Dirac-Hua Problem Including a. Coulomb-like Tensor Interaction

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1 Adv. Studies Theo. Phys., Vo. 5, 0, o., Diac-Hua Pobem Icudig a Couomb-ike Teso Iteactio Mahdi Eshghi Depatmet of Basic Scieces, Ceta Teha Bach Isamic Azad Uivesity, Teha, Ia Eshgi5@gmai.com Abstact I this eseach, we have soved the Diac euatio with the spi ad pseudo-spi symmeties fo Hua potetia icudig a Couomb-ike teso iteactio. The eegy eigevaues euatio ad the coespodig uomaized wave fuctios have obtaied i tems of the Jacobi poyomias. The paametic geeaizatio of the Nikifoov-Uvaov method had used i the cacuatios. Keywods: Diac euatio, Hua potetia, Couomb-ike, Nikifoov-Uvaov Itoductio It is we kow that the exact eegy eigevaues of the boud state pay a impotat oe i uatum mechaics. I paticua, the Diac euatio which descibe the motio of a spi-/ patice has bee used i sovig may pobems of ucea ad high-eegy physics. Recety, thee has bee a iceased i seachig fo aaytic soutio of the Diac euatio. Fo exampe, see [-9]. Cocepts of spi ad pseudo-spi symmeties ad a teso potetia have bee foud iteestig appicatios i the fied of ucea physics [30-3]. O the othe had, teso potetias wee itoduced ito the Diac euatio with the substitutio p p imωβ U ˆ ( [35, 36]. I this way, a spi-obit coupig tem is added to the Diac Hamitoia. Recety, teso coupigs have bee used widey i the studies of ucea popeties. I this egad, see [37-].

2 560 M. Eshghi The iteucea potetia [5] has bee itoduced by Hua, of the fom is e V( = V0, e ( whee,, ad V 0 ae the potetia costats. Teso potetia Couomb-ike [37, 38] is H ZaZbe U(, H, πε 0 R c ( whee Rc = 7.78 fm is the Couomb adius, Z a ad Zb deote the chages of the pojectie a ad the taget ucei b, espectivey. Ou aim the easo fo choosig the Hua potetia i this wok is study the Diac euatio fo this potetia icudig a couomb-ike teso coupig ude the spi ad pseudo-spi symmeties. We have obtaied the eegy eigevaues euatio ad the coespodig spio wave fuctios by usig the paametic geeaizatio of the Nikifoov-Uvaov (NU method. NU method We give a bief desciptio of the covetioa NU method [6]. This method is based o sovig the secod ode diffeetia euatio of hypegeometic-type by meas of specia othogoa fuctios ~ τ ~ σ ψ ( s + ψ + ψ = 0, (3 σ σ whee σ ( ad ~ σ ( s ae poyomias, at the most of the secod degee, ad ~ τ ( s is a poyomias, at most of the fist degee. If we take the foowig factoizatio ψ = φ( y, (3 becomes σ y ( ( s + τ s y + λy = 0, ( whee d π = σ ( ϕ, (5 ds τ = ~ τ + π, τ ( < 0, (6 whee π ( is a poyomia of ode at most oe. The y ( which is a poyomia of degee ca be expessed i tems of the Rodigues eatio a d y = [ σ ρ( ], (7 ρ( ds whee a is a omaizatio costat ad the weight fuctio ρ ( must satisfy the diffeetia euatio

3 Diac-Hua pobem 56 τ ω ω( = 0, ω( = σ ρ(. (8 σ The fuctio π ( ad the paamete λ i the above euatio ae defied as foows ~ ~ σ ( τ σ ( τ π = ± ~ σ + σ, (9 λ = + π (. (0 The detemiatio of is the essetia poit i the cacuatio of π (. It is simpy defied by settig the discimiate of the suae oot which must be zeo. The eigevaues euatio have cacuated fom the above euatio ( λ = λ = ( τ s σ. = 0,,,.... ( Fo a moe simpe appicatio of the method, we deveop a paametic geeaizatio of the NU method vaid fo ay potetia ude cosideatio by a appopiate coodiate tasfomatio s = s(. Thus, we obtai aothe geeaized hypegeometic euatio [39, ] d d s ( 3 + s ( 3( + [ ξs + ξs ξ3] ψ = 0. ( ds ds We may sove this as foows. Compaig ( with (3, yieds τ ( s = s σ ( s = s s, % σ ( S = ξ s + ξ s ξ. (3 %, ( Substitutig these ito (9, we fid 3 5 ( 6 3 ( π( s = + s± k s + + k s+, ( with the foowig paametes ( =, 5 = ( 3, 6 = 5 + ξ, 7 = 5 ξ, 8 = + ξ3. (5 We obtai the paamete k fom the coditio that the fuctio ude the suae oot shoud be the suae of a poyomia k = + ±, (6 (, whee 9 = (7 Fo each k the foowig π s ae obtaied. The fuctio π ( becomes fo the k-vaue We aso have fom τ = % τ + π, π ( s = + s [( + s ], ( ( k = +. ( τ ( s = + ( s [( + s ]. (

4 56 M. Eshghi Thus, we impose the foowig coditio to fix the k-vaue τ ( s = ( ( = 3 ( < 0. ( Whe (0 is used with (0 ad ( the foowig euatio is deived [( 3+ 5] 5 + ( + ( = 0. ( By usig ( ρ( s = s (, (3 ad togethe with (7, we have whee ad ad (, β y ( s = P (, ( 0, 0 3 = + +, (5 0 8 = + ( +, ( P ae Jacobi poyomias. By usig (5, we get 3 3 φ( s s ( 3 =, (7 ad the tota wave fuctio become 3 0, ( s s ( 3 P ( 3 Ψ =, (8 whee = + 8 ad 3 = 5 ( I some pobems the situatio appeas whee 3 = 0. Fo such pobems, the soutio give i (8 becomes as 3 0 ( s Ψ s = s e L (. (9 I some cases, oe may eed a secod soutio of (. I this case, if the same pocedue is foowed, by usig k = ( , (30 the soutio becomes 3 0, Ψ ( s = s ( P (, (3 ad the eegy spectum is [( + ] ( + ( = 0. (3 Pe-defied paametes ae: = +, = + (,

5 Diac-Hua pobem 563 =, 8 = (. ( Soutio of the Diac euatio Accodig to [37-], the Diac euatio of a uceo with mass M movig i a scaa ad a vecto potetias icudig teso iteactio fo spi-/ patices ca be witte as ( h = c =, [ P + β ( M + V ( ˆ s iβ U ( ] ψ k ( = [ E Vv ( ] ψ k (, (3 whee E is the eativistic eegy of the system, P = i is the thee-dimesioa mometum opeato ad ad β the Diac matices which have the foowig foms [7], espectivey 0 σ I 0 =, β =, P = ih, (35 σ 0 0 I whee I deotes the idetity matix ad σ ae thee-vecto paui spi matices 0 0 i 0 σ =, σ =, σ 3 =. (36 0 i 0 0 Fo spheica ucei, the uceo agua mometum J ad Kˆ = βσ ( ˆ. Lˆ+ commute with the Diac Hamitoia, whee σˆ ad Lˆ ae the paui matix ad obita agua mometum, espectivey. The Diac spios ca be witte accodig to thee agua mometum j ad k, F k ( Y ( θ, ψ k ( = ~, (37 igk ( Y ( θ, whee F k ( is uppe ad ( is the owe adia wave fuctios of the G k ~ Diac spios, Y ( θ, ad Y ( θ, ae the spi ad pseudo-spi spheica hamoica fuctios, ad is the adia uatum umbe, ad m is the pojectio of the tota agua mometum o the z-axis. The eigevaues of Kˆ ae k = ±( j + (/ with - fo aiged spi (s /,p 3/, etc. ad + fo uaiged spi (p /, d 3/,etc.. Substitutig (37 ito (3 ad usig the foowig eatios [8] ( σ. A( σ. B = A. B + iσ.( A B, (38 σ. L ( σ. P = σ.ˆ( ˆ. P + i, (39 ad popeties ~ ~ ( σ. L Y ( θ, = ( k Y ( θ,, (0 ( σ. L Y ( θ, = ( k Y ( θ,, (

6 56 M. Eshghi ~ ( σ.ˆ Y ( θ, = Y ( θ,, ( ~ ( σ.ˆ Y ( θ, = Y ( θ,, (3 we obtai the adia pat of the Diac euatio as d k + U ( Fk ( = [ Ek + M Δ( ] Gk (, d ( d k + U ( Gk ( = [ M Ek + Σ( ] Fk (, d (5 whee Δ ad Σ have bee assumed to be adia fuctios, i.e., Δ ( = Vv ( Vs ( ad Σ ( = Vv ( + Vs (. By substitutig Gk ( fom ( ito (5 ad Fk ( fom (5 ito (, we have bee obtaied the foowig two secod-ode diffeetia euatios fo the uppe ad owe compoets, d k( k + k du ( + U ( U ( d d + ( E + M Δ( ( E M Σ( d d k k dδ( + d d k + ( = 0 ( + Δ( Fk, (6 M Ek d k( k k du ( + U ( + U ( d + ( Ek + M Δ( ( Ek M Σ( dσ( d d k + Gk ( = 0 ( M Ek +Σ( d. (7 I the above euatios k( k + = ( + ad k( k = % ( % Spi symmety case Substitutig ( ad ( ito (6, cosideig pseudo-spi symmety, takig Δ( as the Hua potetia ad Σ ( = C ps = cost. ( d Σ( / d = 0 [9, 50], i.e., the euatio have bee obtaied fo the uppe compoet of the Diac spio F k ( becomes d ( k + H ( k + H + d e + ( Ek M ( M + Ek Cs V 0( Ek + M Cs F ( 0 k =. (8 e This euatio is descibes a patice of spi-/ such as the eecto i the

7 Diac-Hua pobem 565 Diac theoy with Hua potetia icudig a teso coupig, that ca ot be soved aayticay because of ( k + H( k + H + / tem, we used the appoximatio scheme suggested by Geee ad Adich [5] ( e C0 +, (9 e whee the paamete C 0 = / is dimesioess costat. By usig a tasfomatio of the fom s = e, we ewite it as foows d s d + + ( bc 0 bb 3 + V% 0b s ds s ( s ds s( [ ] + ( bb 3 + bv % 0+ bc 0 b s+ bb 3 bv % 0 C0b Fk = 0. (50 By compaig (50 with (, we have bee obtaied the paamete set as ξ = bc 0 bb 3 + V % 0b, ξ = bb 3 + bv % 0 + bc 0 b, ξ = ( bb bv % C b, =, =, 3 =, = 0, 5 =, 6 = + ξ, 7 = ξ, 8 = ξ3, 9 = ξ ξ + ξ3+ = V% 0b( + ( b +, 0 = + ξ, 3 = + ξ ξ + ξ3 + + ξ3, = ξ, 3 3 = ξ ξ + ξ3 + + ξ3. (5 Usig (, (6 ad (5, we cacuate the paametes euied fo the method π = s ± + bc0 bb 3 + V% 0b k s + ( bb 3 + bv 0+ bc 0 b + k s ( bb 3 bv 0 C0b, (5 whee k, = bv% 0( b ± ( bb 3+ bv% 0 + C0b bv% 0( + b +. (53 Diffeet k s ead to the diffeet π s. Fo

8 566 M. Eshghi k = bv% 0( b ( bb 3+ bv% 0 + C0b bv% 0( + [ b + ], (5 π ( s becomes π = s Vb % 0 ( + b ( + + bb 3+ bv % 0 + Cb 0 s bb 3+ bv % 0 + Cb 0, (55 ad usig (0, we obtai τ ( s = s s V % 0b( + ( b + + ( s bb 3+ bv % 0+ C 0b, (56 whee τ ( s < 0. Theefoe, usig ( ad (5, we wite the eigevaues euatio as + + Vb % 0 ( + b ( + + bc 0 bb 3+ Vb % 0 + ( + bc0 bb 3+ V% 0b V% 0b( + ( b+ + bc0 bb 3+ V% 0b bc 0 + b + bb 3 V % 0b+ = 0. (57 Now, et us give the coespodig uppe Diac spio. Usig (3, ( ad (7, we wite the coespodig uomaized eigefuctios ae obtaied i tems of the fuctios, ad ad Vb % 0 ( + b ( + bc 0 bb 3+ V% 0b ρ = s ( bc 0 bb 3+ V% 0b, V% 0b( + ( b+, (58 y = P (, (59 + Vb % 0 ( + b ( + bc 0 bb 3+ V% 0b ϕ = s ( ad usig (8, the coespodig wave fuctios to be whee F k k + Vb % 0 ( + b ( + bc bb + V% b F s B s s ( = ( bc 0 bb 3+ Vb % 0, Vb % 0 ( + b ( +. (60 P ( s, (6 B is the omaized costat ad it was detemied by the coditio ds =, ad the owe compoet of Diac spio ca be cacuated by appyig ( as

9 Diac-Hua pobem 567 ad d k Gk ( = + U( Fk (, (6 M + E Δ( d k b = ( E + M C, b = ( k + H( k + H +, k s b E M k 0 3 =, V, 0 = % V (63 3. Pseudo-Spi symmety case Substitutig ( ad ( ito (7 ad cosideig pseudo-spi symmety (the coditio of pseudo-spi symmety d Σ( / d = 0 o Σ ( = cost = C ps [9, 50], we have d ( k + H( k + H d e + ( Ek M C ps ( M + Ek V0 ( Ek M C ps G ( 0 k =, (6 e By usig E. (9 ad s = e, 0 < s <, we obtai d d + + ( bb bv% 5 0+ Cb 0 s ds s ds s ( s + ( bb bv % 5 0 bc 0 b s ( bb 5 6+ bv % 5 0+ Cb 0 Gk = 0, (65 ad usig the NU method ad simia pocedues peseted i subsect. 3.., the eigevaues fo the symmety ae obtaied as by usig ( to be + + Vb % 0 5( + b ( + + bc 0 bb 5 6+ Vb % 0 5 ( + + bc0 b5b6+ V% 0b5 V% 0b5( + ( b+ + bc0 b5b6+ V% 0b5 bc 0 + b + bb 5 6 V % 0b5 + = 0, (66 The owe compoet of the Diac spio ca be cacuated by usig (3 as whee G k k + Vb % 0 5( + b ( + bc 0 bb 5 6+ V% 0b5 ( = ( G s a s s bc 0 bb 5 6+ V% 0b5, V% 0b5( + ( b+ P ( s, (67 a is a omaizatio costat ad it was detemied by the coditio ( s ds =, ad

10 568 M. Eshghi E k + M b = ( k+ H( k+ H, b5 = Ek M C ps, b6 =. (68 Cocusio I this wok, we have bee obtaied aayticay the appoximate eegy euatio ad the coespodig wave fuctios of the Diac euatio fo the Hua potetia couped icudig a Couomb-ike teso ude the coditio of the spi ad pseudo-spi symmeties usig the paametic geeaizatio of the NU method. The eegy eigevaues euatio ad the coespodig uomaized wave fuctios have bee obtaied i tems of the Jacobi poyomias. Ackowedgmet: The autho woud ike to thak the kid my wife Ms. Seyedeh Sogha Moosavi fo thei positive suggestios which have impoved the peset wok. Refeeces [] H. Egifes ad R. Seve, Boud states of the Diac euatio fo PT-symmetic geeaized Huthe potetia by Nikifoov-Uvaov method, Phys. Lett. A 3 (005, 7-6. [] W.C. Qiag, R.S. zhou ad Y. Gao, Supesymmety ad SWKB appoach to the Diac euatio with a couomb potetia, Phys. Lett. A 333 (00, 8-5. [3] H. Motavai, Boud state soutios of the Diac euatio fo the Scaf-type potetia usig Nikifoov-Uvaov method, Mod. Phys. Lett. A (009, [] S.M. Ikhdai ad R. Seve, Soutios of the spatiay-depedet mass Diac euatio with the spi ad pseudo-spi symmety fo the couomb-ike potetia, App. Math. comp. 6 (00, [5] M.R. Setae ad Z. Nazai, Soutio of Diac euatios with five-paamete expoet-type potetia, Acta Phys. Pooica B 0 (009, [6] L.H. zhag, X.P. Li ad C.S. Jia, Aaytica appoximatio to the soutio of the Diac euatio with the Eckat potetia icudig the spi-obit coupig tem, Phys. Lett. A 37 (008, [7] F.Zhou, Y. Wu ad J.Y. Guo, Soutio of Diac euatio fo Makaov potetia with pseudo-spi symmety, Commu. Theo. Phys. (beijibg, chia 5 (009, [8] O. Aydogdu ad R. Seve, Exact soutio of the Diac euatio with the Mie-type potetia ude the pseudo-spi ad spi symmety imit, A. Phys. 35 (00,

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