Communications in Theoretical Physics

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Scattering Phase Correction for Semiclassical Quantization Rules in Multi-Dimensional Quantum Systems

HUANG Wen-Min,1 MOU Chung-Yu,1,2 and CHANG Cheng-Hung2,3,4   


  1. 1Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan

    2Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan

    3Institute of Physics, National Chiao Tung University, Hsinchu, Taiwan

    4Institute of Mathematical Modeling and Scientific Computing, National Chiao Tung University, Hsinchu, Taiwan

  • Received: 2009-02-19 Revised: 2009-10-09 Published: 2010-02-15

Abstract: While the scattering phase for several one-dimensional potentials
can be exactly derived, less is known in multi-dimensional quantum
systems. This work provides a method to extend the one-dimensional
phase knowledge to multi-dimensional quantization rules. The extension is illustrated in the example of Bogomolny's transfer operator method applied in two quantum wells bounded by step potentials of different heights. This generalized semiclassical method accurately determines the energy spectrum of the systems, which indicates the substantial role of the proposed phase
correction. Theoretically, the result can be extended to other semiclassical methods, such as Gutzwiller trace formula, dynamica zeta functions, and semiclassical Landauer-Büttiker formula. In practice, this recipe enhances the applicability of semiclassical methods to multi-dimensional quantum systems bounded by general soft potentials.

Key words: Bogomolny's transfer operator, semiclassical quantization rules, quantum chaos

PACS numbers: 

  • 03.65.Sq