Sine function with a cosine attitude

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1 Sie fuctio with a cosie attitude A D Alhaidari Shura Coucil, Riyadh, Saudi Arabia AND Physics Departmet, Kig Fahd Uiversity of Petroleum & Mierals, Dhahra 36, Saudi Arabia haidari@mailapsorg We give a revealig expose that addresses a importat issue i scatterig theory of how to costruct two asymptotically siusoidal solutios of the wave equatio with a phase shift usig the same basis havig the same boudary coditios at the origi Aalytic series represetatios of these solutios are obtaied I D, oe of the solutios is a eve fuctio that behaves asymptotically as si(x), whereas the other is a odd fuctio, which is asymptotically cos(x) The latter vaishes at the origi whereas the derivative of the former becomes zero there Elimiatig the lowest N terms of the series makes these fuctios vaishigly small i a iterval aroud the origi whose size icreases with N We employ the tools of the J-matrix method of scatterig i the costructio of these solutios i oe ad three dimesios PACS umbers: 365Nk, 3Gp, 365Fd, 3Sa Keywords: scatterig, asymptotic siusoidal fuctios, J-matrix method, orthogoal polyomials, recursio relatios Itroductio: To study the structure ad dyamics of a subatomic system, physicists perform scatterig experimets I such a experimet, a uiform flux of particles (referred to as probes or projectiles) will be icidet o the target uder study The scattered flux will carry the eeded iformatio about the system I potetial scatterig theories [], a cofiguratio potetial fuctio that models the structure of the system ad its iteractio with the probes is proposed This model will be tested agaist the outcome of the experimet Typically, such a potetial fuctio has a fiite rage such that i the asymptotic regio, where the icidet particles are ejected ad the collected, its value is zero Therefore, the icidet ad scattered particles are free ad, thus, represeted by siusoidal wavefuctios The iformatio about the system is cotaied i the differece betwee the two agles ( phase shift ) of these siusoidal fuctios represetig the icidet ad scattered flux Mathematically, this meas that oe eeds two idepedet siusoidal solutios of the free wave equatio such as si( kx ) ad cos( kx δ ), where x is the relevat cofiguratio space coordiate ad k is the wave umber which is related to the icidet beam eergy as k = E The phase shift δ depeds o the eergy, agular mometum, ad model potetial parameters [] Theoretically, the scatterig problem at steady state is described by the solutio of the time-idepedet eergy eigevalue equatio ( H E) χ =, where H is the Hamiltoia operator ad the eergy E is positive ad cotiuous For a geeral dyamical system, the aalytic solutio of this equatio is ofte very difficult to obtai

2 However, for a large class of problems that model realistic physical systems, the Hamiltoia could be writte as the sum of two compoets: H = H V The referece Hamiltoia H is ofte simpler ad carries a high degree of symmetry It is treated aalytically despite its ifiite rage The potetial V that models the system is ot, but it is usually edowed with either oe of two properties Its cotributio is either very small compared to H or is limited to a fiite regio i cofiguratio or fuctio space Perturbatio techiques are used to give a umerical evaluatio of its cotributio i the former case, whereas algebraic methods are used i the latter As for the scatterig problem uder cosideratio, we will oly be cocered with the latter case where the potetial is assumed to vaish beyod a certai fiite rage Thus, the aalytic problem is cofied to the solutio of the referece wave equatio, ( H E) ψ = As discussed above, the solutio to this problem is of fudametal importace i scatterig sice it will be the carrier of iformatio about the structure ad dyamics of the system Typically, there are two idepedet solutios of this problem Both are siusoidal sice they represet free particle motio due to the fact that i the asymptotic regio V = The phase differece betwee the two siusoidal solutios will cotai the scatterig iformatio To obtai a full solutio of the scatterig problem, oe eeds (beside the kiematical solutio of the referece problem) a umerical evaluatio of the dyamics cotaied i the cotributio of the potetial V I the algebraic methods of scatterig, such a evaluatio is carried out by computig the matrix elemets of these operators i a complete set of square itegrable basis Coditios at the iterface betwee the scatterig ad asymptotic regios dictate that the same basis (or liear combiatios thereof) be used for both asymptotic solutios This implies that these two idepedet asymptotic solutios (the sie ad cosie) will have the same symmetry attributed to the basis of the solutio space ad satisfy the same boudary coditios at the origi For example, we shall see that i D the cosie solutio is required to be a odd fuctio that vaishes at the origi whereas the sie solutio is a eve fuctio with a zero derivative at the origi Moreover, lookig at these fuctios far away from the origi, oe could ever predict correctly this curious ad o-trivial behavior Mathematically, it is itriguig to fid a aalytic realizatio of such fuctios Physically, of course, this is fully acceptable sice these solutios have relevace oly outside the scatterig regio, ad whatever happes to them ear the origi is irrelevat I fact, they are eve allowed to diverge there I this work, we use the J-matrix theory of scatterig [3,4], as a example of a algebraic method, i the costructio of these iterestig referece solutios I the followig sectio, we fid eve ad odd solutio spaces for the referece wave equatio i oe dimesio ad demostrate how to costruct a eve (odd) fuctio that behaves as sie (cosie) with cosie-like (sie-like) properties at the origi These will be give as coverget series ivolvig products of Hermite polyomials ad cofluet hypergeometric fuctios I the Appedix, we also costruct the asymptotic siusoidal solutios of the scatterig problem i three dimesios havig these iterestig properties Those will be give as series of products of Laguerre polyomials ad hypergeometric fuctios To illustrate the captivatig ad itriguig properties of these fuctios, graphical represetatios will be give We will also show that by elimiatig the lowest N terms of the series, these fuctios become vaishigly small i a iterval aroud the origi whose size icreases with N

3 The oe-dimesioal case: Oe-dimesioal orelativistic steady flux scatterig is represeted by the solutio of the followig time-idepedet Schrödiger equatio d V( x) χ( x, E) = Eχ( x, E), x, () mdx where V(x) is the scatterig potetial that models the structure of the target ad its iteractio with the probes It is assumed to have a fiite rage but is ot required to be aalytic Thus, the solutios of the referece problem, where V =, are also the asymptotic solutios of Eq () ad carry the scatterig iformatio If we let ψ ( xe, ) stad for the referece wavefuctio, the i the atomic uits = m = we ca write d ( k ) ψ ( x, E) = Measurig legth i uits of λ (eg, the Bohr radius), where λ dx is a positive parameter, we ca write this equatio i terms of dimesioless quatities as d ( dy ) ψ( y, ) =, y, () where y= λx ad = k λ Cosequetly, the two idepedet real solutios of this referece wave equatio are f = Acos( y) ad f = Bsi( y), where A ad B are arbitrary real costats Due to space reflectio symmetry of the referece wave d operator, J =, parity is coserved ad the solutio space of the referece dy problem splits ito two discoected subspaces; eve ad odd f ( f ) belogs to the eve (odd) subspace Therefore, to perform the algebraic calculatios, we eed two idepedet square itegrable bases sets that are orthogoal to each other They must be compatible with the two sub-domais of the referece Hamiltoia, which are ivariat eige-spaces of the parity operator I geeral, the model potetial V(x) does ot have a defiite parity Therefore, both bases will be used i the calculatio of the scatterig phase shift Now, such complete bases that are compatible with the two sub-domais of H ad with the D problem have the followig elemets = A e y H, (3a) φ φ y = A e H, (3b) where =,,, ad Hm( y ) is the Hermite polyomial of order m [5] The ormalizatio costats are take as A = ad A π = Usig! π ( )! the orthogoality relatio of the Hermite polyomials, oe ca easily verify that these two bases are orthogoal to each other (ie, φ φ = ) ad that the elemets of each m oe of them form a orthoormal set (ie, φ φm = δm) Moreover, they are related as φ( x) = φ ( x) The itegratio measure is dy Similar to f, the elemets φ are eve i y ad satisfy the same coditio at the origi; amely, { } = ( df dy) = O the other had, { φ } y= = same coditio at the origi; amely, f () = are odd i y like f ad satisfy the 3

4 Now, the referece solutios f belog to the two liear subspaces spaed by { φ } =, respectively I other words, they are expadable i these two bases sets as the followig ifiite sums (Fourier-like expasio) f = Acos( y) = s φ, (4a) = φ = f = Bsi = s, (4b) where s are the expasio coefficiets that are fuctios of the eergy parameter Due to the completeess of the basis ad that they are square itegrable, oe ca prove that the two series (4a,4b) are bouded ad coverget for all y ad Noetheless, this will also be demostrated umerically i the results ad graphs obtaied below Now, to calculate the expasio coefficiets, we multiply both sides of Eq (4a) ad Eq (4b) by y y e H m y ad e H m y, respectively Itegratig both sides ad usig the orthogoality relatio of the Hermite polyomials ad the itegral formulas [6], y e cos( y) H dy π e = H, (5a) y e y H y dy e = H si( ) π, (5b) we obtai 4 π A s = e H, ad Γ ( ) (6a) 4 π B s = e H Γ ( ) (6b) These are also related as follows: ( i = ( Usig the differetial equatio, recursio relatio, ad orthogoality property of the Hermite polyomials [5], we obtai the followig tridiagoal matrix elemets of the referece wave operator d ( dy ) B A 4 φ φ J m m ( ) ( ) ( ) ( ) = δm, δ m, δ m, Therefore, the matrix represetatio of the referece wave equatio () becomes equivalet to the followig three-term recursio relatio for the expasio coefficiets ( ) ( ) s = s s s where =,, 3,, (8) The iitial relatio ( = ) is s = s s (9) Therefore, we ca i fact determie all of the expasio coefficiets { s } = (7) usig oly s as seed i the three-term recursio (8) ad its iitial relatio (9) These seeds are s A 4 e = π 4, s B e = π () Moreover, usig the differetial equatio of the Hermite polyomials oe ca easily verify that s satisfy the followig secod order differetial equatio i the eergy d d s = ()

5 Now, we are i a positio to search for the complemetary solutios to f As stated i the Itroductio sectio, these solutios must be asymptotically siusoidal with a phase that carries the scatterig iformatio as deviatios from the correspodig phases of f These complemetary solutios, deoted as g, must also belog to the same sub-domais of the referece Hamiltoia as does f That is, we should be able to write them as g = c ( φ ), where c are ew expasio coefficiets that are idepedet from s I the J-matrix method, these ew expasio coefficiets are required to satisfy the followig criteria [4]: () Be a idepedet solutio of the d order eergy differetial equatio (), () Satisfy the same recursio relatio (8) but ot the iitial relatio (9), ad (3) Make the complemetary referece wavefuctios g asymptotically siusoidal ad idetical to f but with a phase differece that depeds o the scatterig potetial parameters ad the eergy I the absece of the potetial, this phase differece is π The first coditio is satisfied by the followig idepedet solutios of Eq (): 3 ( ) ( ) ; ; ; ; c = AD e F, (a) c = BD e F, (b) where F ( a; c; z ) is the cofluet hypergeometric fuctio ad costats, which are determied by the secod coditio as D d ( ) D are ormalizatio = Γ Γ, (3) where d are costats; idepedet of ad They are determied by the third coditio as d = ad d = This could also be verified aalytically ad/or umerically We also fid that the lim g = Asi( y) ad lim g = Bcos( y) y The iitial recursio relatios satisfied by c are ot homogeeous like those of s i Eq (9) They cotai source terms ad read as follows ( ) 4 c c A e y = π, (4a) c c π Be = (4b) Thus, oe eeds oly c as seeds i the three-term recursio relatio (8) ad iitial relatios (4) to obtai all c recursively These seeds are 4 3 π ; ; 4 π ( ; ; ) c = A e F, (5a) c = Be F (5b) Now, we give visual illustratios of the fuctios g showig their itriguig properties For =, Fig a shows f as a solid red curve ad g as the solid blue curve Nothig curious, of course, about the f curve; it is just the cosie fuctio However, it is iterestig to observe that despite the fact that g is a eve 5

6 fuctio of y, it behaves exactly as a sie fuctio away from the origi ad maages to deform itself ear the origi to comply with the cosie-like boudary coditio there (ie, its derivative vaishes) O the other had, Fig b shows a similar curious behavior for the odd fuctio g that behaves as a cosie fuctio far from the origi but vaishes at the origi Lookig at these fuctios far from the origi, oe may ever be able to predict correctly these o-trivial behaviors ear the origi Aother iterestig property of the series represetatio obtaied above for the siusoidal fuctios f ad g goes as follows If the series were to be started ot from = but from = N, for some large eough iteger N, the the resultig fuctios maitai their siusoidal behavior outside a iterval, which is symmetric aroud the origi ad i which the fuctios become vaishigly small The size of the iterval icreases with N I fact, for ay give rage of the scatterig potetial V(y), oe ca choose a N that makes these fuctios vaishigly small withi that rage As a example, we plot the eve fuctios ( f ad g ) show i Fig a ad Fig b for N = ad N =, respectively I the Appedix, we carry out the same developmet i three dimesios The correspodig results are show graphically i Fig 3 ad Fig 4 Give a specific potetial model V, scatterig will itroduce the phase shift agle δ ito the siusoidal fuctios g 3 Coclusio: I oe dimesio, we were able to write the sie ad cosie fuctios as ifiite coverget series of products of odd ad eve Hermite polyomials Complemetary fuctios, which are asymptotically siusoidal ad required by the scatterig theory, were also costructed as series of products of the Hermite polyomials ad the cofluet hypergeometric fuctios We showed that these complemetary fuctios have curious properties ad we gave a graphical demostratio of that Iterestigly, the complemetary sie (cosie) series is a eve (odd) fuctio i space ad satisfy cosie-like (sie-like) boudary coditio at the origi Lookig at these fuctios far away from the origi, oe could ever predict this iterestig ad o-trivial behavior These complemetary solutios of the referece wave equatio were costructed usig the J-matrix method of scatterig I the Appedix, we showed that a similar pheomeo exists i three dimesios I that case, the regular spherical Bessel fuctio, which is asymptotically siusoidal, is writte as a series of products of Laguerre ad Gegebauer polyomials The complemetary solutio, which is also asymptotically siusoidal ad carries the phase shift, is represeted by a series of products of the Laguerre polyomials ad the hypergeometric fuctios I all cases, elimiatig the lowest few terms i the series will result i vaishigly small values of these fuctios withi a fiite regio ear the origi The size of the regio icreases with the umber of elimiated terms 6

7 Appedix: The three-dimesioal case The referece wave equatio for spherically symmetric iteractio i 3D, which is aalogous to Eq () i D, reads as follows d ( ) ψ ( y, ) =, y, dy y (6) where y= λr, r is the radial coordiate, ad is the agular mometum quatum umber Here, the itegratio measure is dy The two idepedet real solutios of this referece wave equatio for E > are well-kow [] They are writte i terms of the spherical Bessel ad Neuma fuctios as follows: ψ (, ) A reg y = y j y π, ad (7a) ψ (, ) irr y = B ( y) ( y) π, (7b) where A ad B are arbitrary real costats The first is regular (at the origi) ad eergyormalized as ψ ψ = δ( ) For >, the secod is irregular ad is ot reg reg A square itegrable Near the origi they behave as ψ reg y ad ψ irr y O the other had, asymptotically ( y ) they are siusoidal: ψ reg A si ( y π π ) ad ψ irr B cos ( y π π ) Therefore, ulike the D case, oly oe of the two idepedet solutios of the referece wave equatio i 3D is regular for all We write it as f = ψreg ( y, ) A complete L basis set that is compatible with this regular solutio ad with the 3D problem has the followig elemets [7] y φ y = ay e L, (8) where L is the associated Laguerre polyomial of order ad the ormalizatio costat is a =!( )! Therefore, expadig the regular solutio i this L basis as = A = f j s π φ ad usig the orthogoality property of = the Laguerre polyomials [8], we obtai the followig itegral represetatio of the expasio coefficiets s y A a y e = L y J y dy, (9) where we have used the origial Bessel fuctios i writig j ( z ) π z J z = This itegral is evaluated i [9] givig = Γ s Aπ ( ) a si θ C (cos θ), () 4 where cosθ =, < θ π, ad C ( z) is the ultra-spherical (Gegebauer) 4 polyomial Usig the differetial equatio, recursio relatio, ad orthogoality property of the Laguerre polyomials [8], we obtai the followig tridiagoal matrix represetatio of the referece wave operator d Jm φ = φm = 4 dy y ( ) cos θδ ( ) δ ( )( ) δ m, m, m, () 7

8 Therefore, the matrix represetatio of the referece wave equatio becomes equivalet to the followig three-term recursio relatio for the expasio coefficiets s ( ) cos θ s = ( ) s ( )( ) s,, (a) ( ) cosθ s = ( ) s (b) Moreover, usig the differetial equatio for the Gegebauer polyomials [] we ca show that s satisfies the followig secod order differetial equatio i the eergy d d ( ) ( x ) x ( ) s =, (3) dx dx x where x = cosθ The complemetary solutio g solutio space ad could, therefore, be writte as that we are iterested i spas the same = = φ g c The ew idepedet expasio coefficiets c are required to comply with three coditios that are aalogous to those i the D case stated above Eq () As a result, oe obtais the followig [] θ c Aπ a (si θ) = Γ F, ; ;si, (4) where F ( a, b; c; z ) is the hypergeometric fuctio Figure 3a ad Fig 3b show f ad g for = ad for = ad = 3, respectively Figure 4a is a reproductio of Fig 3a but after elimiatig the lowest 3 terms i the series The fuctios become very small ear the origi but maitai their magitude ad siusoidal behavior far away from the origi (Fig 4b) Fially, it is worthwhile otig that there exists aother basis for the solutio space of the referece problem i 3D, which carries a similar iterestig represetatio It is referred to as the oscillator basis whose elemets are: Γ ( ) Γ ( 3 ) y φ = y e L ( y ) [] Refereces: [] See, for example, V De Alfaro ad T Regge, Potetial Scatterig (North- Hollad, Amsterdam, 965); J R Taylor, Scatterig Theory (Wiley, New York, 97); R G Newto, Scatterig Theory of Waves ad Particles (Spriger, New York, 98) [] See, for example, A Messiah, Quatum Mechaics (North-Hollad, Amsterdam, 965); E Merzbacher, Quatum Mechaics (Wiley, New York, 97); R L Liboff, Quatum Mechaics (Addiso-Wesley, Readig, 99) [3] E J Heller ad H A Yamai, New L approach to quatum scatterig: Theory, Phys Rev A 9, -8 (974); E J Heller ad H A Yamai, J-matrix method: Applicatio to s-wave electro-hydroge scatterig, Phys Rev A 9, 9-4 (974) [4] H A Yamai ad L Fishma, J-matrix method: Extesios to arbitrary agular mometum ad to Coulomb scatterig, J Math Phys 6, 4-4 (975) 8

9 [5] W Magus, F Oberhettiger, ad R P Soi, Formulas ad Theorems for the Special Fuctios of Mathematical Physics (Spriger, New York, 966) 3rd ed, pp [6] I S Gradshtey ad I M Ryzhik, Tables of Itegrals, Series, ad Products (Academic, New York, 98), page 84 [7] Page 4 i [4] [8] Pages i [5] [9] A D Alhaidari, Evaluatio of itegrals ivolvig orthogoal polyomials: Laguerre polyomial ad Bessel fuctio example, Appl Math Lett, 38-4 (7) [] Page 8-7 i [5] [] A D Alhaidari, H Bahlouli, M S Abdelmoem, F Al-Amee, ad T Al- Abdulaal, Regularizatio i the J-matrix method of scatterig revisited, Phys Lett A 364, (7) [] Page 44 i [4] Figure Captios: Fig : Plot of the regular cosie/sie solutios f (red curve) ad the irregular sie/cosie solutios g (blue curve) i D for = We set A = B = i arbitrary uits Fig : Plot of f (red curve) ad g (blue curve) for = but after elimiatig the lowest N terms i their series represetatio We took N = i (a) ad N = i (b) ad set A = B = i arbitrary uits Fig 3: Plot of the regular solutio f (red curve) ad the irregular solutio g (blue curve) i 3D for = We took = i (a) ad = 3 i (b) ad set A = B = i arbitrary uits Fig 4: (a) Plot of f (red curve) ad g (blue curve) i 3D for = ad = after elimiatig the lowest 3 terms i their series represetatio, (b) The same but far from the origi 9

10 Fig Fig

11 Fig 3 Fig 4

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.

Physics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001. Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;