xprssd n trs of all th xcpt th frst RHS tr s a whr and s thrfor zro. So w ar lft wth d * n whch can b substtutd fro, gvng 5 and so th prturbd nrgy lvl to scond ordr s gvn by 6 Th nd ordr corrcton to th wav functon,, could b calculatd n a slar way to that n whch w got that ld to. Ths t t would b by optng for n 4, but w don't do ths hr. 3 rd ordr prturbaton Hr s th nrgy to 3 rd ordr: n n n n 6a th ordr st ordr nd ordr 3 rd ordr corrcton corrcton corrcton

6 6 If you loo at how th 3 rd ordr tr s an xtnson of th nd ordr tr you can guss how to wrt any hghr ordr trs. Th q th ordr corrcton to th nrgy would b q p n p n n 7 p So ponts concrnng Raylgh-Schrödngr Prtubaton Thory. Although w stoppd at scond ordr, provdd you wr prsstnt nough thr would b no rstrcton to procdng to as hgh an ordr of prturbaton as you wshd, usng th quatons dvlopd n th arlr part of ths account. Th nrgy ntrval Δ n th dnonators of th corrcton trs, 6 or 7 show that succssvly hghr ordr prturbatons a succssvly sallr contrbutons.. All th r th -ordr corrctons to th wav functons, r, and nrgs r, nvolv atrx lnts < > of th prturbaton oprator as n 5 n th nurator and nrgy dffrncs r r n th dnonator, wth th sol xcpton of n 9. Physcally ths ans that th procdur conssts of xng n functons nto, partcularly fro th st of hgh-nrgy unoccupd stats. 3. Bcaus of th lattr pont, RS prturbaton thory cannot b usd f th stat to b xd wth s nrgtcally dgnrat to ths stat. 4. Fro th prturbaton corrctons l thos n qns. 5 and 6 th xng n of hghrordr stats as th dnonator ngatv. Th ffct s thrfor to stablz th lowrnrgy stats. 5. Suppos that w wr nvstgatng th stats of a olcul A that was nfluncd by anothr olcul B at a dstanc R fro t. Thn th prturbaton oprator would consst of thos trs dscrbng th coulob attractons and rpulsons of th partcls of A wth thos of B. Th bass st of functons would b th coplt st { } of functons for both olculs.

7 7 Th xprsson for th prturbd nrgy stats would start off l 6 but would xplor th varous ordrs untl th hghr-ordr trs bco nglgbl. Whn th succssv trs,, 3,... ar xand thy ar found to b of th fors R, R, R 4, R 6, R 8,... whch can b ntrprtd as th utual ntractons of th nt charg, dpols, quadrupols, octupols,... both prannt and nducd that ar cratd on th two olculs. Ths rsult s sots ntrprtd n trs of th non-bondd London or van dr Waals forcs arsng fro th fluctuatng lctronc chargs on th olculs A and B. Whl ths dscrpton s a usful on, you don t nd to thn of fluctuatng lctronc chargs ntrolcular forcs ar th rsult of xtndng or prturbng th haltonan of on olcul by th ffct of th othr. Th ffct would also b dscrbd by a sngl calculaton usng a haltonan for th twoolcul syst. Applcatons of Prturbaton Thory. Th lctronc nrgy of th hlu ato Th hlu ato Haltonan s Z Z H 4 r 4 r 4 r. H = H + W shall trat th lctronc rpulson tr as a prturbaton of th th ordr Haltonan H, so 8 4 r Zroth ordr Ignorng th lctron-rpulson tr 8 w xprss H as a su of th two ranng parts on for ach of th lctrons and H = H + H

8 8 whr H = Z 4 r and H = Z 4 r Ths coponnts H and H ar ach Haltonans for H + whch s a hydrogn-l ato, whos Schrödngr quatons ar xactly solubl: H = H = Rbr w now and xactly. Th ground stat s dscrbd by th hydrognc s atoc orbtals = N xp Zr / a 9 = 4 4 Z nrgy of th H-l ato W wrt th coplt th ordr Haltonan H as H, as a rndr that t nvolvs th coordnats of both lctrons and xprss th rqurd -lctron wav functon, as th spl product, = Opratng on t wth H, w hav H,, = [H + H ] = H + H In th frst tr on th rght hand sd H oprats on gvng and lavng unchangd. Slarly n th scond tr H oprats on gvng and lavng unchangd, so w hav H,, = + = So th gnvalu quaton H,, =,

9 9 tlls us that th H ato th ordr nrgy s,.. twc th nrgy of th H + on. Ths s 54.4 so that th th ordr nrgy s = 8.8 Ths s physcally anngful bcaus ach of th two lctrons s dscrbd as f t wr n a H + ato, whos nrgy s. Th tr 4 r that accounts for thr utual rpulson has bn ottd to for H. But th nrgy s far too ngatv: th fact that th actual ground-stat lctronc nrgy of H s 79. th su of th frst two onzaton nrgs shows that t s ssntal to nclud th ntrlctronc rpulson tr n th Haltonan. Frst ordr Fro q. 9 th frst ordr corrcton to th nrgy s s = <s s> 4, * r, d d Substtutng for fro 9 and prforng th ntgraton lads to = 34.. Ths brngs th nrgy of hlu to frst ordr to = + = = 74.8 [xptl. valu 79. ] Scond ordr In ordr to for th atrx lnts q. 5 w nd all th th -ordr functons for,.. s, s, p and all th corrspondng th ordr nrgs s, s, p,, * d rqurd for substtuton n

10 But ths ar nown xactly snc thy ar th solutons of a hydrogn-l ato. valuaton of fro qs. 5 and 8 gvs 4.3, so th nrgy of hlu to scond ordr s = + + = = 79. [xptl. valu 79. ] Hghr ordrs Th xng of th hghr ordr stats nto th ground stat s should also nclud th contnuu,.. stats wth nrgs gratr than zro whch s th axu nrgy obtand fro th Bohr forula n = 4 Z n. Th dvlopnt of N th ordr prturbaton thory s tdous but routn, as s th nurcal calculaton of all th rqurd atrx lnts. Calculatons hav bn prford up to 3 th ordr gvng = hartr atoc unts of nrgy = 79.6 xptl. valu 79.

11 . Star ffct : Th shftng of th torsonal nrgy lvls of an OH group by an lctrc fld. Consdr a part XOH of a olcul n whch th OH group rotats around th XO bond. OH has an lctrc dpol ont whos coponnt n th plan prpndcular to th XO bond rotaton axs s. If s th dpol ont along O-H thn = cos Whn an lctrc fld F s appld n ths plan.. prpndcular to th rotaton axs, rotats n th xy plan through azuthal angls ontord as so that succssvly cos nto and out of algnnt wth F. Th couplng of th dpol wth th fld producs an nrgy = F = F cos whch s an nrgy prturbaton tr to th haltonan H and s th coponnt of th OH bond dpol ont prpndcular to th rotaton axs. Th total haltonan s H = H +. In th absnc of an lctrc fld F = th soluton of th Schrödngr quaton dscrbng th ntrnal rotaton, H I d d s th falar on of a partcl confnd to a crcl: I

12 whr,,,... W shall xplor th prturbaton of th torsonal nrgy lvls to scond ordr by calculatng th atrx lnts of that ar rqurd n th xprsson fro qn. 5: I ' ' ' Both frst- and scond-ordr prturbaton trs on th rght of ths quaton rqur th atrx lnts whr = < > whr = for th frst-ordr and for th scond-ordr trs. W shall frst valuat th gnral atrx lnt. ' Substtutng for and th zroth ordr functons = F cos = F ½ + th atrx lnt ' bcos ' F cos F d 4 ' d F 4 Both ntgrals ar of th for ' d ' d n d whr n s an ntgr. Now such an ntgral s zro unlss n =, n whch cas t s s footnot. Thn as = thr of th two ntgrals contrbuts and w hav F ' n n n n If n, d cos n sn n Othrws n =, d n

13 3 Ths s our rqurd non-zro atrx lnt of prturbaton. Not that as = thr s no frst ordr corrcton to th nrgy. W can now calculat th scond ordr prturbaton tr of th nrgy. F I h I. F h I Rotatonal lvl = of th rotor s thrfor lowrd by a quantty proportonal to th squar of th fld ntnsty and of th dpol ont of th OH bond. Th - fold dgnracs of th lvls ar not lftd. But thy ar whn hghr-ordr prturbaton thory s appld!. Rcall that s th coponnt of th OH bond dpol ont prpndcular to th rotaton axs, and w should wrt What f th fld wr appld n a drcton othr than prpndcular to th torsonal axs? Thn = F would hav coponnts z and x fro th couplngs along and prpndcular to th ths axs. Th shft fro x would b of th sa for as th on w ust calculatd but sallr bcaus F has a sallr coponnt n th crcl around whch th dpol ont s rotatng, and z would b F cos whr s th constant angl btwn and F. If th lctrc fld wr appld paralll to th rotatonal axs, would stll go fro to but as t rotatd, th OH bond dpol would a a constant angl wth th fld. Th nrgy shft would thn b sply F sn whr sn s th angl ad by th bond dpol wth th torsonal axs and th lctrc fld.

14 4 3. Star ffct : Dgnrat prturbaton thory. nrgy splttngs n th H ato W prturb th th ordr haltonan H of th H ato by addng to t a tr = Fz * Snc z s antsytrc or odd any dagonal atrx lnt s zro.. z dz. For ths rason thr s no Star ffct to st ordr PT. In ordr to go to hghr ordr w ust for off-dagonal atrx lnts n. Ths lnts wll coupl th s ground stat p s to hghr stats such as,,, whch wll rsult n a shftd nrgy lvl of what was th s ground stat, but snc ths stat s non-dgnrat thr wll b no Star splttng. W thrfor a th probl or ntrstng by rplacng th ground stat by th n = stat spannd by th four functons s, p +, p, p -. W should l to now to what xtnt ths 4-fold dgnracy s rovd by th Star ffct. s p p Zroth ordr functons Th four bass functons of th n = shll of th H ato, and thr nrgs all qual, n= ar nown xactly. Th functons can b wrttn as products of functons nvolvng sphrcal coordnats r,, n th for Rr, and th only factor of ths whch wll concrn us wll b whch s of th for whr = for s and,, rspctvly for p +, p, p. Wrttn ths way ths functons ar obvously gnfunctons of th angular ontu oprator l z = wth gnvalus. s s = s l z s = s p + p + = p l z p + = + p + p p = p l z p = p p - p = p l z p = p whr s and p ar th parts of th s and p AO functons wthout th factor.

15 5 Th rason that ths functons ar gnfunctons of l z s bcaus ths oprator couts wth th haltonan. But as l z also couts wth th prturbaton tr = z, atrx lnts ford fro functons functons and corrspondng to dffrnt angular onta and ' ar zro,.. f <l z > = thn < > = also. Ths s bcaus, as l z s a hrtan oprator ts gnfunctons ar orthogonal f thy blong to dffrnt gnvalus,.. atrx lnts < l z > =. Th only non-zro lnts of ar for bass functons corrspondng to th sa gnvalu of l z. Howvr, for th sa rason as n th applcaton of prturbaton thory of a Star lctrc fld to a olcular torson, dagonal atrx lnts of ar zro bcaus th prturbaton = Fz, s not totally sytrc,.. <> = for all >. So w hav th followng. a All dagonal lnts of ar zro b Non-zro lnts of ust b fro pars of functons corrspondng to th sa gnvalu of l z. Ths lavs only on par of functons that fors a non-zro lnt wth : that btwn s and p. Snc thr dgnracy dos not allow th to b usd n prturbaton thory w shall nd to calculat th gnvalus of th nrgy atrx of spannd by th four bass functons s, p +, p, p All th lnts of wll b zro xcpt two, thos ford by s and p, and by p and s. Th zroth ordr nrgy s th nrgy of th s and p orbtals all four hav th sa nrgy, bcaus for all hydrogn-l atos wth on lctron, a subshll wth prncpal quantu nubr n s n -fold dgnrat. Th prturbaton nrgy atrx s thrfor = p s p s Th coplt haltonan atrx, H = H + s thrfor

16 6 H = + = whr s th nrgy of th n = shll rcall that for a hydrogn ato prturbaton atrx lnt <>s assgnd to a paratr v and th s p.. 9 F s z pz that s proportonal to th lctrc fld F. H s bloc-dagonal, th blocs bng, and. Th nrgy gnvalus ar obtand n th standard way by sttng th dtrnant of ach bloc qual to zro: v v v v =.. [ v ] = So th four nrgy gnvalus ar =,, + v, v. Th bloc dagonal for of th nrgy atrx tlls us datly that two of th gnvalus ar zro. In othr words, of th 4 dgnrat stats s, p +, p, p two of th p + and p ar unaffctd by th lctrc fld and so ran doubly dgnrat. Th dgnracy of th two ranng lvls s rovd as th s and p stats cobn to produc two nw stats whos nrgs ar rasd and lowrd by an aount v.

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