Research Article Dirac Equation under Scalar, Vector, and Tensor Cornell Interactions

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1 Hndaw Publhng Copoaton Advance n Hgh Enegy Phyc Volume 0, Atcle ID 70704, 7 page do:0.55/0/70704 Reeach Atcle Dac Equaton unde Scala, Vecto, and Teno Conell Inteacton H. Haanabad, E. Maghood, S. Zankama, and H. Rahmov 3 Phyc Depatment, Shahood Unvety of Technology, Shahood , Ian Young Reeache Club, Gama Banch, Ilamc Azad Unvety, Gama, Ian 3 Compute Engneeng Depatment, Shahood Unvety of Technology, Shahood, Ian Coepondence hould be addeed to E. Maghood, e.maghood84@gmal.com Receved 4 July 0; Reved 7 Augut 0; Accepted 9 Augut 0 Academc Edto: S. H. Dong Copyght q 0 H. Haanabad et al. Th an open acce atcle dtbuted unde the Ceatve Common Attbuton Lcene, whch pemt unetcted ue, dtbuton, and epoducton n any medum, povded the ognal wok popely cted. Spn and peudopn ymmete of Dac equaton ae olved unde cala, vecto, and teno nteacton fo abtay quantum numbe va the analytcal anatz appoach. The pectum of the ytem numecally epoted fo typcal value of the potental paamete.. Intoducton No doubt, the Conell potental altenatvely called Funnel n the lteatue, f not the bet, among the mot appealng nteacton n patcle phyc. The Conell potental contan a confnng tem bede the Coulomb nteacton and ha uccefully accounted fo the patcle phyc data. Unfotunately, to ou bet knowledge, the potental doe not poe exact oluton unde all common equaton of quantum mechanc, that, the nonelatvtc Schödnge equaton, and elatvtc Dac, Klen-Godon, Poca, and Duffn-Kemme- Petau DKP equaton. Hee, we focu on the elatvtc ymmete of Dac equaton, that, pn and peudopn ymmete, whch povde a elable theoetcal ba fo hadonc and nuclea pectocopy. Th ha motvated many tude unde vaou nteacton wthn the pat two decade 3 8 and many efeence theen. Nevethele, none of thee pape ha nvetgated the ymmety lmt unde the Conell potental. Th defntely due to the complcated natue of the eultng dffeental equaton whch cannot be olved by common analytcal technque of quantum mechanc uch a the upeymmety quantum mechanc SUSYQ, Le goup, Nkfoov-Uvaov NU technque, and pont canoncal tanfomaton, In ou tudy, we make ue of the anatz appoach to deal wth th

2 Advance n Hgh Enegy Phyc complcated equaton. A uvey on the applcaton of th technque to othe wave equaton ncludng Schödnge, pnle-salpete, Dac, Klen-Godon, and DKP equaton can be found n 9 7. We oganze the tudy a follow. In the ft tep, we evew the mot eental equaton of the ymmety lmt. We next popoe a phycal anatz oluton to the equaton and, n a ytematc manne, calculate the pectum of the ytem fo any abtay tate. To povde a bette undetandng of the oluton, we povde ome numecal data fo the pectum a well.. Dac Equaton Includng Teno Couplng In phecal coodnate, Dac equaton wth both cala potental S and vecto potental V expeed a, 3 [ α p β M S β α U ] ψ E V ψ,. whee E the elatvtc enegy of the ytem; α and β ae the 4 4 Dac matce and p tand fo the momentum opeato. Fo a patcle n a phecal feld, the total angula momentum opeato J and pn-obt matx opeato K σ L, whee σ and L ae, epectvely, the Paul matx and obtal angula momentum, commute wth the Dac Hamltonan. The egenvalue of K ae κ j / fo the algned pn /,p 3/,etc. and κ j / fo the unalgned pn p /,d 3/,etc.. The complete et of the conevatve quantte can be choen a H, K, J,J z. A hown n 3, the Dac pno condeed a ψ nk fnk g nk G nk F nk Y l jm Y l jm θ, ϕ θ, ϕ,. whee F nk and G nk ae the adal wave functon of the uppe and lowe component, epectvely, and Y l l jm θ, ϕ and Yjm θ, ϕ, epectvely, tand fo pn and peudopn phecal hamonc coupled to the angula momentum j. m the pojecton of the angula momentum on the z-ax. The obtal angula momentum quantum numbe l and l efe to the uppe and lowe component, epectvely. The quadegeneate doublet tuctue can be expeed n tem of peudopn angula momentum / and peudoobtal angula momentum l, whch defned a l l fo algned pn j l / and l l fo unalgned pn j l /. A hown n, 3, ubttuton of. nto. yeld the followng two-coupled dffeental equaton: d d κ U F nk M E nk V S G nk, d d κ U G nk M E nk V S F nk,.3

3 Advance n Hgh Enegy Phyc 3 whch mply gve { d κ κ κ d U du d U dδ /d d M E nκ Δ d κ } U F nκ M E nκ Δ M E nκ Σ F nκ,.4 { d κ κ κ d U du d U dσ /d d M E nκ Σ d κ } U G nκ M E nκ Δ M E nκ Σ G nκ,.5 whee, a the notaton ndcate, Δ V S and Σ V S... Peudopn Symmety Lmt Unde the condton of the peudopn ymmety, dσ /d 0 o equvalently Σ C gp Cont., 3. We chooe Δ a the Conell potental: Δ b p..6 Fo the teno tem, we conde the Conell potental: U A B..7 Subttuton of thee two tem n nto.5 gve { d d κ κ κb B B A a p M E p nκ C p b p M b p E p nκ b p C p M MC p E p nκ } p E nκc p κa A AB.8 G p nκ 0, whee κ l and κ l foκ<0andκ>0, epectvely.

4 4 Advance n Hgh Enegy Phyc.. Spn Symmety Lmt In the pn ymmety lmt dδ /d 0oΔ C g ecton, we conde cont., 3. A the pevou Σ a b, U A B..9a.9b Subttuton of the latte n.4 gve { d d κ κ κb B B A a M a Enκ a C } b M b Enκ b C M MC Enκ p E nκ C κa A AB Fnκ 0,.0 whee κ l and κ l foκ<0andκ>0, epectvely. 3. The Anatz Soluton 3.. Soluton of the Peudopn Symmety Lmt In the pevou ecton, we obtaned a Schödnge-lke equaton of the fom d G p [ nκ ε p d nκ b p cp d p ] κ κ G p nκ 0, 3. whee ε p nκ M MC p E p nκ E p nκc p κa A AB, A, b p M E p nκ C p, c p b p M b p E p nκ b p C p, d p κb B B. 3.

5 Advance n Hgh Enegy Phyc 5 Equaton 3. fal to admt exact analytcal oluton. Theefoe, we follow the anatz appoach wth the tatng quae: G p nκ f p n exp g p κ, 3.3 whee f p n, f n 0, n α n 3.4, f n, g p κ αp β p δ p ln, α p > 0, β p > By ubttuton of f n and g κ nto 3.3, wefnd G p nκ [ g p κ p f p p p ] n g κ f n g κ f p n G p nκ. 3.6 Hee, we conde the cae n 0. Fom we fnd G p 0κ { α p α p β p α p δ p 0 β p β p δ p 0 δp δ p } G p 0κ. 3.7 By compang the coepondng powe of 3. and 3.7, we have α p, ap > 0, δ p ± c p bp β p β p δ p 0, κ 4d p, ± k p, 3.8 ε p 0κ αp δ p 0 β p, whee k p κ 4d p. Actually, to have well-behaved oluton of the adal wave functon at boundae, namely, the ogn and the nfnty, we need to take δ fom 3.8 a δ p k p. 3.9

6 6 Advance n Hgh Enegy Phyc Fom 3., 3.8, the gound-tate enegy atfe M MC p E p 0κ E p 0κ C p κa A AB α p δ p 0 β p 0 3.0a o E p 0κ p E 0κ C p M MC p κa A AB κ 4d p bp 4 0, 3.0b whch moe compactly wtten a E p 0κ { [ ± C p ± Cp 8κA 4A 8AB 4M 4MC p 8 4 4κ 4κ 4d p 4βp ] / }, 3.35, 3. whee the paamete of potental.6 fom 3.8 hould atfy the followng etcton: b p κ 4d p b p k p. 3. Fom 3.3, 3.4, and 3.8, the uppe and lowe component of the wave functon ae g κ bp G p 0κ N 0κ / κ 4d p exp κ 4d p ln, F p 0κ d M E p 0κ C p d κ bp U G p 0κ. 3.3a, 3.3b 3.3c

7 Advance n Hgh Enegy Phyc 7 Fo the ft node n, ungf α and g κ fom 3.5, we ave at b p cp κ κ d p ε p κ α p δp αp β p δ p α p β p δ p α p βp δ p αp β p δ p /. α 3.4 Hee, the conequent elaton between the potental paamete and the coeffcent α p, β p, δ p,andα ae α p, βp bp, δ p k p, ε p κ αp δ p β p, c p βp δ p α δ p, 3.5 c p βp δ p α p α. By olvng the above equaton one can fnd c p and α a c p bp κ 4d p bp 4 κ 4d p, 3.6a α p α β p α δ p 0, β α β p p 4α p δ p α p b p 4 bp 6 κ 4d p. 3.6b The enegy egenvalue theefoe ae E p κ { [ ± C p ± Cp 8κA 4A 8AB 4M 4MC p 6 4 4κ 4κ 4d p 4βp ] / }, 3.7

8 8 Advance n Hgh Enegy Phyc whee the paamete fom 3.8 c p cp kp 3M 4Mk p Mkp c p 3E p 6 4 κ 4Ep κ kp Ep kp κ kp 0 kp kp 3. 3Cp 4C p k p C pk p 3.8 Fo the uppe and lowe component of the wave functon we thu have G p κ N κ α / κ 4d p F p κ d M E p κ C p d κ exp bp U G p κ., 3.9a 3.9b Followng the analytc teaton pocedue fo the econd node n wth f α α and g κ a defned n 3.6, the elaton between the potental paamete and the coeffcent α p, β p, δ p, α,andα ae α p, βp bp, δ p k p, c p βp δ p α p ε p κ αp δ p β p, α, c p βp δ p α α j <j δ p α, 3.0 [ c p βp δ p ] α 4α p <j α α j δ p. The coeffcent α and α ae found fom the contant elaton 0, 7 9 : δ p α α α p α β p α β p j<k δ p 0, 3. α j α k α p j<k α j α k 0. 3.

9 Advance n Hgh Enegy Phyc 9 Theefoe, the enegy egenvalue n th cae E p κ { [ ± C p ± Cp 8κA 4A 8AB 4M 4MC p 4 4 4κ 4κ 4d p 4βp ] / }, 3.3 and the lowe component of the wave functon G p κ N κ α / κ 4d p exp bp Soluton of the Spn Symmety Lmt In th cae, ou odnay dffeental equaton d F nκ d [ εnκ a b c d κ κ ] F nκ 0, 3.5 wth ε nκ M MC E nκ E nκc κa A AB, a A, b a M a E nκ a C, 3.6 c b M b E nκ b C, d κb B B, whch cannot be olved by ou common exact analytcal technque. Let u popoe the anatz oluton: F nκ f n exp g κ, 3.7 whee, f n 0, fn n α n 3.8, f n, g κ α β δ ln, α > 0, β >

10 0 Advance n Hgh Enegy Phyc By ubttuton of f n and g κ nto 3.7, wefnd [ ] F nκ g κ gκ f n g κ f n fn Fnκ Fo the cae of n 0, fom , wefnd F nκ { α α β α δ 0 β β δ 0 δ δ } F nκ. 3.3 By compang the coepondng powe of 3.7 and 3.33, we have α a, β b a, a > 0, c β δ 0, δ ± κ 4d ± k, 3.3 ε 0κ α δ 0 β, whee k κ 4d. To have phycally acceptable oluton, we pck up the value δ k By condeng 3.6, 3.3, the ft node egenvalue atfe M MC E nκ E nκc κa A AB α δ 0 β 0, 3.34 o equvalently E ± 0κ { [ C ± C 4M 4MC 8κA 4A 8AB 8 a ] 4 a 4κ / 4κ 4d 4β, 3.35

11 Advance n Hgh Enegy Phyc whee the paamete a of potental.9a hould atfy the etcton b a a k Fom 3.7, 3.8, and 3.3, the uppe and lowe component of the wave functon ae g κ a b κ 4d a ln, κ F 0κ N 0κ / 4d exp a b G 0κ d M E 0κ C d κ U F 0κ. a 3.37, Secondly, fo the ft node n, ungf α and g κ fom , ou eultng equaton a b c d κ κ ε nκ α δ α β δ α β δ α β δ α β δ /. α 3.40 The elaton between the potental paamete and the coeffcent α,β,δ,andα ae α a, β b, a > 0, δ κ 4d k, ε κ α δ β, 3.4 c β δ α δ, c β δ α α,

12 Advance n Hgh Enegy Phyc whee c and α ae found fom 3.4 a 0, 7 9 c b a κ 4d b 4a a κ 4d, 3.4a α α β α δ 0, 3.4b o β α β 4α δ α b 4a b 6a κ 4d, 3.4c a whch detemne the coepondng enegy a E ± κ { C ± [ C 4M 4MC 8κA 4A 8AB 6 a ] / 4 a 4κ 4κ 4d }, 4β 3.43 wth a a c c k c 6 a a 4 k 0 a k a k 3 3M 4Mk Mk 3E κ 4E κ k E κ k 3C 4C k C k The uppe and lowe component of the wave functon ae then mply found to be F κ N κ α / κ 4d exp a b G κ d M E κ C d κ U F κ. a, 3.45a 3.45b

13 Advance n Hgh Enegy Phyc 3 Fo the econd node n, we chooe f α α and g κ a defned n 3.8 and 3.9. The elaton between the potental paamete and the coeffcent α,β,δ,α,andα ae α a, β b a, δ k, c β δ α ε κ α δ β, α, c β δ α <j α j δ α, 3.46 [ c β δ ] α 4α <j α α j δ, whee δ α α α α β α β j<k δ 0, 3.47 α j α k α j<k α j α k The enegy egenvalue theefoe E ± κ { C ± [ C 4M 4MC 8κA 4A 8AB 4 a ] / 4 a 4κ 4κ 4d }. 4β 3.49 Fo the uppe component of the wave functon, we have F κ N κ α / κ 4d exp a b a We have gven ome numecal value of the enegy egenvalue n Table,, 3, 4, 5, and 6 fo vaou tate. Fo the fnal pont, we wh to emphaze on the degeneacy-emovng ole of the Conell teno potental. A we aleady know, fo vanhng teno nteacton A B 0, the peudopn doublet, that, tate wth quantum numbe n, l, j l / and n,l,j l 3/ ae degeneate. The degeneate tate n the pn doublet ae thoe wth quantum numbe n, l, j l / and n, l, j l /, whee n, l,and j

14 4 Advance n Hgh Enegy Phyc Table : Bound tate fo the peudopn ymmety b, M fm, C p 0fm. l n, κ l, j fm A B 0 A 0.5,B 0. 0, 0S / , 0P 3/ , 3 0d 5/ , 4 0f 7/ , S / , p 3/ , 3 d 5/ , 4 f 7/ , 0d 3/ , 3 0f 5/ , 4 0g 7/ , 5 0h 9/ E p nκ Table : Bound tate fo the pn ymmety b, M fm, C 0fm. l n,κ l, j Enκ fm A B 0 A 0.5,B 0. 0, 0p 3/ , 3 0d 5/ , 4 0f 7/ , 5 0g 9/ , p 3/ , 3 d 5/ , 4 f 7/ , 5 g 9/ , 0p / , 0d 3/ ,3 0f 5/ ,4 0g 7/ , p / , d 3/ ,3 f 5/ ,4 g 7/ C p Table 3: Enege n the peudopn ymmety Lmt fo A 0.5, B 0., b, M fm. E p nκ fm P 3/ f 7/ 0d 3/ 0g 7/

15 Advance n Hgh Enegy Phyc 5 Table 4: Enege n the peudopn ymmety lmt fo A 0.5, B 0., b, C p 0fm. M fm E p nκ fm P 3/ f 7/ 0d 3/ 0g 7/ Table 5: Enege n the pn ymmety lmt fo A 0.5, B 0., b, M fm. C Enκ fm 0f 7/ d 5/ 0P / f 5/ Table 6: Enege n the pn ymmety lmt fo A 0.5, B 0., b, C 0fm. M fm Enκ fm 0f 7/ d 5/ 0P / f 5/ ae the adal, the obtal, and the total angula momentum quantum numbe, epectvely ee Table and. Ou numecal data eveal that, n the peudopn ymmety lmt, the degeneate tate fo A B 0 ae n /, n d 3/ fo l l 0, np 3/, n f 5/ fo l l, nd 5/, n g 7/ fo l 3 l, nf 7/, n h 9/ fo l 4 l 3, and o foth. Fo pn ymmety lmt and A B 0 one can clealy ee that the degeneacy occu n np /,np 3/ fo l, nd 3/,nd 5/ fo l, nf 5/,nf 7/ fo l 3, ng 7/,ng 9/ fo l 4, and o foth. 4. Concluon Becaue of the etablhed ole of the Conell potental and pn, and peudopn ymmete n nuclea and hadon pectocopy, we olved the Dac equaton unde thee ymmety lmt fo vecto, cala, and teno nteacton of Conell-type. In ou calculaton, on the one hand, due to the falue of othe common analytcal technque, and, on the othe hand, the bette nght whch analytcal technque povde u n compaon wth the numecal countepat, we ued the quaanalytcal anatz appoach. By popong novel phycal oluton and afte lengthy calculaton, we could fnd the abtay-tate oluton. Ou eult clealy how the degeneacy-emovng ole of the teno tem and povde the equte undetandng of the oluton fo poble futhe tude. Both the

16 6 Advance n Hgh Enegy Phyc enegy pectum and the egenfuncton can be ued n elated ytem afte the pope phenomenologcal ft done. Refeence D. H. Pekn, An Intoducton to Hgh Enegy Phyc, Cambdge Unvety Pe, 000. J. N. Gnoccho, Relatvtc ymmete n nucle and hadon, Phyc Repot, vol. 44, no. 4-5, pp. 65 6, J. N. Gnoccho and A. Levatan, On the elatvtc foundaton of peudopn ymmety n nucle, Phyc Lette B, vol. 45, no. -, pp. 5, G. Mao, Effect of teno couplng n a elatvtc Hatee appoach fo fnte nucle, Phycal Revew C, vol. 67, no. 4, Atcle ID 04438, page, P. Albeto, R. Lboa, M. Malheo, and A. S. de Cato, Teno couplng and peudopn ymmety n nucle, Phycal Revew C, vol. 7, no. 3, Atcle ID 03433, 7 page, R. J. Funtahl, J. J. Runak, and B. D. Seot, The nuclea pn-obt foce n chal effectve feld theoe, Nuclea Phyc A, vol. 63, no. 4, pp , G.-F. We and S.-H. Dong, Appoxmately analytcal oluton of the Mannng-Roen potental wth the pn-obt couplng tem and pn ymmety, Phyc Lette A, vol. 373, no., pp , G. F. We and S. H. Dong, Algebac appoach to peudopn ymmety fo the Dac equaton wth cala and vecto modfed Pöchl-Telle potental, Euophyc Lette, vol. 87, no. 4, atcle 40004, G. F. We and S. H. Dong, Spn ymmety n the elatvtc ymmetcal well potental ncludng a pope appoxmaton to the pn-obt couplng tem, Phyca Scpta, vol. 8, no. 3, Atcle ID , G. F. We and S. H. Dong, Peudopn ymmety fo modfed Roen-Moe potental ncludng a Peke-type appoxmaton to the peudo-centfugal tem, Euopean Phyc A, vol. 46, no., pp. 07, 00. G. F. We and S. H. Dong, Peudopn ymmety n the elatvtc Mannng-Roen potental ncludng a Peke-type appoxmaton to the peudo-centfugal tem, Phyc Lette B, vol. 686, no. 4-5, pp. 88 9, 00. W.-C. Qang and S.-H. Dong, SUSYQM and SWKB appoache to the elatvtc equaton wth hypebolc potental V 0 tanh /d, Phyca Scpta, vol. 7, no. -3, pp. 7 3, S. Zankama, A. A. Rajab, and H. Haanabad, Dac equaton fo the hamonc cala and vecto potental and lnea plu Coulomb-lke teno potental; the SUSY appoach, Annal of Phyc, vol. 35, no., pp. 5 58, O. Aydoğdu and R. Seve, Peudopn and pn ymmety n Dac-Moe poblem wth a teno potental, Phyc Lette. B, vol. 703, no. 3, pp , 0. 5 Y. Xu, S. He, and C.-S. Ja, Appoxmate analytcal oluton of the Dac equaton wth the Pöchl- Telle potental ncludng the pn-obt couplng tem, Phyc A, vol. 4, no. 5, atcle 5530, M. R. Setae and Z. Naza, Peudopn ymmety n defomed nucle wth axally-ymmetc hamonc ocllato potental, Moden Phyc Lette A, vol. 5, no. 549, K. J. Oyewum and C. O. Akohle, Bound-tate oluton of the Dac-Roen-Moe potental wth pn and peudopn ymmety, Euopean Phycal Jounal A, vol. 45, no. 3, pp. 3 38, E. Maghood, H. Haanabad, and O. Aydogdu, Dac patcle n the peence of the Yukawa potental plu a teno nteacton n SUSYQMfamewok, Phyca Scpta, vol. 86, Atcle ID 05005, 0. 9 H. Haanabad and A. A. Rajab, Enegy level of a phecal quantum dot n a confnng potental, Phyc Lette A, vol. 373, no. 6, pp , H. Haanabad and A. A. Rajab, Relatvtc veu nonelatvtc oluton of the N-femon poblem n a hypeadu-confnng potental, Few-Body Sytem, vol. 4, no. 3-4, pp. 0 0, 007. S.-H. Dong, Z.-Q. Ma, and G. Epoto, Exact oluton of the Schödnge equaton wth nveepowe potental, Foundaton of Phyc Lette, vol., no. 5, pp , 999. S.-H. Dong, Exact oluton of the two-dmenonal Schödnge equaton wth cetan cental potental, Intenatonal Theoetcal Phyc, vol. 39, no. 4, pp. 9 8, S. H. Dong, A new appoach to the elatvtc Schödnge equaton wth cental potental: Anatz method, Intenatonal Theoetcal Phyc, vol. 40, no., pp , 00.

17 Advance n Hgh Enegy Phyc 7 4 H. Haanabad, H. Rahmov, and S. Zankama, Conell and Coulomb nteacton fo the D- dmenonal Klen-Godon equaton, Annalen de Phyk, vol. 53, no. 7, pp , 0. 5 D. Agboola and Y. Zhang, Unfed devaton of exact oluton fo a cla of qua-exactly olvable model, Mathematcal Phyc, vol. 53, Atcle ID 040, 3 page, 0. 6 H. Haanabad, B. H. Yazaloo, S. Zankama, and A. A. Rajab, Duffn-Kemme-Petau equaton unde a cala Coulomb nteacton, Phycal Revew C, vol. 84, Atcle ID , 4 page, 0. 7 S. M. Ikhda, Bound tate enege and wave functon of phecal quantum dot n peence of aconfnng potental model, 8 M. Znojl, Sngula anhamoncte and the analytc contnued facton, Mathematcal Phyc, vol. 30, no., pp. 3 7, M. Znojl, The genealzed contnued facton and potental of the Lennad-Jone type, Mathematcal Phyc, vol. 3, no. 955, 7 page, 990.

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