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arXiv:0806.0801 · American Journal of Physics

Semiclassical scattering in two dimensions

Sadhan K. Adhikari, Mahir Saleh Hussein

Published Nov 18, 2008• 10 citations
arXivHTMLPDF

Abstract

The semiclassical limit of quantum mechanical scattering in two dimensions is developed and the Wentzel–Kramers–Brillouin and eikonal results for two-dimensional scattering is derived. No backward or forward glory scattering is present in two dimensions. Other phenomena, such as rainbows and orbiting, do occur.
DOI
10.1119/1.2970054

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A semiclassical formula that gives scattering cross sections down to an angular resolution of order ${\mathrm{\ensuremath{\Elzxh}}}^{4/5}$ is derived. The formula describes rainbow peaks, which have widths of order ${\mathrm{\ensuremath{\Elzxh}}}^{2/3}$, but not other interference effects, which give structure on a scale of order \ensuremath{\Elzxh}. For cross sections with many rainbow peaks, it can be easier to evaluate than conventional uniform approximations. As an example, the formula is applied to a simple two-dimensional model.
The semiclassical approximation, particularly the Wentzel-Kramers-Brillouin (WKB) method, has been a cornerstone in quantum mechanics for understanding systems where quantum effects are present but allow for an approximate classical description. Despite its widespread success in providing intuitive insights into phenomena like tunneling and energy quantization, the standard WKB expansion often suffers from asymptotic divergence and limited accuracy, especially for lower quantum numbers or near turning points. This paper introduces a novel approach to enhance the precision and convergence of the WKB method by integrating variational principles with advanced resummation techniques. We propose a framework where the WKB expansion for key physical quantities is treated as an asymptotic series, and its coefficients are optimized through a variational procedure derived from the underlying quantum dynamics. Subsequently, these optimized series are subjected to resummation methods, such as generalized Borel transforms or Padé approximants, to extract highly accurate and convergent results. Our methodology aims to transcend the limitations of conventional WKB by systematically incorporating non-perturbative information and improving the analytic properties of the semiclassical series. We present the theoretical foundations of this "Resummed Variational WKB" approach, outlining its potential to yield unprecedented precision in calculating energy eigenvalues, tunneling rates, and other quantum observables, thereby bridging the gap between purely classical and fully quantum mechanical descriptions with enhanced quantitative fidelity.
In this chapter we study another very important and powerful approximation method, the so-called eikonal approximation or Wentzel-Kramer-Brillouin (WKB) method, which is used in studying quantum mechanical systems that are subjected to complicated potentials. This method can be used to find approximate solutions to differential equations where the highest derivative is multiplied by a small parameter. The WKB method gives approximate solution to the Schrödinger equation irrespective of its complicated potential and is mostly used to study one-dimensional systems and also the radial equations for higher-dimensional systems with rotational symmetry.