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Eigenvalues of the Schrödinger equation for a periodic potential with nonperiodic boundary conditions: A uniform semiclassical analysis

Connor, J. N. L. and Uzer, T. and Marcus, R. A. and Smith, A. D. (1984) Eigenvalues of the Schrödinger equation for a periodic potential with nonperiodic boundary conditions: A uniform semiclassical analysis. Journal of Chemical Physics, 80 (10). pp. 5095-6006. ISSN 0021-9606. http://resolver.caltech.edu/CaltechAUTHORS:CONjcp84

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Abstract

A uniform semiclassical expression for the eigenvalues of a one dimensional periodic Schrödinger equation with nonperiodic boundary conditions has been derived. The potential energy function can have any number of symmetric or asymmetric barriers and wells. The treatment is uniform in that the classical turning points can come close together, coalesce, and move into the complex plane as the energy passes through a barrier maximum. A detailed application is made to Mathieu functions of integer order; the equations themselves include the case of fractional order. Approximate semiclassical expressions are derived for the widths of the energy bands and the energy gaps of the periodic Mathieu equation when these quantities are small. The semiclassical results give a physical interpretation to formulas present in the mathematical literature and to the decrease in the splitting of a sequence of avoided crossings with increasing quantum numbers in coupled oscillator systems. Numerical calculations are reported to illustrate the high accuracy of the semiclassical formulas.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1063/1.446581DOIUNSPECIFIED
http://link.aip.org/link/?JCPSA6/80/5095/1PublisherUNSPECIFIED
Additional Information:Copyright ©1984 American Institute of Physics. Received 8 September 1983; accepted 14 October 1983. Support of this research by the National Science Foundation (U.S.A.) and the Science and Engineering Research Council (U.K.) is gratefully acknowledged. We are grateful to NATO for a Senior Scientist Award to J.N.L.C.
Funders:
Funding AgencyGrant Number
National Science FoundationUNSPECIFIED
Science and Engineering Research Council (U.K.)UNSPECIFIED
NATOUNSPECIFIED
Subject Keywords:BOUNDARY CONDITIONS, SEMICLASSICAL APPROXIMATION, POTENTIAL ENERGY, SCHROEDINGER EQUATION, QUANTUM MECHANICS, ONE−DIMENSIONAL CALCULATIONS, EIGENVALUES
Record Number:CaltechAUTHORS:CONjcp84
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:CONjcp84
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11788
Collection:CaltechAUTHORS
Deposited By: Archive Administrator
Deposited On:29 Sep 2008 20:04
Last Modified:26 Dec 2012 10:20

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