Relativistic Scattering States of Coulomb Potential Plus a New Ring-Shaped Potential

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1 Commun. Theo. Phys. Beijing, China pp c Intenational Academic Publishes Vol. 45, No. 5, May 5, 006 Relativistic Scatteing States of Coulomb Potential Plus a New Ring-Shaped Potential CHEN Chang-Yuan, LU Fa-Lin, and SUN Dong-Sheng Depatment of Physics, Yancheng Teaches College, Yancheng 400, China Received July 8, 005; Revised Novembe 30, 005 Abstact In this pape, exact solutions of scatteing states of the Klein Godon equation with Coulomb potential plus a new ing-shaped potential ae studied unde the condition that the scala potential is equal to the vecto potential. The nomalized wave functions of scatteing states on the k/π scale and the calculation fomula of phase shifts ae pesented. Analytical popeties of the scatteing amplitude ae discussed. PACS numbes:.80.-m, Pm, Nk Key wods: Coulomb potential plus a new ing-shaped potential, scatteing states, phase shifts, Klein Godon equation Intoduction In the stong coupling case, elativistic effects have been aely discussed, pimaily due to difficulties involved in solving analytically the Klein Godon equation spin s = 0 paticle o the Diac equation spin s = / paticle. Seveal model potentials have been intoduced ecently to exploe the elativistic enegy specta and wave function behavios. Dominguez Adame and Talukda et al. deived the bound state and the scatteing state solutions of s-wave Klein Godon equation with vecto and scala Hulthén potentials. In the case that the scala potential is equal to vecto potential, Hu and Su 3 obtained the exact solution of the s-wave Diac equation fo Hulthén potentials. Guo et al. 4 and Zhang et al. 5 gave exact solutions of the Diac equation with the Wood Saxon potentials and Scaf-type potentials, espectively. Fo thee-dimensional hamonic oscillato potential, Su et al. 6 and Qiang 7 solved the Klein Godon equation and the Diac equation unde the condition that the scala potential is equal to the vecto potential. Chen et al. 8,9 and Simsek et al. 0 discussed exact solutions of s-wave to the Klein Godon equation with vecto and scala genealized Hulthén potentials. Popeties of the bound states of the Klein Godon equation and the Diac equation with non-cental potential fields, e.g., ing-shaped hamonic oscillato potentials, ing-shaped non-oscillato potentials, and Hatmann potentials, 3,4 have also been exploed. But we have not yet found the elativistic scatteing states elated to non-cental potential fields epoted in pevious liteatue except cental potential fields. Coulomb potential plus a new ing-shaped potential 5 is a kind of non-cental physical potential, which is fomed by eplacing / sin θ in the Hatmann potential with a new tem ctg θ/ except fo the Coulomb-like inteaction tem. This ing-shaped potential is possible application to ing-shaped oganic molecules like cyclic polyenes and benzene. In spheical coodinates, θ, φ, this potential is defined as V, θ = α + β cos θ sin θ, whee α and β ae positive eal constants. When β = 0 and α = Ze, Coulomb potential plus a new ing-shaped potential educes to the Coulomb potential. In the non-elativistic case, popeties of the bound states and scatteing states fo the Coulomb potential plus a new ing-shaped potential have been discussed. 5 Hee we epot solutions of elativistic scatteing states of the Klein Godon equation with Coulomb potential plus a new ing-shaped potential in the case that the scala potential is equal to vecto potential. In Sec., we obtain the exact solutions of scatteing states. The nomalized adial wave functions of scatteing states on the k/π scale and the calculation fomula of phase shifts ae pesented. In Sec. 3, we discuss analytical popeties of the scatteing amplitude in the elativistic case. We find that the enegy of paticles educes to the enegy equation of the bound states, and the adial wave functions of scatteing states educes to the adial wave functions of the bound states, at the poles of the scatteing amplitude. Exact Solutions of Scatteing States In spheical coodinates, θ, φ, the Klein Godon equation with scala potential S, θ and vecto potential V, θ is = c = P + M + S ψ = E V ψ, whee P = i is the momentum opeato, E denotes the enegy, and M denotes the mass. When S, θ, i.e. The poject suppoted by the Pofesso and Docto Foundation of Yancheng Teaches College yctcccy@63.net

2 890 CHEN Chang-Yuan, LU Fa-Lin, and SUN Dong-Sheng Vol. 45 Coulomb potential plus a new ing-shaped scala potential, is equal to the vecto potential V, θ, that is, S, θ = V, θ = α + β cos θ sin θ, 3 equation becomes + M + E α + β cos θ sin θ ψ, θ, φ = E M ψ, θ, φ. 4 As we know, the Coulomb potential plus a new ingshaped potential is non-cental potential field, but it is independent of the angle φ. Theefoe, the solutions of Klein Godon equation fo scatteing states must evidently be axially symmetic about the z-axis, i.e. the diection of the incident paticles. That is to say, the scatteing states can be discussed by the patial-wave method. In analogy to the usual pactice fo a spheical potential, let ψ, θ, φ = u Hθ Φφ. 5 Sepaating the vaiables in Eq. 4, we have d u { d + E M αm + E + sin θ d dθ sin θ dhθ dθ + λ } u = 0, 6a λ βm + E cos θ + m sin θ Hθ = 0, d Φφ dφ + m Φφ = 0. 6c Hee, m and λ ae two sepaation constants. Sine the wave function ψ, θ, φ must be finite in all space fo scatteing states, it follows that the bounday conditions fo Eq. 6a must be u0 = 0 and u is a finite value, and that the bounday conditions fo Eq. 6b must be that H0 is a finite value and Hπ is a finite value, and that the bounday conditions fo Eq. 6c must be Φφ + π = Φφ. It is well known that the solution of Eq. 6c is Φ m φ = π e imφ, m = 0, ±, ±,... 7 So we shall only study solutions of Eqs. 6b and 6a in the following. Let m = βm + E + m, λ + βm + E = l l +, 8 and intoduce a new vaiable x = cos θ. Equation 6b is then eaanged as the univesal associated-legende diffeential equation, 5,6 x d Hx dx x dhx + l l + m dx x Hx = 0, 9 whose bounday condition is that H x=± should be taken as a finite value in tems of the bounday conditions fo Eq. 6b. When l and m ae positive integes o zeo, equation 9 educes to the associated-legende equation. The solution of Eq. 9 has been given by 5,6 whee H l m cos θ = N l m sin θm l m v=0 N l m 6b v Γl v + l v!l m v! Γl cos m θl v, 0 v + = l + l m! Γl + m + is the nomalization constant, and the elation between l and m is given by l = k + m, k = 0,,,... We now study Eq. 6a. Substituting Eq. 8 into Eq. 6a allows us to obtain d u { d + E M αm + E LL + } + u = 0, whee L = + 4k + β M + E + m k + βm + E + m + β M + E, m, k = 0,,,... Fo scatteing states, E > M. Thence letting k = E M /, s = α M + E, 3

3 No. 5 Relativistic Scatteing States of Coulomb Potential Plus a New Ring-Shaped Potential 89 equation can be ewitten as d u d + k + s LL + u = 0. 4 The bounday conditions fo Eq. 4 is u0 = 0 and u is a finite value. Consideing the asymptotic behavio of the adial wave function as 0, we take the wave function with the fom Substituting Eq. 5 into Eq. 4, we have u = Ak L+ f. 5 d f df + L + + ik + ikl + + s f = 0. 6 d d Intoduce a new vaiable z = ik, equation 6 is then e-expessed as z d f df + L + z dz dz L + i s f = 0. 7 k Equation 7 is called the confluent hypegeometic diffeential equation. 7 The paametes ae α = L + i s/k and γ = L +. Analytical solution as 0 is the confluent hypegeometic function, Thus, the adial wave function of the scatteing states is f = FL + i s/k, L +, ik. 8 u k L = A kl k L+ e ik FL + is/k, L +, ik. 9 We now study asymptotic fom of the above expession fo lage, and calculate the nomalization constant of adial wave functions A k L and phase shifts. The asymptotic expession of the confluent hypegeometic function when z is 7 Γα, γ, z Γγ Γα ez z α γ + Γγ Γγ α e± i π α z α, 0 whee + coesponds to π/ < ag z < 3π/, while coesponds to 3π/ < ag z < π/. Fomula 0 is then modified when z = ik = z e iπ/, Γα, γ, z Γγ Γα ez z α γ e i π α γ/ + Γγ Γγ α e i π α/ z α, fom which we have Letting then FL + i s/k, L +, ik whee δ L is a eal numbe. Equation then becomes FL + i s/k, L +, ik Substituting Eq. 5 into Eq. 9 leads to u k L Accoding to Ref. 8, when ΓL + is/k e ik k L++i s/k e i πl++i s/k/, + ΓL + + i s/k e i πl+ i s/k/ k L+ i s/k. ΓL + i s/k = ΓL + i s/k e i δ L, 3 ΓL + + i s/k = ΓL + i s/k e i δ L, 4 e πs/k e i k ΓL + i s/k k L+ i e i k +δ L π L/+sln k/k i e i k +δ L π L/+sln k /k. 5 A kl e πs/k ΓL + i s/k L+ sin k + δ L π L/ + sln k/k. 6 u kl sin k + δ l lπ/ + zln k/k, 7 the adial wave functions of scatteing states fo Coulomb potential in the non-elativistic case ae nomalized on the k/π scale. The new ing-shaped invese squae potential is a shot distance potential in the Coulomb potential plus a new ing-shaped potential, so it has no influence on asymptotic expession of the wave function fo lage. It is

4 89 CHEN Chang-Yuan, LU Fa-Lin, and SUN Dong-Sheng Vol. 45 useful to note that, consideing the asymptotic behavio of the wave function, the scatteing amplitudes ae also valid if we take into account the elativity. 9 That is to say, the asymptotic expession of Coulomb potential plus a new ing-shaped potential in the elativistic case is equal to that of the Coulomb potential in the non-elativistic case when, i.e. u kl sin k + δ l π l/ + s ln k/k. 8 The adial wave functions of the scatteing states fo Coulomb potential plus a new ing-shaped potential in the elativistic case ae nomalized on the k/π scale, too. Hee δ l epesents the phase shifts. Compaing Eqs. 6 with Eq. 8, we may obtain the nomalization constant of scatteing states as A k L = L+ ΓL + i s/k e πs/k. 9 and the phase shifts δ l as δ l = δ L + πl L = agγl + i s/k + πl L. 30 Substituting Eq. 9 into Eq. 9, we obtain the nomalized adial wave functions of scatteing states on the k/π scale as u k L = L+ ΓL + i s/k e πs/k k L+ e ik F L + i s/k, L +, ik. 3 3 Analytical Popeties of Scatteing Amplitude Fom the geneal theoy of the patial-wave method, the scatteing amplitude is defined by e iδ l fθ = l + P l cos θ, 3 ik l=0 whee l is the angula quantum numbe. We now discuss the analytical popeties of the scatteing amplitude, namely, we shall discuss the analytical popeties of the scatteing amplitude in the whole complex k plane by egading the scatteing amplitude as the function of the enegy. To this end, fom Eqs. 30 and 3, we have to discuss the analytical popeties of ΓL + i s/k. Fom the definition of the gamma function Γz + Γz + Γz = = z zz + = Γz + 3 zz + z + =, 33 we know that z = 0,,,... ae the fist-ode poles of the Γz. That is to say, the fist-ode poles of ΓL + i s/k ae situated at then whee n = n + L + i s/k = 0,,, = n, n = 0,,,..., 34 s k = il + + n = in, k + βm + E + m k + βm + E + m + βm + E +, n, m, k = 0,,,. At the poles of phase shifts, the enegy equation of paticles is See Eq. 3 M + E α M E = n + + 4k + β M + E + m k + β M + E + m + β M + E +, 36 n, m, k = 0,,,..., 37 which is exactly the enegy equation of the bound states. Substituting Eq. 34 into Eq. 3 yields αm + E L+ u n L = N n L e αm+e /n αm + E n F n, L +, n, 38 which implies that F n, L +, αm + E/n educes to a polynomial of degee n, since n is a positive intege o zeo. Making use of the elation between the confluent hypegeometic functions and the genealized Laguee polynomials, 7 L µ Γµ + + n nx = F n, µ +, x, 39 n! Γµ +

5 No. 5 Relativistic Scatteing States of Coulomb Potential Plus a New Ring-Shaped Potential 893 and the ecuence elation between the genealized Laguee polynomials n + L µ n+ z + z µ n Lµ nz + µ + nl µ n z = 0, 40 and the othgonality elation of the genealized Laguee polynomials z µ e z L µ nzl µ Γµ + n + n zdz = δ n n, 4 0 n! we have the nomalization constant N n L, N n L = n αm + E Γn + L +. 4 n! Finally we obtain the nomalized adial wave functions as u n L = n α M + E Γn + L + α M + E n! n F n, L +, L+ e α M +E/n αm + E n. 43 The above expession is the exact adial wave function of the bound states of Klein Godon equation with Coulomb potential plus a new ing-shaped potential. In othe wods, at the poles of the scatteing amplitude, the enegy equation of paticles educes to the enegy equation of the bound states, and the coesponding wave functions educes to the adial wave functions of the bound states. 4 Conclusions In this pape, we have discussed chaacteistics of elativistic scatteing states of Klein Godon equation with Coulomb potential plus a new ing-shaped potential. The Coulomb potential plus a new ing-shaped potential is a non-cental potential field, but it is independent of the angle φ, so the scatteing states can been discussed by the patial-wave method. In the case that the scala potential is equal to vecto potential, the nomalized adial wave functions of scatteing states on the k/π scale and the calculation fomula of phase shifts have been pesented. Analytical popeties of the scatteing amplitude have been discussed. We have found that the enegy equation of paticles educes to the enegy equation of the bound states, and the adial wave functions of scatteing states educes to the adial wave functions of the bound states, at the poles of the scatteing amplitude. Refeences F. Dominguez-Adame, Phys. Lett. A B. Talukda, A. Yunus, and M. R. Amin, Phys. Lett. A S.Z. Hu and R.K. Su, Acta Phys. Sin in Chinese. 4 J.Y. Guo, X.Z. Fang, and F.X. Xu, Phys. Rev. A X.C. Zhang, Q.W. Liu, C.S. Jia, and L.Z. Wang, Phys. Lett. A R.K. Su and Z.Q. Ma, J. Phys. A W.C. Qiang, Chin. Phys G. Chen, Mod. Phys. Lett. A G. Chen, Z.D. Chen, and Z.M. Lou, Phys. Lett. A M. Simsek and H. Egifes, J. Phys. A W.C. Qiang, Chin. Phys X.A. Zhang, K. Chen, and Z.L. Duan, Chin. Phys C.Y. Chen, Phys. Lett. A Z.D. Chen and G. Chen, Acta Phys. Sin in Chinese. 5 C.Y. Chen and S.H. Dong, Phys. Lett. A C.Y. Chen and S.Z. Hu, Acta Phys. Sin in Chinese. 7 I.S. Gadshteyn and L.M. Ryzhik, Tables of Integals, Seies, and Poducts, 5th ed., Pegamon, New Yok 994; Z.X. Wang and D.R. Guo, An Intoduction to Special Function, Science Pess, Beijing 979 in Chinese. 8 L.D. Landau and E.M. Lifshitz, Quantum Mechanics non-elativistic theoy, 3d ed., Pegamon, New Yok V.B. Beestetskii, E.M. Lifshitz, and L.P. Pitaevskii, Quantum Electodynamics, nd ed., Buttewoth- Heinemann, Oxfod L.I. Schiff, Quantum Mechanics, 3d ed, McGaw-Hill Pess, New Yok 968. C.Y. Chen, D.S. Sun and F.L. Lu, Phys. Lett. A C.Y. Chen, D.S. Sun, and F.L. Lu, subsimitted to Acta Phys. Sin.