In this paper, the Born series for the s-wave scattering a0 is calculated for a class of central potentials V(r) up to sixth order in a dimensionless coupling strength g. Examples of exponentially decaying potentials as well as truncated potentials involving a single length-scale a are considered. In certain favorable cases, the exact result for the g-dependent s-wave scattering length a0=A0(g)a can be given in terms of special functions. The poles of A0(g) at increasing positive values of g correspond to the thresholds, where s-wave bound-states occur successively. Further scattering problems, where A0(g) is solvable in terms of elementary functions, are also presented.
In this paper, the Born series for the s-wave scattering a0 is calculated for a class of central potentials V(r) up to sixth order in a dimensionless coupling strength g. Examples of exponentially decaying potentials as well as truncated potentials involving a single length-scale a are considered. In certain favorable cases, the exact result for the g-dependent s-wave scattering length a0=A0(g)a can be given in terms of special functions. The poles of A0(g) at increasing positive values of g correspond to the thresholds, where s-wave bound-states occur successively. Further scattering problems, where A0(g) is solvable in terms of elementary functions, are also presented.
We derive two equivalent analytical expressions for an lth partial-wave scattering length al for central potentials with long-range tails of the form V(r)=−ℏ22mBrn−4(rn−2 + Rn−2)2−ℏ22mCr2(rn−2 + Rn−2), (r ⩾ rs, R > 0). For C = 0, this family of potentials reduces to the Lenz potentials discussed in a similar context in our earlier works [R. Szmytkowski, Acta Phys. Pol. A 79, 613 (1991); J. Phys. A: Math. Gen. 28, 7333 (1995)]. The formulas for al that we provide in this paper depend on the parameters B, C, and R characterizing the tail of the potential, on the core radius rs, as well as on the short-range scattering length als, the latter being due to the core part of the potential. The procedure, which may be viewed as an analytical extrapolation from als to al, is relied on the fact that the general solution to the zero-energy radial Schrödinger equation with the potential given above may be expressed analytically in terms of the generalized associated Legendre functions.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world.
For scattering in nonrelativistic quantum mechanics, the range of validity of the perturbation expansion is studied by viewing the expansion as being generated by an iteration procedure, and applying to it a fundamental theorem for iteration processes. The theorem provides a sufficient condition for convergence and, at the same time, gives upper bounds on the error generated in truncating the expansion at a given number of terms. These error bounds for the wave function are in turn used to find upper bounds on the truncation error for the S-matrix expansion. Bounds for the exact S matrix are also given. These results are illustrated by applying them to the simple case of one-particle potential scattering, for both the plane-wave and partial-wave analyses.
We detect an omission in the paper ``Conditionally exactly soluble class of quantum potentials'' by A. de Souza Dutra [Phys. Rev. A 47, R2435 (1993)]. There, two strongly singular s-wave bound-state problems have been claimed completely solvable in closed form. Unfortunately, all the displayed wave functions are merely asymptotically correct solutions which do not satisfy the appropriate threshold boundary condition. We show that the incorporation of the threshold boundary conditions only leads to a very partial exact solvability at a single energy and for special couplings.
For potentials with attractive tails, as occur in typical atomic interactions, we present a simple formula for the s-wave phase shift ${\ensuremath{\delta}}_{0}$. It exposes a universal dependence of ${\ensuremath{\delta}}_{0}(E)$ on the potential tail and the influence of effects specific to a given potential, which enter via the scattering length a, or equivalently, the noninteger part ${\ensuremath{\Delta}}_{\mathrm{th}}$ of the threshold quantum number nth. The formula accurately reproduces ${\ensuremath{\delta}}_{0}(E)$ from threshold up to the semiclassical regime, far beyond the validity of the effective-range expansion. We derive the tail functions occurring in the formula for ${\ensuremath{\delta}}_{0}(E)$ and demonstrate the validity of the formula for attractive potential tails proportional to 1/r6 or to 1/r4, and also for a mixed potential tail consisting of a 1/r4 term together with a non-negligible 1/r6 contribution.
Accurate scattering lengths and effective ranges for hydrogen atomic collisions are determined from effective range theory. The precise ab initio b3Σ+u potential energy curve values of H2 calculated by Kolos and Wolniewicz and C6, C8, C10 long-range expansion coefficients are used in obtaining accurate representation of the two-body interaction potential. The s-wave scattering length obtained for this potential (A =1/33455 ao) is used to calculate the zero temperature energy as a function of density of electron spin-polarized ("spin-aligned") atomic hydrogen (H+) using previous many-body perturbation theory results for bosons. Neglecting small potential shape dependent terms, one obtains E/N (energy per particle ) as a function of density, which may be compared with similar results obtained from Monte-Carlo or variational calculations. The scattering lengths and effective ranges for the other hydrogen isotopic systems are also presented. It is shown that the first bound state for the b3Σ+u potential (within the Born-Oppenheimer approximation) only exists for a hypothetical isotopic species with mass larger than a value m = 3.258176 g/mole (where the scattering length becomes infinite). In terms of the quantum parameter, η ≡ [MATH]2/ (2µεσ2), two-body binding occurs in the b3Σ+u potential only for η < 0.17094. The second order liquid-to-gas transition (corresponding to the onset of infinite-body binding) occurs at the accurately determined critical value of the quantum parameter ηCB = 0.43891. This corresponds to zero scattering length.