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Revista cubana de física

The quasiclassical approximation for problems with position dependent mass. The transfer matrix method

Horacio Coppola, Rodríguez Álvarez

Published Jan 1, 1989• 0 citations
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We have obtained a set of coupled differential equations from the continuous limit of the transfer matrix method. Decoupling such a set of equations yields an extension to the Wentzel-Kramers-Brillouin (WKB) approximation for the Schrödinger equation with the position-dependent effective mass (PDEM). In the classically allowed region, the decoupling is to ignore the reflection resulting from the variations of both the potential and effective mass. By considering an infinite-well example with the PDEM, it is shown that the extended WKB approximation can provide not only an estimation to eigenenergies, but also an analytic form to approximate wavefunctions.

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Energy eigenvalues for the systems with position-dependent effective mass

Y C Ou, Zhuangqi Cao, Qishun Shen

Based on the analytical transfer matrix method (ATMM), the universal dispersion equation determining the energy spectrum for an arbitrary potential function experienced by a particle with arbitrary position-dependent effective mass distribution is obtained. As an illustrative example, for a given mass distribution m ( x ) and a potential function V ( x ), our predicted results are in complete agreement with the exact ones. The energy spectrum of a harmonic oscillator with nonconstant mass is obtained. Since the exact solutions to Schrödinger equation with nonconstant effective mass are a valuable means for checking and improving models and numerical methods in the study of electronic properties, the ATMM will find wide applications in material science and condensed matter.

Journal of Physics A Mathematical and General•Mar 23, 2004•39 citations
We have obtained a set of coupled differential equations from the continuous limit of the transfer matrix method. Decoupling such a set of equations yields an extension to the Wentzel-Kramers-Brillouin (WKB) approximation for the Schrödinger equation with the position-dependent effective mass (PDEM). In the classically allowed region, the decoupling is to ignore the reflection resulting from the variations of both the potential and effective mass. By considering an infinite-well example with the PDEM, it is shown that the extended WKB approximation can provide not only an estimation to eigenenergies, but also an analytic form to approximate wavefunctions.
Based on the analytical transfer matrix method (ATMM), the universal dispersion equation determining the energy spectrum for an arbitrary potential function experienced by a particle with arbitrary position-dependent effective mass distribution is obtained. As an illustrative example, for a given mass distribution m ( x ) and a potential function V ( x ), our predicted results are in complete agreement with the exact ones. The energy spectrum of a harmonic oscillator with nonconstant mass is obtained. Since the exact solutions to Schrödinger equation with nonconstant effective mass are a valuable means for checking and improving models and numerical methods in the study of electronic properties, the ATMM will find wide applications in material science and condensed matter.