Modified WKB approximation

TL;DR: This work studies a modified WKB approximation in which the S term in Schrodinger's equation dominates (after a guess for S is supplied), which produces only the nodeless ground-state wave function, but unlike the W KB approximation it can be applied straightforwardly to problems having many degrees of freedom.

Abstract: In the WKB approximation the ${\mathrm{\ensuremath{\nabla}}}^{2}$S term in Schr\"odinger's equation is subordinate to the |\ensuremath{\nabla}S${\mathrm{|}}^{2}$ term. Here we study a modified WKB approximation in which the ${\mathrm{\ensuremath{\nabla}}}^{2}$S term dominates (after a guess for S is supplied). Our approximation produces only the nodeless ground-state wave function, but unlike the WKB approximation it can be applied straightforwardly to problems having many degrees of freedom. As a test, we apply the method to potential problems, including the hydrogen and helium atoms, and to ${\mathrm{\ensuremath{\varphi}}}^{4}$ field theory. We show that good numerical results for the bound-state energy can be obtained even when the initial guess for S is manifestly inaccurate. Our method supplies the wave function as well as the energy. \textcopyright{} 1996 The American Physical Society.