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American Journal of Physics

Use of ‖Ψ‖2 and flux to simplify analysis of transmission past rectangular barriers or wells

James E. Draper

Published Jun 1, 1979• 7 citations
Publisher

Abstract

The transmission coefficient for a completely general one-dimensional rectangular potential barrier or well, or series of steps, is analyzed simply by concentrating on ‖Ψ‖2 rather than Ψ. Graphs of ‖Ψ‖2 for all cases are given, whereas only two textbooks graph anything related to Ψ, and those plots of Re(Ψ) can be misinterpreted.

DOI
10.1119/1.11900

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The transmission coefficient for a completely general one-dimensional rectangular potential barrier or well, or series of steps, is analyzed simply by concentrating on ‖Ψ‖2 rather than Ψ. Graphs of ‖Ψ‖2 for all cases are given, whereas only two textbooks graph anything related to Ψ, and those plots of Re(Ψ) can be misinterpreted.
The transmission coefficient for quantal tunneling through a rectangular, one-dimensional potential barrier is analyzed. By employing continuity of ‖ ψ ‖2 and flux the results, even with an asymmetrical barrier, are obtained easily. The asymmetrical barrier has the same transmission in either direction. Considerable insight into the form of the most physical quantity, ‖ ψ(x) ‖2, is afforded with little algebra. Graphs of transmission are given for general barriers, and properties of the solutions are discussed.
The transmission coefficient for a one-dimensional system of N potential barriers of general shapes (unequal or equal) is expressed exactly, by means of the phase-integral method of N. Fröman and P. O. Fröman, in terms of quantities characterizing the separate barriers and wells of the system. The exact formula, being of interest for further rigorous studies of transmission properties, can for evaluation readily be converted into a useful approximate formula by insertion of available phase-integral expressions (of an arbitrary order) for the characteristic quantities mentioned above. Error estimates are given. The sub- and superbarrier cases are treated in a unified way. Some key facts about the phase-integral method are given in Appendix A.