Schrödinger Equation Via Laplace-Beltrami Operator

Transcription

1 IOSR Jourl of Mthemtics (IOSR-JM) e-issn: , p-issn: X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu, Mehmet Açil (Mthemtics, Yüzücü Yıl Uiversity, Turkey) Correspodig Author: Esi İ Eskitşçioğlu Abstrct: I this work firstly we cosider Schrödiger differetil equtio which stisfies to fid wve fuctios tht re importt for microscopic systems. The we geerlize these equtios by usig Lplce- Beltrmi opertor d compre results we obtied with solutios of Schrödiger differetil equtios. Keywords: Differetil Equtio, Bessel Fuctio, Potetil Dte of Submissio: 5--7 Dte of cceptce: I. Itroductio As we kow, i the qutum mechics microscopic systems re lyzed by mes of wve fuctio tr, will be defied which is obtied by solvig Schrödiger tr, h i t, r V r t, r () t m for some potetil V r where m, d i re mss, Plck costt d the squre root of (-) respectively []. I fct, differetil equtios which stisfy coditios of the foudtio of qutum mechics my be foud. But we will cosider Schrödiger equtio becuse of the fct tht this equtio is comptible with experimets. Schrödiger equtio c be expressed s follows: tr, i Ht, r t Where H V r m is clled Hmiltoi opertor []. Note tht we look for wve fuctios such tht member of { t, r L : D is cotious }. Now we defie ew Hmiltoi H V r m by mes of Lplce-Beltrmi opertor L. Here fuctio []. I this cse for modified Schrödiger equtio we hve tr, i H t, r. t By usig seprtio of vribles we obti time-depedet term T s T e pr where i t h p r is oegtive like Schrödiger equtio. Here vlues which will be determied lter is ot eergy vlues s we will see i the lst prt of the pper d it will be used to fid eigefuctio from below. H r r () Equtio () is idepedet of time modified Schrödiger equtio. Whe cosiderig equtio (), it is esy to see tht opertor H is Hermiti opertor. So if we obti rel eigevlues, the we c sy tht eigefuctios correspodig to differet from orthoorml set with i dditio to usig ormliztio. DOI:.979/ Pge

2 Schrödiger Equtio Vi Lplce-Beltrmi Opertor Here we first try to determie eigevlues of (). To do this oe my be expected to solve modified Schrödiger equtio, but becuse of the fct tht to obti both solutios of modified Schrödiger equtio d eigevlues from these solutios is ot esy lwys. Therefore we use differet tretmet. This icludes the followig equivlet H r E r (3) i modified Schrödiger equtio where E is eergy vlue. I this cse, H. V m h. V m m h h V.. m m I the secod term bove coefficiet of is like the right side of (3). Our im is to solve () by usig (3). Geerlly, we my eed to use perturbtio theory but eve some physicl cse like the ifiity squre well my ot eed. I cses like Coulomb potetil or Newtoi potetil we obti ew potetil like iverse squre potetil by usig r. This potetil hve cosidered i [3,4,5].So we c pproch eigevlues of () by usig results we obtied. Cosequetly, we c fid eigefuctios correspodig eigevlues we obti d compre these fuctios with fuctios foud from (3). Here i order to void to use perturbtio theory, we cosider the ifiite squre well..the Ifiity Squre Well potetil d, x Vx, other cses p x x x Asi kx i oe dimesio. So solutio of (3) is (4) me where A d k, so s follows: h d d H. m dx dx By usig (3) d (4) the equtio becomes d dx dx A k si kx dx d d where d we hve E m m h x x A k e e x si kx dx Ak si kx x k x k k 4 k kx k 4 x 4 k k 3 4 k 3 x 3 k 3 cos Cosider Modified Schrödiger equtio is DOI:.979/ Pge

3 Schrödiger Equtio Vi Lplce-Beltrmi Opertor s solutios. Now for both to show the cosistecy of opertio d to determie eigevlues, we look for Ak 6 k cse. Whe goes to, si 3 kx A si kx. Hece k or E. m grphics d coditio of cotiuous. Now by usig tretmet These vlues of bsolutely pproch, we c see tht from both E if we cosider modified Schrödiger equtio h d d x, m dx m dx we hve solutios with the idex s x BesselI, i m xc BesselK, i m xc. BesselK, z give the modified Bessel fuctio of first kid I z d secod where BesselI, z d kid K z respectively. But the secod term described bove pproch ifiity s x goes to for big Therefore we tke C. Cosequetly, we obti C x C I i m x, C I fct, we kow tht Bessel fuctio I hs oscilltio o the x -xes so hve ifiity roots. If we c fid these roots esily we c determie truth vlues of from I i m. I order to determie s C we c use ormliztio, such follows: From dx dx Filly, we hve C IO i m dx. C I dx i x I dx I x d. d compre these solutios with for some vlues of, or more explicitly from DOI:.979/ Pge

4 Schrödiger Equtio Vi Lplce-Beltrmi Opertor As we see whe icreses, curves pproch ech other more d more. Therefore the expecttio vlues d other will be computed for tx, for computtio, we c do this by usig the fuctio. I fct, eve though we eed geerl wve fuctio x x dx 3.4, x x dx 5 for smll. So for,,, We c see tht there is big differece betwee d. Actully, our im is ot to obti close vlues. If so we c tke s more smll. Ad stdrd devitios re s follows: 8.6 d 8. x x From here we c sy tht fuctios hve erly sme dispersio s we will see from figure. It is cler tht vlues foud bove re more close ech other for lrger.we see lso tht from the followig tble 5 x x x x DOI:.979/ Pge

5 Schrödiger Equtio Vi Lplce-Beltrmi Opertor Refereces []. Chrles Z. Mrti, Vritiol formuls for the Gree fuctio. Al. Mth. Phys.:893, []. Dvid J. Griffiths, Itroductio to qutum mechics(secod editio), Perso Eductio Ic.,5 [3]. E. A. Guggeheim, The iverse squre potetil field, Proc.Phys.Soc.;966, Vol.89 [4]. Elise Guillumi-Esp et l., Clssicl d qutum dymics i iverse squre potetil. Jourl of mthemticl physics 55,359, 4 [5]. Wolfgg Bieteholz et l., Commet o '' the two-dimesiol motio of prticle i iverse squre potetil: clssicl d qutum spect'', Jourl of mthemticl physics 56, 4,5 Esi İ Eskitşçioğlu."Schrödiger Equtio Vi Lplce-Beltrmi Opertor." IOSR Jourl of Mthemtics (IOSR-JM) 3.6 (7): DOI:.979/ Pge