Accuracy of the Transmission Coefficient across Parabolic Barriers as Obtained from a Generalized WKB Approach

Abstract A comparison is carried out in the case of parabolic potential barriers between the transmission coefficient T obtained from the solution of Schrödinger's equation and the approximate transmission coefficient T(WKB) which is derived from a generalised WKB approach. It is shown that T(WKB) is the first term of the asymptotic expansion of T in powers of 1/X2, with X2 ∼ s \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {U_B } $\end{document} UB, where s is the width of the barrier and UB is the barrier height. Expressions are obtained which give an upper limit \documentclass{article}\pagestyle{empty}\begin{document}$ |\overline {T - T(WKB)}| $\end{document} /T(WKB) of the relative error which occurs when using T(WKB) instead of T for the transmission coefficient. It is seen that in most practical cases this error is fairly small.