The exact wave functions that describe scattering of a charged particle by a confined magnetic field (Aharonov-Bohm effect) and by a Coulomb field are analyzed. It is well known that the usual procedure of finding asymptotic forms of these functions which admit a separation into a superposition of an incident plane wave and a circular or spherical scattered wave is problematic near the forward direction. It thus appears to be impossible to express the conservation of probability by means of an optical theorem of the usual kind. Both the total cross section and the forward scattering amplitude appear to be infinite. To address these difficulties we find a new representation for the asymptotic form of the Aharonov-Bohm wave function that is valid for all angles. Rather than try to define a cross section at forward angles, however, we work instead with the probability current and find that it is quite well behaved. The same is true for Coulomb scattering. We trace the usual difficulties to a nonuniformity of limits.
A simple plane wave solution of the Schrödinger–Helmholtz equation is a quantum eigenfunction obeying both energy and linear momentum correspondence principles. Inclusion of the outgoing wave with scattering amplitude f obeys unitarity and the optical theorem. By closely considering the standard asymptotic development of the plane wave, we show that there is a problem with angular momentum when we consider forward scattering at the point of closest approach and at large impact parameter given semiclassically by ( l + 1/2)/ k where l is the azimuthal quantum number and may be large (J Leech et al , Phys. Rev. Lett. 88 257901 (2002)). The problem is resolved via non-uniform, non-standard analysis involving the Heaviside step function, unifying classical, semiclassical and quantum mechanics, and the treatment is extended to the case of pure Coulomb scattering.
A well known example in quantum electrodynamics (QED) shows that Coulomb scattering of unpolarized electrons, calculated to lowest order in perturbation theory, yields a results that exactly coincides (in the non‐relativistic limit) with the Rutherford formula. We examine an analogous example, the classical and perturbative quantum scattering of an electron by a magnetic field confined in an infinite solenoid of finite radius. The results obtained for the classical and the quantum differential cross sections display marked differences. While this may not be a complete surprise, one should expect to recover the classical expression by applying the classical limit to the quantum result. This turn not to be the case. Surprisingly enough, it is shown that the classical result can not be recuperated even if higher order corrections are included. To recover the classic correspondence of the quantum scattering problem a suitable non‐perturbative methodology should be applied.
One recipe for mathematically formulating a relativistic quantum mechanical scattering theory utilizes a two-Hilbert space approach, denoted by
H and
H0, upon each of which a unitary representation of the Poincaré Lie group is given. Physically speaking,
H models a complicated interacting system of particles one wishes to understand, and
H0 an associated simpler (i.e., free/noninteracting) structure one uses to construct 'asymptotic boundary conditions" on so-called scattering states in
H. Simply put,
H0 is an attempted idealization of
H one hopes to realize in the large time limits
t→±∞. The above considerations lead to the study of the existence of strong limits of operators of the form
eiHtJe−iH0t, where
H and
H0 are self-adjoint generators of the time translation subgroup of the unitary representations of the Poincaré group on
H and
H0, and
J is a contrived mapping from
H0 into
H that provides the internal structure of the scattering asymptotes. The existence of said limits in the context of Euclidean quantum theories (satisfying precepts known as the Osterwalder-Schrader axioms) depends on the choice of
J and leads to a marvelous connection between this formalism and a beautiful area of classical mathematical analysis known as the Stieltjes moment problem, which concerns the relationship between numerical sequences
{μn}n=0∞ and the existence/uniqueness of measures
α(x) on the half-line satisfying \begin{equation*} \mu_{n}=\int_{0}^{\infty}x^{n}d\alpha(x). \end{equation*}
For Pt. IV see abstr. A41769 of 1969. The classical and quantal problem of a charged particle moving in an attractive Coulomb potential with total energy zero is considered. A complete correspondence identity is obtained by expressing the kernel of the quantum-mechanical spectral operator IE= delta (E-H) exactly in terms of the classical action function in momentum representation. Use is made of the dynamical symmetry properties exhibited by both the classical and quantal problems, and a path-integral method of evaluating the spectral operator kernel is also presented.