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Accuracy of the Transmission Coefficient across Parabolic Barriers as Obtained from a Generalized WKB Approach
Author(s) -
Berz F.,
Morice J. A.
Publication year - 1987
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.2221390223
Subject(s) - wkb approximation , transmission coefficient , limit (mathematics) , transmission (telecommunications) , mathematical analysis , mathematics , physics , mathematical physics , quantum mechanics , telecommunications , computer science
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/ X 2 , with X 2 ∼ s \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {U_B } $\end{document} U B , where s is the width of the barrier and U B is the barrier height. Expressions are obtained which give an upper limit \documentclass{article}\pagestyle{empty}\begin{document}$ |\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.