Scattering theory without large-distance asymptotics

Tong Liu 1 Wen-Du Li 1 Wu-Sheng Dai 0 1 0 LiuHui Center for Applied Mathematics, Nankai University & Tianjin University , Tianjin 300072, P.R. China 1 Department of Physics, Tianjin University , Tianjin 300072, P.R. China In conventional scattering theory, to obtain an explicit result, one imposes a precondition that the distance between target and observer is infinite. With the help of this precondition, one can asymptotically replace the Hankel function and the Bessel function with the sine functions so that one can achieve an explicit result. Nevertheless, after such a treatment, the information of the distance between target and observer is inevitably lost. In this paper, we show that such a precondition is not necessary: without losing any information of distance, one can still obtain an explicit result of a scattering rigorously. In other words, we give an rigorous explicit scattering result which contains the information of distance between target and observer. We show that at a finite distance, a modification factor the Bessel polynomial appears in the scattering amplitude, and, consequently, the cross section depends on the distance, the outgoing wave-front surface is no longer a sphere, and, besides the phase shift, there is an additional phase (the argument of the Bessel polynomial) appears in the scattering wave function. Contents 1 Introduction 2 3 Rigorous result of scattering without large-distance asymptotics 2.1 Phase shift 2.2 Asymptotic boundary condition 2.3 Scattering wave function 2.4 Outgoing wave-front surface 2.5 Differential scattering cross section 2.6 Total scattering cross section 2.7 Condition on potentials Conclusions and outlook Rl (r) = Clhl(2) (kr) + Dlhl(1) (kr) In conventional scattering theory, which is now a standard quantum mechanics textbook content, to seek an explicit result, one imposes a precondition that the distance between target and observer is infinite. As a result, the conventional scattering theory loses all the information of the distance and the result depends only on the angle of emergence. In this paper, we will show that without such a precondition, one can still achieve a rigorous scattering theory which, of course, contains the information of distance that is lost in conventional scattering theory. The dynamical information of a scattering problem with a spherical potential V (r) are embedded in the radial wave equation, + k2 V (r) Rl = 0. The scattering boundary condition in conventional scattering theory is taken to be where hl(1) (z) and hl(2) (z) are the first and second kind spherical Hankel functions, e2il = Dl/Cl defines the scattering phase shift l, and Al = 2ClDl. 2) Replace the plane wave expansion in the boundary condition with its asymptotics: where jl (z) is the spherical Bessel function. Technologically speaking, the above two treatments in conventional theory are to replace the spherical Hankel function, hl(1) (kr) and h(2) (kr), and the spherical Bessel funcl tion, jl (kr), with their asymptotics, and, thus, inevitably lead to the loss of information of the distance r. In this paper, we will show that the above two replacements is not necessary; without these two replacements, we can still obtain a rigorous scattering theory which contains the information of the distance between target and observer. A systematic rigorous result of a scattering with the distance between target and observer is given in section 2. The conclusion and outlook are given in section 3. Rigorous result of scattering without large-distance asymptotics In this section, a rigorous treatment without large-distance asymptotics for short-range potentials is established. The scattering wave function, scattering amplitude, phase shift, cross section, and a description of the outgoing wave are rigorously obtained. Phase shift In conventional scattering theory, as mentioned above, one replaces the solution of the free radial equation, Rl (r), given by eq. (1.3) with its asymptotics, eq. (1.4), using the asymptotics of the spherical Hankel functions h(1) (kr) ik1r ei(krl/2) and hl(2) (kr) l ik1r ei(krl/2). Obviously, such a replacement will lose information. In the following, with Rl (r) given by eq. (1.3), rather than its asymptotics, eq. (1.4), we solve the scattering rigorously. The first step is to rewrite Rl (r) given by eq. (1.3) as Rl (r) = Clhl(2) (kr) + Dlhl(1) (kr) Akrl sin kr 2 + l + l ikr l 1 where e2il = Dl/Cl and Ml (x) = |yl (x)| and l (x) = arg yl (x) are the modulus and argument of the Bessel polynomial yl (x), respectively. In order to achieve eq. (2.1), we prove the relation 1 Clhl(2) (x) + Dlhl(1) (x) = Ml ix Axl sin x 2 + l + l ix l 1 By the Bessel polynomial [2], we can rewrite hl(1) (x) and hl(2) (x) as hl(1) (x) = eix Xl ikl1 (l + k)! k=0 2kk! (l k)!xk+1 , hl(2) (x) = eix Xl (i)kl1 (l + k)! 2kk! (l k)!xk+1 . k=0 hl(1) (x) = ei(xl/2) i1x yl i1x , hl(2) (x) = ei(xl/2) i1x yl i1x . Using eq. (2.6), we have " ei(xl/2) Clhl(2) (x) + Dlhl(1) (x) = Cl Writing the Bessel polynomial as yl = Mleil, we prove the relation (2.2). The wave function, then, by (r, ) = Pl=0 Rl (r) Pl (cos ), can be obtained immediately from eq. (2.1), When the distance r is finite, the coefficient becomes MlAl and the phase becomes l + l, where Ml and l both depend on r. While, in conventional scattering theory, r , the coefficient is Al and the phase is l, and they are both independent of r. It should be emphasized that l here is the same as that in conventional scattering theory. This is because l is determined only by the coefficient Cl and Dl and yl ik1r r= 1. Thus when r , Cl, Dl, and, accordingly, l remains unchanged. The modification factors, l and Ml, are independent of potentials. When r , Ml (r ) = 1 and l (r ) = 0. 2.2 Asymptotic boundary condition The outgoing wave is no longer a spherical wave when the observer stands at a finite distance from the target, other than that in large-distance asymptotics. The outgoing wave now becomes a surface of revolution around the incident direction, determined by the potential and the observation distance. Because the outgoing waves are different at different distances, there is no uniform expression of the asymptotic boundary condition like eq. (1.2). Here, we express the boundary condition as , r By the relations hl(1) (x) = jl (x) + inl (x) and h(2) (x) = jl (x) inl (x), the spherical l Bessel function jl (x) can be rewritten as jl (x) = 21 hh(1) (x) + hl(2) (x)i, where nl (x) is the l spherical Neumann function [2]. By eq. (2.6), we have jl (kr) = Ml ik1r k1r sin kr l2 + l ik1r . (2.12) Substituting this result into eq. (2.11) proves eq. (2.10). The plane wave expansion (2.10) is exact, rather than the asymptotic one, eq. (1.6), used in conventional scattering theory. In conventional scattering theory, the spherical Bessel function jl (kr) given by eq. (2.12) is replaced by its asymptotics: jl (kr) k1r sin (kr l/2), i. (...truncated)