Higher-order calculation of transmission below the potential barrier

Abstract

In extending the Miller-Good modified WKB approximation to include the higher-order terms, a divergence was introduced. Because of this divergence, the approximation was limited to energies above the potential barrier. With this divergence removed, the modified WKB method is no longer limited to energies above the potential barrier. In order to demonstrate this method, we calculate the transmission coefficients for energies below the peak of the potential barrier and show that the higher-order terms are essential to the approximation.