Exact scattering and bound states solutions for novel hyperbolic potentials with inverse square singularity

Transcription

1 Exact scatterig ad boud states solutios for ovel hyperbolic potetials with iverse square sigularity A. D. Alhaidari Saudi Ceter for Theoretical Physics, P. O. Box 37, Jeddah 38, Saudi Arabia Abstract: We use the Tridiagoal Represetatio Approach (TRA) to obtai exact scatterig ad boud states solutios of the Schrödiger equatio for short-rage iverse-square sigular hyperbolic potetials. The solutios are series of square itegrable fuctios writte i terms of the Jacobi polyomial with the Wilso polyomial as expasio coefficiets. The series is fiite for the discrete boud states ad ifiite, but bouded, for the cotiuum scatterig states. PACS: 3.65.Ge, 3.65.Nk, 3.65.Fd, 3.65.Ca Keywords: sigular hyperbolic potetials, phase shit, eergy spectrum, tridiagoal represetatios, orthogoal polyomials, recursio relatio. Itroductio I quatum mechaics, a physical system ad its iteractio with the surroudigs is described by a wavefuctio, which is a solutio of the wave equatio that cotais a potetial fuctios that models the structure ad dyamics of the system. Therefore, solutios of the wave equatio (e.g., the Schrödiger, the Dirac, etc.) with as may potetial models as possible remaied oe of the prime iterest sice the early iceptio of quatum mechaics. Several methods for obtaiig exact solutios of the wave equatio were itroduced. These iclude, but ot limited to, supersymmetry, shape ivariace, group theory, operator algebra, factorizatio, path itegral, poit caoical trasformatio, asymptotic iteratio, etc. Some of these are equivalet to each other but most address the same class of exactly solvable potetials. Recetly, the Tridiagoal Represetatio Approach (TRA) was developed to hadle a larger class of exactly solvable potetials. It is a algebraic approach ispired by the J-matrix method [] ad based o the theory of orthogoal polyomials ad their close associatio with tridiagoal matrices. A recet short review of the TRA is foud i []. I this work, we use the TRA to obtai exact solutios for the oe-dimesioal timeidepedet Schrödiger equatio for ovel short-rage iverse-square sigular hyperbolic potetials. I additio to the wavefuctio, we write explicitly the discrete boud states eergy spectrum ad the cotiuous eergy scatterig phase shift. I sectio, we formulate the problem withi the TRA ad i sectio 3, we obtai the eergy spectrum ad phase shift.. TRA formulatio of the problem I the atomic uits m, the time-idepedet oe-dimesioal Schrödiger equatio for a poit particle of mass m uder the ifluece of a potetial V(r) reads as follows d V() r E () r dr, ()

2 where r, E is the particle eergy ad () r its wavefuctio. For boud states, the wave fuctio vaishes at the origi ad at ifiity, whereas for scatterig states it is oscillatory at ifiity with a bouded amplitude. Now, we make a trasformatio to a dimesioless coordiate x() r cosh( r), where is a real positive scale parameter. Thus, x ad Eq. () is mapped ito the followig secod order differetial equatio i terms of x ( ) d d x x U( x) r( x) dx dx, () where E ad U( x) V( r). I accordace with the TRA, we search for a complete set of square itegrable basis ( x ) to expad the wavefuctio i a series as () r f ( x), where f are the expasio coefficiets. Additioally, we require that the basis carries a tridiagoal matrix represetatio for the wave operator J d V() r E. dr Cosequetly, the matrix wave equatio becomes a three-term recursio relatio for f. The basis elemets ( x ) are writte i terms of classical orthogoal polyomials whose argumet is compatible with the rage of the cofiguratio space coordiate (i.e., x ). Moreover, the differetial equatio of these polyomials must have the same structure of the part px ( ) d qx ( ) d dx dx as that of the differetial wave equatio (). The properties of the Jacobi polyomial show i Appedix A suggest that we ca choose the followig set of fuctios as basis elemets where, N, (3) (, ) ( x) c( x ) ( x ) P ( x),,,,..., N ad N is a o-egative iteger. Therefore, must be egative whereas the ormalizatio costat is suggested by the orthogoality si ( ) ( ) ( ) relatio (A5) as c si ( ) ( ). The parameters ad will be determied below by the tridiagoal represetatio requiremet. I terms of the variable x, the wave operator J is writte usig Eq. () as follows d d J ( x ) x U( x) dx dx. () Usig the differetial equatio of the Jacobi polyomial (A), we obtai the followig actio of the wave operator o the basis elemets (3) d J( x) c( x) ( x) x dx ( ) ( ) (, ) ( ) U( x) P ( x) x x (5)

3 Usig the differetial property of the Jacobi polyomials (A), this actio becomes J( x) c( x) ( x) x x ( ) ( )( ) ( ) P P P ( )( ) ( )( ) ( )( ) (, ) (, ) (, ) ( ) ( ) (, ) ( ) U( x) P x x (6) To produce a tridiagoal represetatio, the right side of this equatio must be a combiatio of ( x ) ad ( ) x with costat (x-idepedet) factors. The recursio relatio of the Jacobi polyomials (A3) shows that (6) will cotai terms iside the curly brackets proportioal to P (, ) ad P with costat factors oly i oe of three cases: (, ) (a) ( ) ad ( x ) U( x) ABx. (7a) x (b) ( ) ad ( x ) U( x) ABx. (7b) x (c), ( ) ( ) ad U ( x ) x x A Bx. (7c) where A ad B are arbitrary real dimesioless costats. The terms with ad o the right side of the equatios for U(x) are eeded to cacel the correspodig terms i Eq. (6) that destroy the tridiagoal structure. Cosequetly, the potetial fuctio correspodig to the case (7a) read as follows: V V V V V() r, (8a) sih ( r) cosh( r) sih ( r) cosh ( r ) where V, A V ad to make the potetial vaish at ifiity we took B. This is a short-rage sigular potetial with sigularity r at the origi, which is comig from the first term oly ad with stregth V. Reality requires that V. This potetial ca support scatterig states, ad if V V 8 the it ca also support boud states as we shall see i the followig sectio. The basis parameter, which must be egative, is determied by the umber of boud states, which is less tha or equal to the basis size N ad it is costraied by the coditio that N. I Appedix B, we show that the symmetry of the three-term recursio relatio that results from (6) gives 3. O the other had, the potetial fuctio associated with the case (7b) is: V V V V V() r, (8b) sih ( r) cosh( r) sih ( r) sih ( r ) where V, A V take B. This is also a short-rage sigular potetial with a sigularity ad to make the potetial vaish at ifiity we had to 3 r at the origi of

4 stregth ( V V ). Similarly, reality dictates that V. This potetial ca also support scatterig states, ad if V V 8 the it ca support boud states as well. I fact, V V 8 is the critical sigularity stregth for this iverse square potetial below which quatum aomalies appear [3-]. The basis parameter, which must be greater tha, is determied by the umber of boud states ad it is costraied by the coditio that N. I Appedix B, we show that the symmetry of the three-term recursio relatio resultig from (6) gives 3. Fially, the potetial fuctio for the last case (7c) has a richer structure ad reads as follows: V V cosh( r) Vr () Vcosh( r) V, (8c) sih ( r) where ( ) V V, ( ) V V, A V ad B V. Reality i this case requires that V V 8. Without the V term, this potetial becomes the wellkow hyperbolic Rose-Morse potetial, which has a well-established exact solutio. Moreover, usig the idetities: cosh x cosh x sih x, sih x (cosh x)(sih x) ad cosh x sih x where W V V we ca show that potetial (8c) is idetical to W W Vr () V cosh( r) V, (8d) sih ( r ) cosh ( r ). The first two terms are the hyperbolic Pöschl-Teller potetial with half the argumet. Now, potetial (8d) has bee treated recetly by Assi, Bahlouli ad Hamda usig the TRA [5]. Therefore, we will ot ivestigate this potetial here but refer the iterested reader to the cited work. Note that the potetial fuctio (8b) is obtaied from (8a) by the map x x ad V V. Moreover, the associated bases (3) are obtaied from each other by the additioal parameter exchage ad alog with x x. We will use this exchage symmetry below to ecoomize o calculatio ad search oly for the solutio of the problem with the potetial (8a). Applyig the said map to this solutio will produce the other solutio associated with the potetial (8b). Hece, the total map becomes as follows x x, V V,,. (9) 3. TRA solutio of the problem To obtai the exact solutio of the problem, we eed to idetify all igrediets i the wave fuctio () r f () x. Sice the basis elemets ( x ) as give by Eq. (3) are ow fully determied as show above, we oly eed to fid a exact realizatio for the expasio coefficiets f. To do that, we substitute each of the two potetial fuctios U(x) alog with their associated parameters ito Eq. (6). Due to the tridiagoal requiremet, the matrix wave

5 equatio J becomes a three-term recursio relatio for the expasio coefficiets that will be solved exactly i terms of orthogoal polyomials. I Appedix B, we obtai these symmetric three-term recursio relatios associated with the potetials (8a) ad (8b). We fid that the two resultig recurrece relatios, (B5) ad (B6), are equivalet to each other uder the parameter map (B7), which is equivalet to the map (9) above. Therefore, we cosider oly oe of them, say (B5) associated with the potetial (8a). We idetify the orthogoal polyomial associated with the recurrece relatio (B5) ad use the aalytic properties of this polyomial to write the phase shift for the scatterig states ad eergy spectrum for the boud states. We compare (B5) to the recursio relatio of the ormalized versio of the Wilso polyomial W ( ;,,, ) z a b c d that reads (see Eq. A7 i [6]) ( ab)( ac)( ad)( abcd) ( bc)( bd)( cd) zw a W ( a b c d)( a b c d ) ( a b c d )( a b c d ) ( ab)( cd)( ac)( ad)( bc)( bd)( abcd) W abcd (abcd3)(abcd) ( )( ab)( cd)( ac)( ad)( bc)( bd)( abcd) W ( )( ) a b c d abcd abcd () This ormalized versio of the Wilso polyomial is writte as (see Eq. A6 i [6]) W z a b c d ( ;,,, ) abcd ( ab ) ( ac ) ( ad ) ( abcd ) 3, abcd, a i z, a i z F ( ) ( ) ( )! a b, a c, a d abcd bc bd cd () abcd,,, where F3 z ( a) ( b) ( c) ( d) z efg,, ( e) ( f ) ( g)! ad ( a) ( a) a( a)( a)...( a). It ( a) is also required that Re( abcd,,, ) with o-real parameters occurrig i cojugate pairs ad that z. However, the compariso of (B5) to () shows that either Re( ab, ) or Re( cd, ) ad z is pure imagiary. Thus, we defie a ocovetioal Wilso polyomial, W ( ;,,, ) z abcd, as a polyomial of degree i z with,,,..., N, Re( ab) or Re( ab) N, Re( ac) N ad Re( a d) N. The correspodig three-term recursio relatio for W ( ;,,, ) z a b c d is as follows ( a b)( a c)( a d)( a b c d ) ( b c )( b d )( c d ) zw aw ( a b c d)( a b c d ) ( a b c d )( a b c d ) ( ab)( cd)( ac)( ad)( bc)( bd)( abcd) W abcd (abcd3)(abcd) ( )( ab)( cd)( ac)( ad)( bc)( bd)( abcd) W ( )( ) a b c d abcd abcd () Comparig () to () gives W ( z; abcd,,, ) ( ) W( z; abcd,,, ). Thus, W ( ;,,, ) z abcd, a b c d, a za, z will be writte i terms of the hypergeometric fuctio F3 ab, ac, a d. Moreover, comparig (B5) to () gives f as the ocovetioal Wilso polyomials modulo a overall factor idepedet of. Sice W, the we ca write f fw. 5

6 Additioally, the compariso gives the polyomial parameters ad argumet i terms of the physical parameters as follows ab i cd i z V V (3) 6 where, i the compariso process, we have used the followig idetity, which is valid for all ad real parameters,, Therefore, we obtai f (, V, V) f(, V, V) W z ; a, a, c, c (). Normalizability of the wave fuctio for boud states ad boudedess of scatterig states makes the positive defiite weight fuctio for W ( ;,,, ) z abcd equal to [ f(, V, V )] [7-8]. The eergy spectrum of the discrete boud states for the potetial (8a) is obtaied from the spectrum formula of W ( ;,,, ) z abcd, which is obtaied i tur from the spectrum formula of the covetioal Wilso polyomial by the map z z. Now, the spectrum formula of the Wilso polyomial is obtaied from its asymptotics ( ) ad is give by formula (C) i Appedix C of Ref. [8]. That formula uder the map z z becomes z ( k a) or z ( k c) depedig o whether a or c equals to i, givig k V VV k A k (5) ( ) where k,,.., kmax ad k max is the largest iteger less tha or equal to A. O the other had, we ca evaluate the eergy spectrum idepedetly usig a umerical procedure that starts by writig the recurrece relatio (B5) as the matrix eigevalue equatio T f R f, where T is the tridiagoal matrix obtaied by settig i (B5) ad R is the tridiagoal matrix multiplyig i (B5). The basis parameters costrait is satisfied by the assigmet V. O the other had, the basis parameter is still arbitrary but must have a upper boud as max N. Therefore, we vary the value of betwee mi ad max util a plateau of computatioal stability of the eergy spectrum is reached for a coveietly chose accuracy. We give the results of such computatio i Table showig the rate of covergece of the eergy spectrum as the size of the basis icreases. The Table also demostrates a excellet agreemet with the exact results obtaied from the eergy spectrum formula (5). We foud that the plateau (rage of values of with o sigificat chage i the 6

7 result withi the chose accuracy) is larger for lower boud states, which is typical for such calculatios. Specifically, we foud that the plateau of computatioal stability i the basis parameter for the Table is withi the rage 3 N, N ad the values give i the Table correspod to the middle of the plateau. Figure is a plot of the boud states wavefuctios correspodig to Table, which are computed usig the followig series k ( r) sih( r) sih( r ) cosh( r ) f(, V, V ) k (, ) cw (6) z ; aa,, cc, P cosh( r) Now, the phase shift for the cotiuum scatterig states is obtaied from the asymptotics ( ) of W ( ;,,, ) z a b c d, which is obtaied i tur from the asymptotics of the covetioal Wilso polyomial ( ;,,, ) W z abcd by the map z z (i.e., z iz). Now, the phase shift for the Wilso polyomial is give by formula (C) i Appedix C of Ref. [8]. Uder the map z z that formula becomes ( E) arg z i (7) Mathematically, this is a o-trivial result because the usual asymptotics of orthogoal polyomials is take as while keepig the argumet of the polyomial ad its parameters fiite. However, i this case two out of the four parameters of the polyomial also ted to ifiity as. As see from (3), those are the parameters that are equal to sice it is required that N while N.. Coclusio I this work, we used the Tridiagoal Represetatio Approach to obtai the exact solutio of the oe-dimesioal Schrödiger equatio for a short-rage iverse-square sigular hyperbolic potetial. The boud (scatterig) state wavefuctio is writte as a fiite (ifiite) sum of square itegrable basis fuctios i cofiguratio space. The basis fuctios are writte i terms of the Jacobi polyomials ad chose such that the matrix represetatio of the wave operator is tridiagoal ad symmetric. Cosequetly, the matrix wave equatio becomes a symmetric three-term recursio relatio for the expasio coefficiets of the wavefuctio, which is solved i terms of a modified versio of the Wilso polyomial. Usig the asymptotics of this polyomial we were able to obtai aalytic closed form expressios for the discrete boud states eergy spectrum ad the cotiuous eergy scatterig states phase shift. This study is costitute aother demostratio of the advatage of usig the TRA as a viable alterative method for the solutio of quatum mechaical problems. Ackowledgemets This work is partially supported by the Saudi Ceter for Theoretical Physics (SCTP). We are grateful to Prof. H. Bahlouli for fruitful discussios. 7

8 Appedix A: The Jacobi polyomial For ease of referece, we list here the basic properties of the versio of the Jacobi polyomial that we used i this work. It is defied i the usual way, as follows (, ) ( ) (, ) ( ) ( ) ( ) (, ; ; x P x F ) ( ) P ( x). (A) However, here,,,..., N, ad N for x. It satisfies the followig differetial equatio d d (, ) x x P ( x) dx, (A) dx It also satisfies the followig three-term recursio relatio xp ( )() (, ) (, ) P ( )( ) ( )( ) P P ( )() ()() (, ) (, ) (A3) ad the followig differetial relatio d ( ) ( x ) P P dx ( )( ) (, ) (, ) ( )( ) ( ) ( )( ) ( )( ) (, ) (, ) P P (A) The associated orthogoality relatio reads as follows [9] (, ) (, ) ( ) ( ) ( x ) ( x ) P ( ) ( ) si x Pm x dx ( ) ( ) si ( ) m where m,,,,..., N. Equivaletly,, (A5) (, ) (, ) ( ) ( ) ( ) ( ) ( x) ( x) P ( x) Pm ( x) dx ( ) ( ) ( ) m. (A5) Appedix B: Three-term recursio relatios for the expasio coefficiets of the wavefuctio Substitutig the potetial fuctio (8a) alog with its associated parameters ito (6) ad defiig the parameter q, we obtai 8

9 J( x) c( x) ( x) q ( ) ( )( ) ( ) P P P ( )( ) ( )( ) ( )( ) q A ( x ) ( ) q (, ) P (, ) (, ) (, ) (B) Usig the three-term recursio relatio of the Jacobi polyomial (A3) ad writig the result i terms of the basis fuctios ( x ), we get after some maipulatios the followig J( x) F qd ( x) F q D( x) x q A q FC ( x) ( )( ) (B) where C ( )( ), D ( )( )( )( ) ( )( 3) ad F ( ) q. Now, sice the wave operator J is Hermitia the its real matrix represetatio must by symmetric. Therefore, the term that multiplies ( ) x i Eq. (B) must be idetical to the oe that multiplies ( ) x but with the replacemet. This is true oly if q makig 3 ad mappig Eq. (B) ito the followig J ( x) D ( x) D ( x) x C A ( x) (B3) where we have used the idetity C. (B) Now, we isert the wavefuctio series f i the wave equatio J ad use (B3) to obtai the followig symmetric three-term recursio relatio for the expasio coefficiets of the wavefuctio A f C f D f Df (B5) 9

10 The potetial parameter A V is show explicitly whereas the other parameter V is implicit i as V. Repeatig the same calculatio for the potetial (8b), we fid that 3 ad ed up with the followig symmetric three-term recursio relatio for the expasio coefficiets of the wave fuctio A f C f D f Df (B6) where here A V ad V. This relatio could, i fact, be obtaied from the recursio relatio (B5) associated with the potetial (8a) by the followig map, A A, f ( ) f. (B7) The first two parts of this map are equivalet to the potetial parameter exchage V V. The last part of the map has its origi i the potetial map (9) give at the ed of sectio where x x. Usig this map ad the parameter exchage, the property of the Jacobi polyomial (A) that reads map (B7). (, ) (, ) P ( x) ( ) P ( x) is the reaso behid the last part of the Refereces [] A. D. Alhaidari, E. J. Heller, H. A. Yamai, ad M. S. Abdelmoem (Editors), The J- Matrix Method: Developmets ad Applicatios (Spriger, Netherlads, 9). [] A. D. Alhaidari, Solutio of the orelativistic wave equatio usig the tridiagoal represetatio approach, J. Math. Phys. 58 (7) 7 [3] S. A. Coo ad B. R. Holstei, Aomalies i quatum mechaics: the /r^ potetial, Am. J. Phys. 7 () 53 [] A. M. Essi ad D. J. Griffiths, Quatum mechaics of the /x^ potetial, Am. J. Phys. 7 (6) 9 [5] I. A. Assi, H. Bahlouli ad A. Hamdad, Exact solvability of two ew 3D ad D orelativistic potetials withi the TRA framework, Mod. Phys. Lett. A (8) i productio [6] A. D. Alhaidari ad T. J. Taiwo, Wilso-Racah Quatum System, J. Math. Phys. 58 (7) [7] A. D. Alhaidari ad M. E. H. Ismail, Quatum mechaics without potetial fuctio, J. Math. Phys. 56 (5) 77 [8] A. D. Alhaidari, Quatum mechaics with orthogoal polyomials, arxiv:79.765v [quat-ph] [9] M-P Che ad H. M. Srivastava, Orthogoality Relatios ad Geeratig Fuctios for Jacobi Polyomials ad Related Hypergeometric Fuctios, Appl. Math. Comput. 68 (995) 53

11 Table Captio Table : The complete fiite boud states eergy spectrum (i uits of (8a) with the parameter values V ad V 8 (i uits of ) for the potetial ). The Table shows the rate of covergece of the calculatio with the basis size N. It also demostrates very good agreemet with the exact spectrum give by Eq. (5). Table N = N = 3 N = 5 N = Exact Figure Captio Fig. : The u-ormalized boud states wavefuctios associated with the potetial parameters i Table. The horizotal axis is the cofiguratio space coordiate, which is measured i uits of. Fig.