Approximate Solutions to the One-Dimensional Schrödinger Equation by the Method of Comparison Equations

Abstract

The method of "comparison equations" for obtaining approximate solutions to the one-dimensional Schr\"odinger equation is discussed. In this method, one Schr\"odinger equation, ${\ensuremath{\psi}}^{\ensuremath{'}\ensuremath{'}}(x)+{k}^{2}(x)\ensuremath{\psi}(x)=0$, is transformed into another, ${v}^{\ensuremath{'}\ensuremath{'}}(z)+{K}^{2}(z)v(z)=0$, by a simultaneous change of independent and dependent variables, $x\ensuremath{\rightarrow}z$, $\ensuremath{\psi}\ensuremath{\rightarrow}v$. Then $\ensuremath{\psi}(x)$ and $v(z)$ are related by $\ensuremath{\psi}(x)=v(z){(\frac{\mathrm{dz}}{\mathrm{dx}})}^{\ensuremath{-}(\frac{1}{2})}$ whenever ${k}^{2}(x)$ and ${K}^{2}(z)$ satisfy the relation ${K}^{2}(z){(\frac{\mathrm{dz}}{\mathrm{dx}})}^{2}={k}^{2}(x)\ensuremath{-}(\frac{1}{2}) 〈z;x〉$. Here $〈z;x〉=\frac{{z}^{\ensuremath{'}\ensuremath{'}\ensuremath{'}}}{{z}^{\ensuremath{'}}}\ensuremath{-}(\frac{3}{2}){(\frac{{z}^{\ensuremath{'}\ensuremath{'}}}{{z}^{\ensuremath{'}}})}^{2}$ is the Schwarzian derivative of $z$ with respect to $x$ and the prime indicates $\frac{d}{\mathrm{dx}}$. A set of "best" criteria for the transformed potential ${K}^{2}(z)$ is obtained, where by "best" is meant that we can completely neglect $〈z;x〉$ in first approximation and yet not have the turning-point problems that plague the WKB method (which is a special case of the comparison-equation method). The WKB method sets ${K}^{2}(x)\ensuremath{\equiv}1$ and neglects $〈z;x〉$; the result is that the transformation $x\ensuremath{\rightarrow}z$, $\ensuremath{\psi}\ensuremath{\rightarrow}\ensuremath{\nu}$ is singular at the turning points, where ${k}^{2}(x)=0$. We choose ${K}^{2}(z)$ to have the proper asymptotic behavior far from the turning points, so that ${z}^{\ensuremath{'}}\ensuremath{\approx}1$; hence $〈z;x〉\ensuremath{\approx}0$ in these regions, and we match the zeroes of ${k}^{2}(x)$ and ${K}^{2}(z)$ in order to keep the transformation regular. This method is applied to various potentials with one and two turning points. Transmission and reflection coefficients $T$ and $R$ and transmitted and reflected phase shifts $\ensuremath{\mu}$ and $\ensuremath{\nu}$ are calculated for potentials with one turning point and potential barriers, and expressed in terms of the energy $E$ and the quantity $W=|\ensuremath{\int}{{x}_{1}}^{{x}_{2}}k\mathrm{dx}|$, where ${x}_{1,2}$ are the possibly complex turning points, ${k}^{2}({x}_{1,2})=0$. Quantization rules, in terms of the classical action, are derived for various types of potential wells.