Modified WKB approximation

In the WKB approximation the ${\mathrm{\nabla }}^2$S term in Schrödinger’s equation is subordinate to the |∇S${\mathrm{|}}^2$ term. Here we study a modified WKB approximation in which the ${\mathrm{\nabla }}^2$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 ${\mathrm{\phi }}^4$ 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. © 1996 The American Physical Society.