2. Time-Independent Perturbation Theory
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1 M., /7 Tus. /8 Wd.,/9 Fri., / M., /4 Tus. /5 6. Ti-idpdt Prturbti, -dgrt 6. Ti-idpdt Prturbti, dgrt 6. Fi Structur f ydrg Dily.M Wkly Dily.W Dily.F Z & ypr-fi Dily.M Wkly Dily: Kyl Jcb Spcr Gigj At Jssic S Atwi Jth Csy Jry Mrk Cr Brd quipt Griffith s tt Chck dilis Aucts Dily.M Mdy /7 Griffiths 6. Ti-idpdt Prturbtis, -dgrt Ths t fw dys, w ll b lkig t wys t gt pprit slutis t prbls tht r t difficult t ctly slv.. Ti-Idpdt Prturbti Thry Th id f prturbti thry is tht, if yu chg th ilti ly littl bit, yu d pct th wvfuctis d thir rgis t chg ly littl bit t. Nw, w r usd t usig Tylr sris t pd ut cplictd fucti rud s giv pit, d ly usig s y trs s w d fr th lvl f ccurcy dsird. S i th s wy, w c igi tht th w wvfuctis d rgis shuld b prssibl i trs f sris f subsqutly lss sigifict crrctis.. N-dgrt Prturbti Thry 6.. Grl Frulti Igi yu hd syst, t b ccrt, sy prticl i b, d iitilly th b flr ws prfctly sth. S yu hv yur sipl ilti, Ĥ, d yur sipl wvfuctis tht g with it,, Which hv thir sily fud rgis,
2 Th yu g d ss it ll up by pushig up dt i th iddl f th wll s flr. Nw, th w ilti will b th ld plus w tr fr th dditi f th dt,. Ccptul: Wht is th pit f i quti 6.8? Ccptully, ths t fw stps r sy t udrstd if yu thik f this dyiclly; s yu push up, th ilti slwly chgs, it ll d up s, but lg th wy it s t Whr t strts ut s s yu v t yt strtd pushig d ds up s wh yu v fiishd kig th dt. w ds th w wvfucti dpd this turig? Wh kws, but i pricipl, it ds dpd it, d w c writ ut its dpdc s pwr sris: Whr th s r th crrctis if it k yu fl r cfrtbl, w t rlly b usig tht fct. j j j! j, but w "S f th tti, ily th suprscripts d subscripts, wt vr y hd bit. Thir prsc d th scti hrd t fllw d diffrtit th usful ifrti fr th rst." Brdly W Siilrly, th w rgy ust dpd up hw turd th prturbti is; hw, wh kws? But prsubly tht t c b prssd s pwr sris: Agi, if it ks yu fl r cfrtbl, yu c igi w v gt Tylr sris hr j j d, but w w t us tht fct. j j! Th fct w will us is this th ld filir: But if w rphrs this i trs f ur prturbti fctr,, d ll th crrcti trs,
3 .. Multiplyig this ut, w gt 4 Cllctig trs by pwrs i, Nw, this is plyil i d it ust su up t rgrdlss f th vlu f. S, wht ust b tru f ch idividul cfficit f? ch ust b idividully qul t. 6.. First-Ordr Thry S, th th -rdr quti is th w lrdy kw: if thr wr prturbti, w d hv th rigil ilti which is slvd by th rigil wvfuctis with thir rigil rgis. Th st -rdr quti thugh, is ws. Lt s s wht w c gt fr it. st -rdr rgy Crrcti. If w ir-prduct it with th kw, w hv Of curs, th rgis r just ubrs, s w hv But, d th rigil ilti is st ssurdly riti, s w hv
4 4 Ad w kw tht th rigil ilti prtig th rigil rgy igstt rturs, s Nw w c ccl th first tr fr lft d right t b lft with il! W w kw th first-rdr crrcti t th rgy. Why Prturbti thry? Nw yu y wll sy but wit, is this rsult f y us? Aftr ll, w psitd tht Ad w r rlly itrstd i th cs wh th prturbti is cpltly turd, s =, d tht s ll crrcti trs r qully iprtt! Tht culd idd b th cs; d i tht cs, prturbti thry wuld t b th wy t tckl th prbl. Th gi, if it s t just tht ws c-up--ti sll wh w first strtd turig th prturbti t th ilti, but it s tht th prturbti is itslf sll, th w c hp tht th st -rdr tr will b biggr th th d -rdr which will b biggr th Th Prturbti pprch is usful. rcis: This wrks just s wll whthr w r tlkig but ilti prtr prssd i diffrtils d wvfuctis s fuctis r th ilti s tri d th wvfuctis s vctrs. First ff, wht r th rigil ilti s igvctrs? d Th wht s th st -rdr crrcti t th rgy fr ch? S th rgy f th first stt is, t st -rdr crrcti,
5 5 "Culd w g vr th scd prt f pl 6." Atwi Th fllwig rcis is lik th scd prt f pl 6., but with littl r t. rcis: A littl r cplictd, Lt s g bck t tht ifiit squr wll d tckl th s scri w d csidrd lst ti with th WKB: lctr i -D slid with vltg pplid, i.., slpd btt. d d d d d cs si si si si si si S, th rgy fr ch stt, t first rdr crrcti, is. Strtig Wkly W: Griffiths 6. fr prt yu d this, but with dlt-pttil i th iddl f th wll. st -rdr Wvfucti Alright, s w v gtt littl prctic fidig th first-rdr crrcti t th rgy. Wht but th wvfucti, hw is tht ssd up bcus f prturbti? "I' cfusd why griffiths uss quti 6. i th drivti?" Jssic I'd ls lik t kw hw Griffiths uss 6. fr th rst f th drivti.spcr Ditt Jth Ms. I fllw th tti d such, but I' still cfusd. Gigj Nw, th trick is tht ur ld slutis frd cplt bsis st fr this spc, s th crrcti shuld b prssibl i trs f tht bsis st. wvr, cvt sic this is suppsd t b crrcti tr i, tht s crrcti tp f th th rdr tr,, ur su will stp vr this br f th bsis st., c Of curs, s usul, c,
6 6 S, w ll us th st -rdr quti t gt rlti fr th st -rdr wvfucti which w c th ir-prduct t gt th cfficits. It gs lik this. Th st -rdr quti is Rwritig with th thigs w kw th right-hd sid Ir prductig tht with, Ad tht is c, S t: t s usful if tw rigil stts hv th s rgy s th stt yu r itrstd i. Wdsdy s rdig is but hdlig tht kid f situti.. Ccptul: C w us quti 6. t dtri wvfuctis fr th liu t bsd prturbti t ydrg t? Why r why t? "I bit cfusd but th scd hwrk qusti, ily with wht ctly "prturbti t hydrg t" ctully tils." S M Wll, s first stp, yu d wt t k th sipl chg tht Z=, d w d d t grliz th wvfucti t b tw-lctr wvfucti. Tht s ll sy ugh d ws d i Chptr 4 but th th qusti is, c w trt th lctr-lctr itrcti trs s prturbti? W culd g ftr th grud stt sic, i th uprturbd cs, this is t dgrt. wvr, ftr tht, thr wuld b dgrcis i th thr stts. Fr pl, th =, l = d th =, l = hv th s rgis i th u-prturbd pttil, s yu culd t g ftr crrctis fr thr withut th su blwig up wh th thr ws ddd t th su.
7 7 Athr issu is tht th prturbti f th lctr-lctr itrcti is t s sll; if I rbr, it s yb /5 th f th lctr-prt itrcti, s st rdr y t b ugh t stisfctrily ccdt fr ddig i this prturbti. Lt s giv it spi. rcis: Rturig t ur tri ilti, wht s th first-rdr crrcti t th first igvctr? First ff, w hv with with d with S th tw ir prducts r S, th su is Th, t first rdr,
8 8 rcis: Or, fr th r cplictd, diffrtil-ilti cs, with th slpd btt i squr wll,, si, d S, d d si si si si Whr cs cs si si cs cs si cs si cs cs cs d d Fr v +, th csis r which kills th whl thig; fr dd +, thy r -, s 8 fr + dd; fr + v. Th dd dd 4 8 si 6 si Nt prtty, but thr it is.. Strtig Wkly W: Griffiths 6. fr prt b yu d this, but with dlt-pttil i th iddl f th wll.
9 9 6.. Scd-Ordr Thry Nw, i prctic, if usig pwr sris psi is gd id t ll, it s bcus th highr rdr trs r gligibl, s yu ly d fw trs; i.., i this ctt, if yu r dlig with r prturbti thus th. S, w ll g ftr th d -rdr prssi fr th rgy, but lv ff thr. Lkig t th d -rdr quti tht w d rigilly grtd, st -rdr rgy Crrcti. Oc w v gt th st rdr rgy crrcti d th st -rdr wv crrcti, th w kw st vrythig i this prssi. Tkig th ir prduct f this with will islt th rgy crrcti: Th lft st trs bth sids r qul d s ccl But w lrdy dcidd tht th br f ur cplt bsis st f rigil slutis tht ds t g it buildig th highr rdr crrctis fr is. Frlly, w culd prss, c. S. W c lv ff thr, sic w v lrdy fud, th gi, subbig tht i, w gt
10 . Strtig Wkly W: Griffiths 6.4. d -rdr rgy crrctis fr th dlt-spik pttil d fr th prturbd hric scilltr chgd k Nt: yu r t pctd t prv Griffiths ct prt. Fr prt b it: Yu shuld t hv t ctully d y itgrls if yu rcll tht d qutis.66 tll yu th rsult f th risig d lwrig prtrs prtig wvfucti. I fct, yu shuld fid tht, i s ld, chptr hwrk yu v lrdy slvd st f th ky itgrls.
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