Elementary Potential Scattering

Ajoy Kumar Ghatak, S. Lokanathan

R. Gagnon

Scattering theory for extended elementary particles in stochastic phase space is studied. It is shown that the interacting Hamiltonian is equivalent to an effective potential in configuration representation. Asymptotic completeness can be studied by investigating the behavior of the effective potential. The sharp-point limit of the extension of these particles is studied. It is also shown that scattering theory can also be studied directly in stochastic phase space in the optimal case.

Norbert Kaiser

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.

Nikolay Leonidovich Chuprikov

We present a renewed wave-packet analysis based on the following ideas: if a quantum one-particle scattering process and the corresponding state are described by an indivisible wave packet to move as a whole at all stages of scattering, then they are elementary; otherwise, they are combined; each combined process consists from several alternative elementary ones to proceed simultaneously; the corresponding (normed) state can be uniquely presented as the sum of elementary ones whose (constant) norms give unit, in sum; Born's formula intended for calculating the {\it expectation} values of physical observables, as well as the standard timing procedure are valid only for elementary states and processes; only an elementary time-dependent state can be considered as the quantum counterpart to some classical one-particle rajectory. By our approach, tunneling a non-relativistic particle through a static one-dimensional potential barrier is a combined process consisting from two elementary ones, transmission and reflection. In the standard setting of the problem, we find an unique pair of solutions to the Schrödinger equation, which describe separately transmission and reflection. On this basis we introduce (exact and asymptotic) characteristic times for transmission and reflection.