Laboratory of Physics and Material Chemistry, Physics Department, Sciences Faculty, University of M'sila-M sila Algeria * a

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1 Intnatonal Font Scnc Ltts Submttd: 7--5 ISSN: , Vol., pp 9-44 Accptd: 7--4 do:.85/ Onln: ScPss Ltd., Swtzl Invstgatons on th Rlatvstc Intactons n On-lcton Atoms wth Modfd Yuawa Potntal fo Spn / Patcls Abdlmadd Mach, a * Laboatoy of Physcs Matal Chmsty, Physcs Dpatmnt, Scncs Faculty, Unvsty of M'sla-M sla Alga * a abmach@gmal.com Kywods: Yuawa potntal, noncommutatv spac, sta poduct, Bopp s shft mthod Dac quaton. Abstact. ngy lvls of on lcton atoms hav bn -xamnd by applyng an altnatv ptubatv schm n solvng th modfd Dac quaton (m.d..) fo th modfd Yuawa potntal modl wth an abtay spn-obt quantum numb l, l, l, l by mans Bopp s shft mthod nstad to solvng (m.d..) wth sta poduct, n th famwo of noncommutatvty th dmnsonal al spac (NC: D-RS). It s obsvd that th obt coctons of ngs a dpndd on th nw dsct atomc quantum numbs, m und spn-symmty psudospn symmty two nfntsmal paamts,, whch nducd by poston-poston noncommutatvty. Futhmo, n lmt of paamts,,, th nw ngy quatons fo modfd Yuawa potntal a consstnt wth th sults of odnay latvstc quantum mchancs fo odnay Yuawa potntal.. Intoducton In vy cnt yas th ngy lvls of atoms, nucl, hadons hav bn studd by sval analytc mthods n th cas th two latvstc quatons Kln-Godon Dac fo ctan potntals of th physcal ntst [-], som of ths potntals a nown to play mpotant ols n many flds, on of such potntal s th Yuawa potntal has bn a subct of many studs, t s a cntal potntal of nucla shll modl n addton to usd n nonlatvstc quantum mchancs to study th ntactons btwn nonlatvstc patcls, n atomc molcula physcs, t psnts a scnd Coulomb potntal du to th cloud of lctonc chags aound th nuclus, on th oth h, t has also bn usd to dscb th ntacton btwn chagd patcls n plasma, sold collodal suspnsons []. It s wll nown that th odnay quantum stuctus oby th stad Wyl-Hsnbg algba n both Schödng Hsnbg (th opatos a dpndd on tm) pctus ((HP) (SP)), spctvly, as (Thoughout ths pap th natual unt c a mployd): x, p x t, p t x, x p, p x t, x t p t, p t wh th opatos x t, p t n (HP)a latd to th cospondng opatos x, p n (SP) fom th followng poctons latons: h com x p x t, p t xp( H t t ) x, p xp( H t ), com com t () () H dnot to th odnay quantum Hamltonan opato fo Yuawa potntal. In addto fo spn ½ patcls dscbd by th Dac quato xpmnt tlls us that must satsfy Fm Dac statstcs oby th stcton of Paul, whch mply to gvs th only non-null qualtm ant-commutato fo fld opatos as follows: ScPss appls th CC-BY 4. lcns to wos w publsh:

2 Volum t,, t, ' -' () Vy cntly, many authos hav wod on solvng th fundamntal quatons (Schödng, Kln-Godon Dac quatons) wth physcal potntal n th nw stuctu of quantum mchancsnown by NC quantum mchancs whch nown fstly by H. Snyd, t s shown a dct latonshp btwn stng thoy noncommutatv gomty to obtanng pofound nw applcatons fo dffnt aas of matt scncs n th mcoscopc nano scals wthn th famwo of quantum mchancs quantum fld thoy [-7]. It s mpotant to notcng that, th nw quantum stuctu of NC spac basd on th followng NC canoncal commutatons latons (NCCRs) n both (HP) (SP)), as follows [4-6]: x, p x (4) t, p t, x, x x t, x t p, p p t, p t th nw opatos x t, p t n (HP) a latd to th cospondng nw opatos x, p n (SP) fom th nw poctons latons [7-]: wth Hnc x, p x t, p t xp( H t t )* x, p *xp( H t ) nc nc t (5) bng th Hamltonan opato of th xtndd quantum systm dscbd on (NC: D- RS) symmts. Th vy small paamts (compad to th ngy) a lmnts of antsymmtc al matx of dmnson lngth dnot to th nw sta poduct (th Moyal- Wyl poduct), whch s gnalzd btwn two abtay functons f x f x gx g x to f x g x f gx nstad of th usual poduct fg x n odnay th dmnsonal spacs [-7]: x x fg x, p f x g x f gx ( fg f g O x x (6) wh ) f x g x x x a th nw functon n (NC: D-RS), th followng tm ( f x gx s nducd by (spac-spac) noncommutatvty popts O sts fo th scond hgh od tms of, a Bopp s shft mthod can b usd, nstad of solvng any quantum systms by usng dctly sta poduct pocdu [-7]: x, x x t, x t p, p p t, p t (7) Th th-gnalzd coodnats x x, y x, z x n th NC spac a dpndd wth cospondng th-usual gnalzd postons x, y, z th momntums coodnats p, p, p by th followng latons, as follows [8-4]: x y z x x x x p y p z, p y y p, x p z (8) z z p x p y.

3 Intnatonal Font Scnc Ltts Vol. Th non-vansh-commutatos n (NC-D: RS) can b dtmnd as follows: x, p y, p z, p x y, x, y, x, z, y, z z (9) whch allow us to gttng th opato on NC th dmnsonal spacs as follows [8-4]: L () th couplng L Lx Ly Lz wth /, Lx z zp y, L y zp x - xpz Lz xp y x. Futhmo, th nw qual-tm ant-commutato fo fmonc fld opatos n NC spacs can b xpssd as [8-4]: * t, t, ' - ', * t,, t, ' t,, t, ' - ' * () Th pupos of th psnt wo s to xtnd psnt th soluton of th Dac quaton wth spn-/ patcl movng n modfd Yuawa potntal of th nw fom: L L fo th spn symmtc cas Vmy () L L fo thp - spn symmtc cas In (NC: D-RS) usng th gnalzaton Bopp s shft mthod to dscov th nw symmts a possblty to obtan anoth applcatons to ths potntal n dffnt flds. Ths wo basd ssntally on ou pvously wos [-5]. Th plan of cntly wo s as follows: In nxt scto w bfly vw th Dac quaton wth Yuawa potntal on basd to fs. [8-]. In scton th, w gv a dscpton of th Bopp s shft mthod fo (m.d..) wth modfd Yuawa potntal. Thn n scton fou, w apply stad ptubaton thoy to stablsh xact modfcatons at fst od of nfntsmal paamts, fo th ptubd Dac quaton n (NC-D: RS) fo spn-obtal (psudo-spn obtal) th latvstc magntc spctum fo modfd Yuawa potntal. Scton ffth s dvotd fo dscusson th global spctum cospondng NC Hamltonan fo modfd Yuawa potntal. Fnally, w gv a bf concluson n last scton.. Rvw of th Dac quaton fo Yuawa potntal n latvstc quantum mchancs Lt us stat ths scton by th vw of a latvstc patcl n sphcally symmtc fo th potntal V, whch nown by Yuawa potntal on basd to th man fnc []: V () wh dnot to th ang of nucla foc stngth of potntal, spctvly. Th Dac quaton dscbng a fmonc patcl (spn-/ patcl) wth scala S (, ) vcto V, potntals s gvn by [8-]: P ( M S(, )),, V,,, (4)

4 Volum h M a th fmons mass th latvstc ngy whl (, I ) I a th usual Dac matcs, th spno,, can b xpssd as [8-]:, n f, n gn l Y, Fn m, (5) G n l Y, m wh a s latd to th total angula momntum quantum numbs fo spn symmty l p-spn symmty l as [8-]: th Paul matcs whl l f - /, s/, p /, tc, l, algnd spn (6) l f l, p,,,, unalgnd spn / d / tc l l f - /, s/, p /, tc, l, algnd spn (7) l f l, p,,,, unalgnd spn / d / tc l Th adal functons ( F n, G n ) a obt by solvng th followng dffntal quatons [8- ]: d d d d d d ( ) d d M n M n Fn M (8) n d d ( ) d d M n M n G M n (9) n Th xact spn symmty cospondng followng l Schödng quaton []: d d, thus th adal functon F n satsfyng th Th latvstc ngy n d d ( ) M M Fn (), adal upp wav F n a gvn by []: n Av s s s v () / L / 4 n C v C v 4 M C M M C s

5 Intnatonal Font Scnc Ltts Vol. F L / 4 L / 4 Nn! n n Ln, xp, () n wh L n sts fo th assocatd Lagu functons L ( ) d symmty whch cosponds, th latvstc ngy adal low wav a gvn by []: n C p C d p 4 M C p n. Fo, th xact psudospn / L / 4 n G n A pv M A v () p p G L / 4 L N / 4 n n! n Ln, xp, (4). NC latvstc Hamltonan fo modfd Yuawa potntal:. Fomalsm of Bopp s shft Mthod In ths scton I fst hghlght n bf th bascs of th concpts of th quantum noncommutatv quantum mchancs n th famwo of latvstc Dac quaton fo modfd V Yuawa potntal ao on basd to ou wos [8-4]: a) Odnay Dac Hamltonan opato H p, x plac by NC Dac Hamltonan opato H, p x, nc b) Odnay spno plac by nw spno, c) Odnay latvstc ngy n placs by nw latvstc ngy nc odnay poduct plac by nw sta poduct. Thus, th Dac quaton n odnay quantum mchancs wll chang nto th modfd Dac quaton n xtndd quantum mchancs fo modfd Yuawa potntal as follows: H p, x (5) nc nc Th Bopp s shft mthod pmuts to duc th abov NC quaton to smplst fom wth usual poduct tanslatons appld to n spac phas opatos: H p, x (6) nc nc Wh th nw Hamltonan opato H nc p, x can b xpssd n th gnal vats: both NC spac NC phas (NC-D: RSP), only NC spac (NC-D: RS) only NC phas (NC: D-RP) as, spctvly: H nc (7) p, x H p p x ; x x p fo NC -D:RSP

6 4 Volum H nc p, x H p p ; x x p fo NC -D:RS (8) H nc p, x H p p ; x, x x fo NC -D:RP (9) In cntly wo, w a ntst wth th scond vaty whch psnt by q. (8) by th mans of th auxlay two vaabls x p, th nw modfd Hamltonan H nc p, x may b wttn as follows wh th modfd Yuawa potntal H nc V p, x P ( M S()) V s gvn by: V Th Dac quaton n th psnc of abov ntacton mthod as follows: () () V P ( M S()),, V can b wttn accodng Bopp shft,, () Th adal functons ( F n, ) a obt by solvng two quatons: G n d d d d nc M G Fn nc n () M G Gn nc n (4) wth V S V S, lmnatng F n G n fom qs. () (4), w can obtan th followng two Schödng-l dffntal quatons n (NC-D: RS) symmts as follows: d d ( ) M M F nc nc n (5) d d ( ) M M Aft staghtfowad calculatons on can obtans th tms follows: xp G nc nc n (6) n (NC-D: RS) as L fo th spn symmtc cas (7) L fo thp - spn symmtc cas L L fo fo th spn symmtc cas thp - spn symmtc cas (8)

7 Intnatonal Font Scnc Ltts Vol. 5 whch allow us to wtng th modfd Yuawa potntal V V n (NC-D: RS) as follows: L V p,,, L fo th spn symmtc cas (9) L V p,,, L fo thp - spn symmtc cas It s claly that, th fst tm psnt th odnay Yuawa potntal whl th st pats V p,,, V p,,, a poducd by th dfomaton of spac, ths allows wtng th modfd Yuawa potntal n th NC cas as an quaton smlaly to th usual Dac quaton of th commutatv t wth a non local potntal. Futhmo, usng th unt stp functon (also nown as th Havsd stp functon o smply th thta functon) w can wt th modfd Yuawa potntal to th followng fom: wh V nc V p,,, nc V p,,, (4) fo x x (4) fo x W gnalzd th constant fo th psudospn (p-spn) symmty V C ps constants whch psntd n fs. [8-] nto th nw fom V C constants n (NC-D: RS) nstng th potntal n q. (9) nto th two ps Schödng-l dffntal quatons (5) (6), on obtans: d d d d ( ) ( ) nc ps nc ps n (4) M M C M C M C L F nc nc ps M M C M C M C L G nc nc ps V nc ps nc ps n (4) It s claly that, th addtv two pats V p,,, V p,,, a popotonal wth nfntsmal paamt, thus w can consdd as a ptubatons tms. 4. Th xact latvstc spn-obtal Hamltonan th cospondng spctum fo th modfd Yuawa potntal n (NC: D- RS) symmts fo n xctd stats fo on-lcton atoms: 4. Th xact latvstc spn-obtal Hamltonan fo modfd Yuawa potntal n (NC: D- RS) symmts fo on-lcton atoms: In ths pat of th pap, aga th two ptubatv tms V p,,, V p,,, can b wttn to th quvalnt physcal fom fo modfd Yuawa potntal as follows: V V p p,,,,,, LS LS fo fo th spn symmtc thp - spn symmtc cas cas (44)

8 6 Volum Futhmo, th abov ptubatv tms V p,,, V p,,, can b wttn to th followng nw quvalnt fom fo (m.a.o.) potntal: V V p p,,,,,, J J L L S S fo fo th spn symmtc thp - spn symmtc To th bst of ou nowldg, w ust plac th two spn-obtal couplng S L L S by th xpsson J L S J L S, n latvstc quantum mchancs. Th st ( H nc p, x, J, L, S J z ) foms a complt of consvd physcs quantts th spn obt quantum numb ( ) s latd to th quantum numbs fo spn symmty l p-spn symmty l as follows [8-]: Wth l l, H so H l f - /, s/, p /, tc, l, algnd spn (46) l f l, p/, d /, tc, l, unalgnd spn l f - /, s/, p/, tc, l, algnd spn (47) l f l, p/, d /, tc, l, unalgnd spn ll, whch allows us to fom two dagonal matxs so, n (NC: D-RS) as: cas cas (45) H f - /, s, p, tc, l, algnd spn so H so f l, p/, d /, tc, l, unalgnd spn H so H f - /, s, p, tc, l, algnd spn so H so f l, p/, d /, tc, l, unalgnd spn H so / / / / (48) (49)

9 Intnatonal Font Scnc Ltts Vol Th xact latvstc spn-obtal spctum fo modfd Yuawa potntal symmts fo th n xctd stats fo on-lcton atoms n (NC: D- RSP) symmts: nc p Ou goal n ths sub scton s to fnd th modfcatons to th ngy lvls : u,, n,, nc p: d, n,, fo ( /, s, p, tc, l, algnd spn - / / spn-up) ( l, p, d, tc, l, unalgnd spn spn down), spctvly, at fst / / th od of nfntsmal paamt, fo xctd n stat, obt by applyng th stad ptubaton thoy, usng qs. (), (44), (45), (46) (45) as:,,,, V,,, V,,, dd n nc p nc ao p n * * F V nc ao n p,,, Fn d ncao G V p G n,,, n Th fst pats psnt th modfcatons to th ngy lvls fo th spn symmtc cass nc p: u, n,, nc p: d, n,, whl th scond pat psnt th modfcatons to,,,,,, ) fo th spn spn-symmty, thn th ngy lvls ( nc nc p: d n, w hav xplctly: nc p: u nc p: d Instng th adal functon obtan: nc p: u nc p: d,,,, n,, p: u n,,,,, F n F n d (5) * nc n d (5) * nc n d (5),,,, F n F n F n gvn by q. () nto th abov two qs. (5) (5) to L / 4 4 n! N xp 4 L n,, nc L / 4 4 n! N xp 4 L nc Now, w ntoduc th followng nw facto T T n,, L as follows: n, n L / 4,, L xp L n 4 w apply th followng spcal ntgaton [7]:. t. n n d d (5) (54) d n n (55) n m! xp tlm t Ln t dt F, ; n, ; (56) m! n wh F, ; n, ; obt fom th gnalzd th hgomtc functon p Fq,..., p,,..., q, z fo p q whl x dnot to th usual Gamma functon. Aft staghtfowad calculatons, w can obtan th xplctly sults fo th facto T, L n,, :

10 8 Volum T n,, L 4 F n n! L / 4 n L / 4 4n 4 L / 4 L / 4, L / 4 ; n L / 4 ; whch allow us to obtan th modfcatons to th ngy lvls fo th n xctd stats:,,, n! N T,, L nc p: u, n nc n (57) (58),,, n! N T,, L (59) nc p: d, n nc n 4.. Th xact latvstc magntc spctum fo modfd Yuawa potntal fo th n xctd stats fo on-lcton atoms n (NC: D- RS) symmts: Havng obt th xact modfcatons to th latvstc ngy lvls nc p: u, n,,,, n xctd stats whch poducd wth latvstc NC spn-obtal nc p: d n, fo th Hamltonan opato, ou obctv now, n ths scto w consd anoth ntstd nw physcally manngful phnomna, whch also can b poduc fom th ptubatv tms of Yuawa potntal latd to th nflunc of an xtnal unfom magntc fld, t s suffcnt to apply th followng th placmnts to dscbng ths phnomna: B L L BL BL fo fo th spn symmtc thp - spn symmtc h s nfntsmal al popotonal s constants, w choos th magntc fld smplfy th calculatons, whch allow us to ntoduc th modfd nw magntc Hamltonan H,,,, on th (NC: D-RS), as: mag wh H n, mag,,, n cas cas (6) B B fo BJ S B fo th spn symmtc cas (6) BJ S Bfo thp - spn symmtc cas S B, S B dnots to th two odnay psudo Hamltonans of Zman ffct. To obtan th xact NC magntc modfcatons of ngy,, fo modfd Yuawa mag - n, potntal und spn-symmty cas whch poducd automatcally fom th ffct of opato H,,,,, w ma th followng two smultanously placmnts: mag n, m (6)

11 Intnatonal Font Scnc Ltts Vol. 9 Th th latvstc magntc modfcaton of ngy,, cospondng gound mag - n, stat on th (NC-D: RS) symmts, can b dtmnd fom th followng laton: mag-, n,, n! nc mb N T n,, L,, Th latvstc magntc modfcaton of ngy,,, can b obt as: mag- (6) m n fo psudospn symmty mag -, m,,,, n! mb N T,, L n nc (64) wh m m a psnt th angula momntum quantum numbs satsfyng th ntvals l m l l m l, whch allow us to fxng ( l ) ( l ) valus. 5. Th man sults of xact modfd global spctum fo modfd Yuawa potntal fo on-lcton atoms und spn-symmty p-spn symmty n (NC: D-RS): Ths pncpal pat of th pap s dvotd to th psntaton of th sval sults obt n th th pvous sctons, w sum th n xctd stats gnngs ( nc u,, n,,, nc,,, ) of modfd Dac quaton cospondng fo ( - /, d n, s/, p /,tc, l, algnd spn spn-down) ( l, p/, d /, tc, l, un algnd spn spn up), spctvly, at fst od of paamt, fo modfd Yuawa potntal n (NC: D-RS), spctvly, on basd to th obt nw sults (58), (59), (6) (64), n addton to th ognal sults () () of ngs n commutatv spac, w obtan th followng ognal sults: ncu ncd,,, n N T L,, n n! nc n,, n! mb N T,, L nc,,, n N T L,, n n! nc n,, n! mb N T,, L nc As t s mntond n f. [8], n vw of xact spn symmty n commutatv spac, V V F G ), w nd to gnalz th abov ( n n n tanslatons to th cas of NC th dmnsonal spacs, thn th ngatv valus nc,,, nc,,, a obt as: u n, d n, ncu,,, n m n,, ncu,,,,, n,, n N T L n! nc n,, n! nc mb N T n,, L, ncd,,, n,, nc d,, n,,,, n! N T,, L n n! mb N T,, L nc nc n n n n n n (65) (66) (67)

12 4 Volum It s claly, that th obt gnvalus of ngs a al; consquntly, th modfd quantum Hamltonan opato H nc p, x s Hmtan. Rgadng pvously obt sults (45) (6), t s asy to achv th followng quaton fo Hamltonan opato H p, x nc : H ncao p, x H p, x com S L S L wh H p, x s th odnay Hamltonan opato: com ao BJ S B fo BJ S B fo th spn symmtc cas th p - spn symmtc cas (68) H com p, x P ( M S( )) (69) Dnot to th odnay Hamltonan opato n th commutatv spac. In ths way, on can obtan th complt ngy spcta fo modfd Yuawa potntal n (NC: D-RS) symmts. Th followng accompanyng constant latons: a) Th two quantum numbs m satsfd th two ntvals: l m l l m l, thus w hav l l valus fo ths quantum numbs, b) W hav also two valus fo p-spn symmty l l two valus fo spn symmty l l, c) Fo ach gnvalus of n w hav two physcal valus. n Allow us to dduc th mpotant ognal sults: vy stat n usually th dmnsonal spac wll b plac by 8l 8l sub-stats und p-spn symmty spn symmty, whch n pmts to fxng th dgnatd stats to th 8 l 8n valus n (NC: D-RS) symmts. It s asy to s that th obt ognally sults duc to th odnay sults dscbd on quantum mchancs whn th noncommutatvty of spac dsappas,,, quatons (65), (66), (67) (68) ducs to (), () (69) on covs th stad txtboo sults. Fnally, on concluds; ou obt sults a suffcntly accuat fo pactcal puposs. 6. Th mpotant concludng mas Lt us summaz ou sults as follows: () Th soluton pocdu psntd n ths pap s basd on th both of Bopp s shft th mthod stad ptubaton thoy, w nvstgat th bound stat ngs of n xctd stats fo modfd Yuawa potntal dscbd on (NC: D-RS). () It s found that th ngy gnvalus dpnd on th dmnsonalty of th poblm th nw atomc quantum numbs th two angula momntum quantum numbs m, m n addton to th nfntsmal paamts,. () W hav also constuctng th cospondng NC Hmtan Hamltonan opato H nc p, x whch psntd by q. (68). Th ngy gnvalus a n good agmnt wth th sults pvously. Fnally, w pont out that ths xact sults (65), (66) (67) obt fo ths nw poposd fom of th modfd Yuawa potntal may hav som ntstng applcatons n th study of dffnt quantum V mchancal systms, atomc molcula physcs.

13 Intnatonal Font Scnc Ltts Vol. 4 Acnowldgmnts Ths wo was suppotd wth sach laboatoy of: Physcs Matal Chmsty, n Physcs dpatmnt, Scncs faculty, Unvsty of M'sla, Alga. Rfncs [] M.K. Baha, F. Yasu, Fmonc patcls wth poston-dpndnt mass n th psnc of nvsly quadatc Yuawa potntal tnso ntacto Pamana Jounal of Physcs. 8() () [] M. Hamzav, A. A. Raab, H. Hassanabad, xact psudospn symmty soluton of th Dac quaton fo spatally-dpndnt mass Coulomb potntal ncludng Coulomb-l tnso ntacton va asymptotc taton mthod, Physcs Ltts A. 74(4) () [] H. Acay, C. Tzca xact solutons of th Dac quaton wth hamonc oscllato potntal ncludng a Coulomb-l tnso ntacto Intnatonal Jounal of Modn Physcs C. (6) () [4] H. Acay, Dac quaton wth scala vcto quadatc potntals Coulomb-l tnso potntal, Physcs Ltts A. 7(6) (9) [5]. Maghsood, H. Hassanabad, S. Zanama, xact solutons of th Dac quaton wth Poschl-Tll doubl-ng shapd Coulomb potntal va th Nfoov-Uvaov mthod, Chns Physcs B. () () [6] C. Bd Y. F. Chng, On th xact solutons of th Dac quaton wth a novl angldpndnt potntal, Physca Scpta. 79() (9) 5-5. [7] B.I. Ita, A.I. Iuba, Solutons of th Dac quaton wth gavtatonal plus xponntal potntal, Appld Mathmatcs. 4 () -6. [8] J. Mawh A. Ronvaux, Schödng Dac quatons fo th hydogn atom Lagu polynomals, Ach. Hst. xact Sc. 64 () [9] B. Bswas, S. Dbnath, Bound stats of th Dac-Katz-Fus poblm wth spn psudospn symmty va Laplac tansfom appoach, Bulg. J. Phys. 4 (6) [] F. Padl, A.A. Raab, Invstgaton of th nucla systm usng th D-dmnsonal wav quato Chns Jounal of Physcs. 54 (6) [] K.J. Oywu C.O. Aoshl, Bound-stat solutons of th Dac-Rosn-Mos potntal wth spn psudospn symmty, u. Phys. J. A. 45 () 8. [] H. Hassanabad,. Maghsood, S. Zanama, Rlatvstc symmty of th Dac quaton Ttz potntal, u. Phys. J. Plus. (). [] M. R. Sta, S. Hada, Spn symmty of th Dac quaton wth th Yuawa potntal, Phys. Sc. 8 () 65. [4] A. Soylu, O. Baya, I. Boztosu An appoxmat soluton of th Dac-Hulthn poblm wth psudospn spn symmty fo any stat, J. Math. Phys. 48(8) (7) 8. [5] C.A. Onat, J.O. Oonubah, Rlatvstc nonlatvstc solutons of th gnalzd Pӧschl-Tll hbolcal potntals wth som thmodynamc popts, Int. J. Mod. Phys.. 4 (5) 55. [6] S.M. Ihda, R. Sv, Appoxmat analytcal solutons of th gnalzd Woods-Saxon potntals ncludng th spn-obt couplng tm spn symmty, Cnt. u. J. Phys. 8 () 665. [7] S.M. Ihda, R. Sv, Appoxmat gnvalu gnfunctons solutons fo th gnalzd Hulthén potntal wth any angula momntu J. Math. Chm. 4() (7)

14 4 Volum [8] M. shgha, S.M. Ihda, Rlatvstc ffct of psudospn symmty tnso couplng on th M-t potntal va Laplac tansfomaton mthod, Chn. Phys. B. () (4) 4. [9] S.M. Ihda; M. Hamzav, Appoxmat latvstc bound stat solutons of th Ttz Hua otatng oscllato fo any κ-stat, Fw-Body Syst. 5 () [] M. Hamzav, A.A. Raab, Soluton of Dac quaton wth Kllngbc potntal by usng wav functon ansatz mthod und spn symmty lmt, Commun. Tho. Phys. 55() () 5 7. [] H. Goudaz, M. Sohbat, S. Za Soluton of Dac quaton wth spn psudospn symmty fo an anhamonc oscllato, Jounal of Mathmatcal Physcs. 5 () 56. [] M.R. Shoa t al., ngy lvls of spn-/ patcls wth Yuawa ntacto Jounal of Modn Physcs. 5 (4) [] A. Mach, A study of Schödng quaton wth nvs sxtc potntal n -dmnsonal non-commutatv spac, Af. Rv Phys. 9 (4) [4] A. Mach, Spctum of Schödng quaton wth H.L.C. potntal n non-commutatv two-dmnsonal al spac, Af. Rv Phys. 9 (4) [5] A. Mach, Dfomd bound stats fo cntal facton pow potntal: non latvstc Schödng quato Af. Rv Phys. (5) 97-. [6] A. Mach, Spctum of hydogn atom gound stat countng quadatc tm n Schödng quato Af. Rv Phys. (5) [7] A. Mach, Atomc spctum fo Schödng quaton wth atonal sphcal t potntal n non commutatv spac phas, Af. Rv Phys. :46 (5) 7-8. [8] A. Mach, A complt analytcal soluton of th M-t potntals n non- commutatv -dmnsonal spacs phass symmts, Af. Rv Phys. :5 (6) -7. [9] A. Mach, A nw latvstc study fo ntactons n on-lcton atoms (spn ½ patcls) wth modfd M-t potntal, J. Nano- lcton. Phys. 8(4) (6) [] A. Mach, A nw nonlatvstc nvstgaton fo ntactons n on-lcton atoms wth modfd nvs squa potntal: noncommutatv two th dmnsonal spac phas solutons at Planc s nano-scals, J. Nanomd. Rs. 4() (6) 9-5. [] A. Mach, Invstgatons on th latvstc ntactons n on-lcton atoms wth modfd anhamonc oscllato, J. Nanomd. Rs. 4(4) (6) 97-. [] A. Mach, A nw nonlatvstc nvstgaton fo ntactons n on-lcton atoms wth modfd vbatonal-otatonal analyss of supsngula plus quadatc potntal: xtndd quantum mchancs, J. Nano- lcton. Phys. 8(4) (6) [] A. Mach, Nw xact ngy gn-valus fo (MIQYH) (MIQHM) cntal potntals: non-latvstc solutons, Af. Rv Phys. (6) [4] A. Mach, Nw xact soluton of th bound stats fo th potntal famly V()=A/- B/+C (=,-,-) n both noncommutatv th dmnsonal spacs phass: non latvstc quantum mchancs, Intnatonal Ltts of Chmsty, Physcs Astonomy. 58 (5) [5] A. Mach, A nw appoach to th non latvstc Schödng quaton fo an ngydpndd potntal V, V n both noncommutatv th dmnsonal spacs l l phass, Intnatonal Ltts of Chmsty, Physcs Astonomy. 6 (5) -9. [6] A. Mach, A nw study to th Schödng quaton fo modfd potntal -4-6 V() = a + b + c n nonlatvstc th dmnsonal al spacs phass, Intnatonal Ltts of Chmsty, Physcs Astonomy. 6 (5) 8-48.

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