Quantum Schrödinger Equation with Octic Potential In Non Commutaive Two-Dimensional Complex space. Maireche Abdelmadjid

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1 Life Science Jounal 4;() Quantu Schödinge quation with Octic Potential In Non Coutaive Two-Diensional Coplex space Maieche Abdeladjid Physics Depatent, Sciences Faculty, Univesity of M sila Algeia abaieche@yahoo.f Abstact: In this wok, The effect of the non coutativity is studied, on the with Octic potential (Haonic, 4 8 Quadatic, sextic...) V a b c d e, by applying the Boopp's shift ethod to fist ode in the non-coutativity paaete, we shown that the Hailtonian will be epesented into two atixes. The odified of the enegies levels at the fundaental state deteined only with a nueical solution. One can conclude fo this wok that the non-coutativity applying on the Octic potential, poduced two types of inteactions between a paticle with spin / an extenal agnetic field, the fist one epesent the odinay Zeean effect (anisotopic inteaction), the new inteaction epesent a cobbling between the total onuent extenal agnetic field (Isotopic inteaction).. [Maieche Abdeladjid. Quantu Schödinge quation with Octic Potential In Non Coutaive Two- Diensional Coplex space. Life Sci J 4; ():5-59] (ISSN: 97-85).. 49 Keywods: Hydogen ato, Sta poduct, coplex space, the Octic potential noncoutative space. Pacs nubes:..nx,.-,.5-w -Intoduction: The divegence poble in quantu field theoy (QFT), unified of gavitational inteaction with stad odel,ae a fundaental easons to intoduce a new concepts of space-tie, known by noncoutative spaces,,5,9,,,5,, the noncoutativity is intoduced by any ways, the siple appoach, it conside the position oentu opeatos obeys to the Heisenbeg coutation elation, that is siilaly to quantie space-tie coodinates 9, when the coutato x i, x j, the atheatical foalis of sta poduct, Boopp's shift ethod the Seibeg-Witten ap ae plays a fundaental oles in this new theoy. The ich atheatical stuctue of the noncoutative theoy gives a ise to the hop to get a bette undesting of physics phenoena at sallness distances to solve above-entioned pobles. The physics idea of a noncoutative space satisfied by anew atheatical poduct which eplaces the old odinay poduct known by sta poduct, noted by [, 7, 8,9,,,, 5, ]: x i, x j xi * x j x j * xi i * ij () Thoughout this pape the natual unites c ae eployed. A Boopp's shift ethod will be used in ou pape, Instead of solving the non coutative Schödinge equation by using stat poduct pocedue: x i, x j x ix j x jx i i ij () We eplace the sta poduct with usual poduct togethe with a Boopp's shift 5,, 7, 8,9, : x i x i ij p j p i p i whee i,n.() ij The paaete is an antisyetic eal atix of diension squae length in the noncoutative canonical-type space. The sta poduct between two abitay functions f x g x, in the fist ode of, as follow [,7,8,9,,,,5, ]: i ij f x gx f x. gx i f x j gx (4) We eaked that the esult of equation () satisfied by applying the notion of the sta poduct epesented by eq.(4). Actually thee ae any attepts to study noncoutative space tie in thee diensional space, but a liited physics phenoenon s ae studied in two diensional space, the hydogen ato consideed a ideal odel in physics, we want to study this ato in two diensional (D) space, by applying the new concepts of space, based on the coplex 5

2 Life Science Jounal 4;() coodinate p ˆ ˆ xˆ iyˆ coesponding oentus p ip /,7 x y.the ai to this wok, is to study an to discove the physical effect of the noncoutativity two diensional coplex space of the Octic potential. This pape is oganied as follows. In section we pesent the Coplex non coutative space. In section, we pesent Octic potential in odinay two diensional coplexes. In section 4 we deive the defoed Hydogen ato with Octic potential in noncoutative coplex space. We solve this equation obtain the non-coutative odification of the enegy levels. Finally, in section 5, we daw ou conclusions. -Coplex non-coutative two diensional spaces: In D space, the coplex coodinates syste p,, defined by, thei oentus p,7 : x iy, x iy p p x ip y, p p p x ip y.(5) The two diensional coplex space, foulated by the following opeatos coodinates thei oentus,, p p, as follows: xiŷ i xy p x ip y ip xiŷ i xy p x ip y i xy p p p, p p.() One can show that, the squae of the position opeato, in D coplex space, can be deteined, fo two possibility ethods, the fist one, using the algeba (), one show that, in the fist ode of the paaete : L.(7) While the second values of, afte a staightfowad calculation, one get this values, in the fist ode of the paaete : L (8) Futheoe, one can show that the two coutatos,,p expessed as, afte a staightfowad calculation:,, p i Fo a kinetic te p (9) D Coplex space, as follows: px ip y px ip y te, defined into odinay p ip p ip x y x y () Whee is the est asses, which take the sae values, in D coplex space: p p p p p..() This esults satisfied because the non coutativity of space-tie epesented by eq.() iposed that pi p i. -The Octic potential in odinay two diensional coplex spaces: We expessed the Schödinge equation coesponding the Octic potential V in odinay two diensional spaces, in the pola coodinate, 5,, 7 : V.() Whee is Laplacien in a pola coodinate in D odinay space: 4 8 V a b c d e..() The paaete d, the noalied coplete V descibed Octic potential wave eq. will be witing to the fo: R..(4) Whee R ae the adial function the angle function satisfied the following fo espectively: 54

3 d d V d Life Science Jounal 4;() 4 R.. (5) d Whee is the enegy eigenvalues is the obital angula oentu quantu nubes, the stad solution of given by : exp i whee,,... The adial function state by7 : R exp p Whee.() R deteined fo the gound p given by: (7) 4 p e k ln..( 8) Whee e c, e d e a, 5.(9) Then, the adial function the fundaental state e k e b R the enegy of 7 ae espectively : k 4 N exp R 4 k d4ce ( 8e ) Whee N is the noalied constant deteined fo the noaliation elation: R d () 4-The Octic potential in two diensional coplex spaces: The Hailtonian opeato associated with Octic potential H in the D coplex space, deteined fo the elation: H p p V..() Whee V is the opeato of Octic potential in D coplex space. The Schödinge equation n D coplex space: V () Whee is coplete wave function, the kinetic te p p eplaced by, while is the eigenvalues of enegy associated with Octic potential. With a Boopp's shift ethod, the above eq. will be, as follows: V x i x i ij p j R (4) V x Whee i x i ij p j, is the odinay potential in D space, as a function of x i x i ij p j instead of odinay position x i, then, the Octic potential V, will be witten to the fo: V a b 4 c d 8 e..(5) The adial equation the angle function in D coplex space satisfied the following fo espectively: d d d V d 4 R.() we have seen that in section the position opeato in two diensional D coplex space have two possible values, the fist one is L, the diffeent tes of Octic potential V ae: a a al b b 4 b L c c c 4 L d 8 d 8 4d L e e 5e 8 L (7) Which follow to wite the Octic potential as follows: V VV pet...(8) 55

4 Life Science Jounal 4;() Whee V V pet Is given by: b 4 8 a c 4d 5e L pet.(9) We eaked that the te V pet is popotional to the sallness paaete, then we consides as a petubaive te. Know using the second values of opeato L to get the tes of Octic potential: a a al b b 4 b L c c c 4 L d 8 d 8 4d L e e 5e 8 L Winch follow to wite:.() V VV pet.. ( ) Whee V pet Is given by: b V.() Also we eaked that, the te 4 8 a c 4d 5e L pet V pet is conside a petubaive te. A staightfowad calculation leads to get the two adials functions coesponding V pet as follow: V pet d 4 V pet d d V 4 pet d..() We obseved that the two opeatos V R V R V pet V pet ae popotional to L espectively. Its ipotant to notice that will be ewite to the two L L L fos L s L s which ae coesponding a paticle with spin up spin down espectively, the odification to the enegy associate with spin up spin down at fist ode of, obtained by the applying the petubation theoy: n fl n p V pet p ds n fl n p V pet p ds Whee..(4) ds dd instead of dxdyd, the new adial function f, defined as follows: b 4 8 f a c 4d 5e.(5) The non-coutative odification of the enegy levels associated with spin up at the fist ode of coesponding the gound state deteined by using equation two eqs. () (4):.() This can be witten as follow: k 4 N exp f d 4 N at bt ct 4dT 5 (7) Whee et T, T, T, 4 T 5 T ae given by : k 4 T exp d k5 4 T exp d k5 4 T exp d..(8) 4 k7 4 T exp d 5 k9 4 T exp d Concening the non-coutative odification of the enegy levels associated with spin down at the fist ode of coesponding the gound state deteined by using equation two eqs. k 4 4 N exp 4 : f d 5 5

5 Life Science Jounal 4;()..(9) This can be witten as follow: 4 N at bt ct 4dT et.(4) It is ipotant to notice that, the tes: T, T, T, 4 5 T T epesented by equation (8), ipossible to deteined, because the integal fo: k 4 exp d is vey difficult task, as its obseved in page: at efeence, the only solution to this poble will be nueically. Know we suaie the obtained esults of the enegy of fundaental state associated with spin up spin down paticle espectively:..(4) Winch follows to wite: k d 4ce 4 5 N at bt ct 4dT 5eT 8e k d 4ce N 4 5 at bt ct 4dT 5eT 8e..(4) Regading the two eqs. (9) (), one can show that, the coesponds two Hailtonian opeatos H H espectively, deteined fo the elation: H pp VfL sh whee s / spin down H pp VfL sh whee s / spin down..(4) These esults confied that, the Hailtonian H, in D coplex space will be epesented by a diagonal atix of ode ( ), as follow: H Ĥ Ĥ (44) The explicit fo of above diagonal atix deteined fo: H ppv fl s / ppv fl s /.(45) Regading to the eq.(4), the eigenvalues of enegies ae eal, then the Hailtonian H is Heitian, epesented a paticle with spin ( / ) in a unifo extenal agnetic field. The atix (45) can be ewitten as follows: H H o H ag.(4) Whee H o, is epesented by a following atix, as follows: H o p p VI.(47) Whee the united identity atix is I in D space, H ag is epesented by a following atix, as follows: H ag f L s / L s /..(48) Futheoe, on based to the efeences 8, 9,,, one can wite: L L s L L s.(49) J B sb J B sb if if s s Whee B Bk (the vecto of a agnetic field, which oiented with (O.) axes), B ( is a popotional constant), ( s is the agnetic oent, as its eaked in the intoduction, s, then ), one can wite the Hailtonian epesented the Zeean effect H Zeean as follows : H Zeean B sb..(5) Whee J L s, using two eqs. (49) (5), one can wite the atix H ag as follows: H ag fj BI fh Zeean I.(5) Physically, the atix (47), epesented a paticle with spin ( / ) inteacted exactly with the Octic potential in odinay D space 7, the 57

6 Life Science Jounal 4;() coesponded wave, the enegy ae deteined fo the eq., while the atix (48) epesented two inteactions between a paticle with spin ( / ) a extenal agnetic field, the fist one epesent the odinay Zeean effect the new inteaction epesent a cobbling between the total onuent J extenal agnetic field B. The adial function f, taduced the physics paticulaity of the studied potential. Futheoe, instead to eq.(4) one can wite: Ĥ pp VfL s Ĥ whee s / spin down Ĥ pp VfL s Ĥ whee s / spin up..(5) Which follow to give anothe fo of the second atix to the Hailtonian opeato Ĥ, in D coplex space, as follows: Ĥ H o fl s / Futheoe, one can wite: H o fl s / (5) L L s J B sb if s L if L s J B sb s..(54) The atix (5) epesents the Hailtonian opeato Ĥ : Ĥ Ho fj B s / B Ho fj B s / B (55) Also, afte a staightfowad calculation, one can wite H ag : H ag fh Zee an I fj BI.(5) Which epesented a paticle feionic inteacted with a agnetic field in two types, the fist one is the effect of Zeean the second is a new cobbling between the total onuent J extenal agnetic field B. Futheoe, egading to the two eqs. (5) (5), we obseved that the fist te f H I ) in eq. (5) the fist te ( Zeean (fh Zeean I ) in eq. (5) changed in the values the diection oientation of diection to pojection of spin with B, while, the second agnetic pats ( f J BI eqs. Thus, the fist pat ( f H ZeeanI the fist pat ( f H ZeeanI ) is invaiant in two ) in eq. (5) ) in eq. (5) is consideed an anisotopic inteaction, while the second agnetic pats ( f J BI ) is consideed an isotopic inteaction. The global inteaction between the extenal agnetic field odified the enegy of odinay space by : 4 5 N at bt ct 4dT 5eT 4 5 N at bt ct 4dT 5eT (57) Then, we can talk that, the popety of the noncoutativity of space ceated autoatically two types of the inteactions between the paticle extenal agnetic field B. Its woth to ention that, when the paaete ( ), we get odinay D space esults, as follows: Li H ag H ag.(58) 5-Conclusion: The effect of the non coutativity is studied, on Octic potential by applying the Boopp's shift ethod to fist ode in the non-coutativity paaete, we deived the two Hailtonian atixes Ĥ Ĥ, we show that the odified of the enegies levels at the fundaental state deteined only with a nueical solution. One can conclude fo this wok that the non-coutativity applying on the Octic potential, poduced two types of inteactions between a paticle with spin ( / ) an extenal agnetic field, the fist one epesent the odinay Zeean effect the new inteaction epesent a coupling between the total onuent J extenal agnetic field B. The siple physical liited ( ) poved that ou coect obtained esults. Acknowledgents: This wok was suppoted with seach laboatoy of: Physique et Chiie des atéiaux, in univesity of M'sila, Algeia. 58

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