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American Journal of Physics

Quantum scattering from cylindrical barriers

Sean McAlinden, Janine Shertzer

Published Sep 21, 2016• 5 citations
Publisher

Abstract

We solve the two-dimensional Schrödinger equation for particles with momentum px=ℏk scattering off of a hard circular cylinder using the finite element method; we compare our results with the exact analytic solution. The quantity of interest to experimentalists is the differential cross section σ(ϕ)=|fk(ϕ)|2, which represents the final angular distribution of only the scattered particles. Here, we are also interested in the interference between the incident and scattered wave, which can be seen in the probability density for the total wave function, ρ(x,y)=|ψk(x,y)|2. We also apply the finite element method to the problem of particles scattering off of a hard rectangular cylinder, for which there is no analytic solution.

DOI
10.1119/1.4960021

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We solve the two-dimensional Schrödinger equation for particles with momentum px=ℏk scattering off of a hard circular cylinder using the finite element method; we compare our results with the exact analytic solution. The quantity of interest to experimentalists is the differential cross section σ(ϕ)=|fk(ϕ)|2, which represents the final angular distribution of only the scattered particles. Here, we are also interested in the interference between the incident and scattered wave, which can be seen in the probability density for the total wave function, ρ(x,y)=|ψk(x,y)|2. We also apply the finite element method to the problem of particles scattering off of a hard rectangular cylinder, for which there is no analytic solution.
Time-dependent scattering of finite duration wave packets on a rigid cylinder is studied using exact solutions of the corresponding Schrodinger equation. The similarity between the time evolution of the total wavefunction and evolution of the corresponding classical ensemble of particles is exposed. Special attention is paid to a discussion of the differences between the total wavefunction dynamics and the dynamics of the scattered part of the wavefunction, the part which is commonly used for the evaluation of scattering cross-sections. This discussion emphasizes the necessity of using the total wavefunction in the description of scattering.
Introduction.We give three applications of probability to scattering theory.The first question we consider is that of the "scattering length."We obtain a probabilistic expression for scattering length and use this expression to prove that in the case of an impenetrable obstacle ("hard" potential) the scattering length T is the same as the electrostatic capacity.Next, we briefly touch upon a problem which is inherently probabilistic-namely, scattering in a medium with random boundary.We find that if a large number of small hard spheres ("dust particles") are distributed at random in the medium, the effect is to shift all the eigenvalues by a constant.Our final topic is the so-called "inverse problem" of scattering theory: given the scattering operator, to compute the potential.For certain spherically symmetric potentials, this problem was solved in a famous paper of GelTand and Levitan.Here we show that their result can be included in the circle of potential-theoretic problems which can be solved by means of Brownian motion and Wiener integrals.We quickly sketch some of the physical background.We are concerned with the Schrodinger equation of non-relativistic quantum mechanics.In its stationary or time-independent form, this equation isThe "wave function" I/J depends on both the position vector fiEfi 3 and the wave number k.The real parameter k 2 is proportional to the energy.The function q(R) is called the "potential."It represents the interaction of the quantum-mechanical "particle" with some force field.The "scattering problem" is, roughly speaking, to determine the effect of this interaction asymptotically for large R.More precisely, we seek a solution of (1) which at |fl| = » is of the asymptotic form e iV2kz + fk(e)e iV ^k> ^l\\R\\ where e is the unit vector in the direction of R. The first term, e iV2kz , is an incident wave in the direction of the zaxis; the second term /fc(e)e iV ^I^H/||R||, is the "scattered wave."
The concept of a generalized scattering amplitude is introduced and applied to quantum potential scatterings. After an outgoing spherical wave is factored out from a wave function, the remaining wave envelope is defined as the generalized scattering amplitude. The transformed Schr\"odinger equation, in terms of the generalized scattering amplitude, can be solved numerically by using a finite-difference method over the entire scattering domain. Example problems of scatterings by a spherical potential well and the (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$,H) static field interaction potential are solved to validate the theoretical formulation and numerical method. The far-field solutions for the ordinary scattering amplitude and differential cross section agree very well with those obtained from the partial-wave analysis. The radial profiles of the generalized scattering amplitude and particle density function over the entire scattering region are also presented, and their properties are discussed. These results demonstrate that the method can be used to yield a complete and accurate solution to scattering problems involving arbitrary interaction potentials.
The scattering of a monoenergetic beam of atoms by a hard hemisphere on a hard flat surface is formulated quantum mechanically. Differential and integral cross sections are calculated as a function of incident beam energy and incident angle and compared with the familiar results of scattering from a free hard sphere. Two types of interference processes are identified. In addition to the previously discussed off-specular reflection symmetry interferences, Rayleigh oscillations are found in the integral cross sections. They result from the interference of specular scattering from the hemisphere and from the flat surface. Their phases are found to be independent of the angle of incidence when plotted versus the normal component of the incident wave vector.
The scattering amplitude of a charged particle from a long hard cylinderical solenoid is derived by solving the time independent Schrödinger equation on a double connected plane. It is a summation over the angular momentum quantum number (partial wave summation). It is shown that only negative mechanical angular momenta contribute to the amplitude when the radius of the solenoid goes to zero limit without varying the magnetic induction flux (Flux line). Original Aharonov-Bohm result is obtained with this limit.