N LTERNTIE TRETMENT FOR YUKTYPE POTENTILS B. Göü K. Kösa ad E. Baı Depatet of Egieeig Pysics Uivesity of Gaziatep 7 GaziatepTuey bstact e popose a ew appoxiatio scee to obtai aaytic expessios fo te boud state eegies ad eigefuctios of Yuawa ie potetias. Te pedicted eegies ae i exceet ageeet wit te accuate ueica vaues epoted i te iteatue. Keywods: Yuawa potetia Petubatio teoy PCS No:.65.Ge. INTRODUCTION Sice te pioeeig wo of Yuawa [] te potetia exp as bee extesivey ivestigated i te iteatue. Tis is due to te specia oe of tis potetia i diffeet baces of pysics. I pasa pysics it is ow as te DebyeHüce potetia i soidstate pysics ad atoic pysics it is caed te ToasFei o sceeed Coub potetia. so tis potetia is we ow i ucea pysics as te doiat ceta pat of uceouceo iteactio aisig out of te oepioexcage ecais. Tus te paaetes ad ae give by diffeet expessios depedig o te pobe ude cosideatio. I a tese cases a owedge of te vaious boud state eegies is essetia fo udestadig ad coeatig te popeties of diffeet systes. Sice te Scödige equatio fo suc potetia does ot adit exact aaytic soutios vaious ueica ad appoxiate aaytica etods e.g. [6] ave bee deveoped i te past. Moe ecety a ew etodoogy [7] as bee itoduced based o te decopose of te adia Scödige equatio i two pieces avig a exacty sovabe pat wit a additioa piece eadig to eite a cosed aaytica soutio o a appoxiate teatet depedig o
te atue of te additioa/petubed potetia. Te appicatios [8] of tis ove teatet to diffeet pobes i bot boud ad cotiuu egios ave bee pove te success of te foais. it te cofidece gaied fo tese appicatios ad beaig i id te sigificace of a eiabe agebaic soutio fo Yuawatype potetias tat is ceay epoted i [6] we deostate ee ow suc iteactio potetias ca be sipy teated witi te faewo of te peset foais. Te outie of te pape is as foows. I sectio we setc te ai steps of te pocedue. e peset oy tose fouae tat ae ecessay fo udestadig te foowig cacuatios; a detais ad te ateatica foudatios ca be foud i [78]. I sectio we appy te appoac to te Scödige equatio wit te Yuawa potetia ad peset te esuts obtaied aayticay wic aes cea tat te peset scee gives quite good accuacy fo eegy vaues despite its aaytica atue. Fia sectio ivoves soe cocudig eas.. FORMLISM Fo te cosideatio of speicay syetic potetias te coespodig Scödige equatio i te boud state doai fo te adia wave fuctio as te fo ψ ψ [ E ] wee is a petubig potetia. Let us wite te wave fuctio ψ as ψ i wic weeas is te ow oaized eigefuctio of te upetubed Scödige equatio is a odeatig fuctio coespodig to te petubig potetia. Substitutig ito yieds E. Istead of settig te fuctios wic eads to ad we wi set tei ogaitic deivatives 5
6 wee is te eigevaue of te upetubed ad exacty sovabe potetia ad 7 i wic is te eigevaue fo te petubed potetia ad. E If Eq. 7 wic is te ost sigificat piece of te peset foais ca be soved aayticay avig a cosed fo as i 6 te woe pobe Eq. sipy educes to E 8 wic is a we ow teatet witi te fae of supesyetic quatu teoy SSQT [9]. Obviousy tis faous pesciptio of te eated iteatue appeas atuay as a subset of te peset scee. I additio it is stessed tat Eq. 7 povides a uc bette faewo i teatig suc pobes ta te oe i te supesyetic quatu ecaics. Because i SSQT oe soud stat fo coespodig to te goud state wavefuctio ad use te iea opeatos ± to obtai wavefuctios/supepotetias of te excited states. Howeve witi te fae of peset scee oe ows expicity te soutio of Eq. 6 aey te woe spectu ad coespodig eigefuctios of te upetubed iteactio potetia so tat oe ca easiy cacuate te equied fo ay state of iteest eadig to diect coputio of eated coectios to te upetubed eegy ad wavefuctio. Fo te petubatio tecique peseted i tis ette we ave iitiay assued tat we coud spit te give potetia i two pats Eq.. Te ai pat coespods to a sape ivaiat potetia Eq. 6 fo wic te supepotetia is ow aayticay ad te eaiig pat is teated as a petubatio Eq. 7. Teefoe it is obvios tat Yuawa ie potetias ca be teated usig ou pesciptio. I tis case te zeotode te coespods to te Couob potetia wie igeode tes cositute te petubatio. Howeve te petubatio te i its peset fo caot be soved exacty toug Eq. 7. Tus oe soud expad te fuctios eated to te petubatio i tes of te petubatio paaete ; ; 9
wee deotes te petubatio ode. Substitutio of te above expasio ito Eq. 7 by equatig tes wit te sae powe of o bot sides yieds up to O. Eq. 7 ad its expasio Eqs. give a fexibiity fo te easy cacuatios of te petubative coectios to eegy ad wave fuctios fo te t state of iteest toug a appopiatey cose petubed supepotetia uie te ote petubatio teoies. e wi sow i te ext sectio toug te appicatios tat tis featue of te peset ode eads to a sipe faewo i obtaiig te coectios to a states witout usig copicated ad tedious ateatica pocedues.. PPLICTION e ow appy tis etod to a Yuawatype potetia wit te agua oetu baie exp wee te fist piece is te zeotode ad sape ivaiat exacty sovabe piece coespodig to te upetubed potetia wit wie is te petubatio te 6 L toug te expasio of te expoetia te. Ou caefu cacuatios ave caified tat te ai cotibutios coe fo te fist tee tes. Hece te peset cacuatios ae pefoed up to te secodode ivovig oy tese additioa potetia tes wic supisigy povide igy accuate esuts... Goud state cacuatios I te igt of Eq. 6 te zeotode cacuatios eadig to exact soutios ca be caied out eadiy wit te coice of a suitabe yiedig te Couob potetia K
L exp!! i wic Γ Γ L!! is a associate Laguae poyoia. Tese aaytica soutios ae aeady exist i te iteatue povidig a supeioity to te peset cacuatios. Fo te cacuatio of coectios to te zeotode eegy ad wavefuctio oe eeds to coside te expessios eadig to te fist ad secodode petubatio give by Eqs.. Mutipicatio of eac te by i tese equatios ad eepig i id te eatio i Eq. 5 oe ca obtai geea expessios fo te coectios i te fist dz z z d 5 ad secodode d 6 dz z z z 6 6 fo ay state of iteest. ccodig to tese fouas we ca cacuate ad expicity oy we we ow wat te eegy coectios ad ae fo wic te woe of te petubed wavefuctio ca be cacuated i a cosed fo by Eq. 5. It is aso oted tat te owe iit of te itegatio fo eegy cacuatios soud be caged fo to to accoodate te fact tat is aways positive. Tus te goud state cacuatios witi te fae of Eqs. 5 ad 6 give 6 6 [ ] [ ] { } 7 5
Te aaytica expessios fo te owest eegy ad adia wavefuctio of a Yuawatype potetia ae te give by E ψ exp 8 Tese expicit expessios suppot te siia wos i [6]. Tabe sows ueica vaues of te petubed eegies fo a few vaues of ad. Te esuts obtaied ae copaed wit tose of [5] togete wit te esuts of Roges ad is cowoes [] wo soved te Scödige equatio ueicay. Ou esuts ae i eaaby good ageeet. Tabe ad iustate aote copaiso of ou cacuatios wit tose of [6] wo caied out tei cacuatios i a diffeet uit. Tese two diffeet copaiso ae cea te sesitivity of peset cacuatios atoug te pocedues i [6] to epoduce te coectios aayticay see siia to ous. I paticua Tabe caifies tat te peset etod is a vey usefu oe fo age potetia paaetes fo wic ueica soutio of te Scödige equatio is exteey difficut. Because fo a age stegt te Yuawa potetia is vey deep ad te wavefuctio becoes vey sapy peaed ea te oigi. Tis causes a geat dea of difficuty i te ueica soutio of te Scödige equatio wic is efected i te istabiity of te wavefuctio tus obtaied atoug te eegy eigevaue is faiy stabe ad accuate... Excited state cacuatios Te pocedue eads to a ady ecusio eatios i te case of goud states but becoes exteey cubesoe i te desciptio of adia excitatios we odes of wavefuctios ae tae ito accout i paticua duig te ige ode cacuatios. toug sevea attepts ave bee ade to by pass tis difficuty ad ipove cacuatios i deaig wit excited states e.g. [5] witi te fae of supesyetic quatu ecaics ad te eated efeeces teei egadig te Logaitic Petubatio Teoy tey ave ot esuted i a desiabe sipe agoit. Hece as a aote objective of tis pape ee a expicit teatet is itoduced witi te fae of te peset foais ad descibed a staigtfowad pocedue fo obtaiig te petubatio coectios toug ady ecusio fouae avig te sae fo bot fo goud ad excited states. dz 6
Usig Eqs. 5 ad te fuctio eated to te excited states ca be cacuated expicity fo te coputio of petubatios expessed by 5 ad 6. So te fistode coectios i te fist excited state ae [ ] [ ] 5. 9 Howeve igeode cacuatios ave siguaity pobes duig te itegatios because of te odes appeaig i te above equatio ad wit soe cofidece suggest tat. To ovecoe tis pobe we focus o a idde eatio i [ ] [ ] 5 wic tasfos Eq. 9 ito. Use of te appoxiated i 6 gives te eegy coectio i te secodode as 6 7 7 6. Teefoe te appoxiate eegy vaue of te Yuawa potetia coespodig to te fist excited state is E. Te eated adia wavefuctio ca be expessed i a aaytica fo i te igt of Eqs 5 6 ad 8 if equied. Oe ca easiy cec te siiaities as i betwee te tes of ay of iteest to by pass te oda difficuty as i te fist excited state. Ou caefu ad exausted ivestigatios ave eveaed tat te atio betwee tese siia tes i fo ay state is appoxiatey wic eas tat te appoatio used ee woud ot affect cosideaby te sesitivity of te cacuatios. Futeoe tese ivestigatios put fowad a iteestig ieaacy betwee tes of diffeet quatu states i te fist ode cicuvetig te oda difficuties eegaty 7
wic fo istace fo te secod excited state 7 7 eads to 6 6. Cacuatios fo ige excited states ca be caied out i te sae ae witout epoyig tedios itegas esuts of wic ae faiy i good ageeet wit te accuate ueica itegatio esuts see Tabes ad we copaed to te ote teoetica wos. Fiay toug te copaiso of ou esuts wit tose of [] fo age vaues yieds ideed exceet esuts we do ot iusate tese tabes ee fo caity wic ay be epoduced easiy witi te scee descibed i tis sectio. ad. CONCLUDING REMRKS I cocusio a ew usefu tecique fo sovig te boud state pobe fo Yuawatype potetias witi te fae of Riccati equatio ave bee obtaied ad te copaiso of cacuatio esuts wit te accuate ueica vaues as bee pove te success of te foais. voidig te disadvatages of te stadad oeativistic petubatio teoies te peset fouae ave te sae sipe fo bot fo goud ad excited states ad povide i picipe te cacuatio of te petubatio coectios up to a abitay ode i aaytica o ueica fo. dditioay te appicatio of te peset tecique to Yuawa potetia is eay of geat iteest eadig to aaytic expessios fo bot eegy vaues ad wavefuctios. Of paticua ipotace is te appeace of goud state eegy i a sipe fo. Copaig vaious eegy eves wit diffeet wos i te iteatue we fee tat ou aaytic teatet quite eiabe ad fute aaytic cacuatio wit tis opetubative scee woud be usefu. I paticua ou etod becoes oe eiabe as te potetia stegt iceases wie te ueica soutio of te Scödige equatio gets ustabe ad ueiabe i cacuatig especiay te wavefuctio. Tus te peset etod icey copeets te existig ueica etods. 8
Fiay it is oted tat te peset esuts ca be exteded easiy to N diesio wit te cosideatio of te wo i [] by te epaceet of te agua oetu te wit Λ M wee M N. REFERENCES [] Yuawa H 95 Poc. Pys. Mat. Soc. Jp. 7 8; 97 9 8 [] Roges F J Gabose H C J ad Hawood D J 97 Pys. Rev. 577 [] La C S ad asi Y P 97 Pys. Rev. 875; McEa J Kisse L ad Patt R H 976 Pys. Rev. 5 ; Pati S H 98 J. Pys. : Mat. Ge. 7 575; Yuaov I Yuaova E P ad Oiveia F 998 J. Pys. : Mat. Ge. 7; ad te efeeces teei. [] Cattejee 986 J. Pys. : Mat. Ge. 9 77 [5] Lee C Pys. Lett. 67 [6] Caabati B ad Das T K Pys Lett. 85 [7] Göü B Ci. Pys. Lett. 685 [8] Öze O ad Göü B Mod. Pys. Lett. 8 58; Göü B Ci. Pys. Lett. ; Göü B ad Koça M 5 Mod. Pys. Lett. 55; Göü B Çei N ad Oğa E 5 Mod. Pys. Lett. 68; Göü B ad Koça M to appea i Mod. Pys. Lett. quatp/555; Göü B ad Koça M subitted to Pys. Lett. quatp/6 [9] Coope F Kae ad Suate U P 995 Pys. Rep. 5 67 [] Gou B Oze O Koca M Tutcu D ad Cacei Y J. Pys. : Mat. Ge. 87 9
TBLES Tabe. Eegy eigevaues of te Yuawa potetia i uits of. Fo copaiso we set ad g. State g Peset Cacuatios Ref. [] Exact Ref. [] Ref. [5] s..5...5.5.996.99.985.9659.959.96.996.99.98.966.959.96.996.99.985.9659.959.96.996.98.966.96 s..6.6.6.6.5.5..5..59.6.58.6....9..5.55.6.5.5.65.65.65.65 p..6.6.6.6.5.....9.5.9.5..9.9.9..5.99..99.5.6.65.655.65 p..76.7.76.7.5.....9.9.96.96..757.757.756.76.5.68.686.6799.5.79.7.86.6 d..75.7.75.7.5...7..9.9.9.96..75.75.75.75.5.67.675.678.5.88.8.77.76
Tabe. Te sae as i Tabe but. f ad. Peset Cacuatios Ref. [6] Nueica Ref. [6] aytica.56.565.99 8.58.58.57.586.99. 6 6.859.998 6.859.99 6.89.875 9.59.97.595 9.59.98.5959 9..85.56 Tabe. Te sae as i Tabe but f Peset Cacuatios Ref. [6] Nueica Ref. [6] aytica 6.7.97.6.7.7.8.6.57.886..699 5.8.9.665 5..98.95.7..6998 5...665 5.5.77.956..56.79 5.59.7.689 5.5. 5.85.8