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arXiv:cond-mat/0412326

Semiclassical methods for multi-dimensional systems bounded by finite potentials

Wenmin Huang, Cheng-Hung Chang, Chung‐Yu Mou

Published Dec 13, 2004• 0 citations
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Abstract

This work studies the semiclassical methods in multi-dimensional quantum systems bounded by finite potentials. By replacing the Maslov index by the scattering phase, the modified transfer operator method gives rather accurate corrections to the quantum energies of the circular and square potential pots of finite heights. The result justifies the proposed scattering phase correction which paves the way for correcting other semiclassical methods based on Green functions, like Gutzwiller trace formula, dynamical zeta functions, and Landauer-Büttiker formula.

DOI
10.48550/arxiv.cond-mat/0412326

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This work studies the semiclassical methods in multi-dimensional quantum systems bounded by finite potentials. By replacing the Maslov index by the scattering phase, the modified transfer operator method gives rather accurate corrections to the quantum energies of the circular and square potential pots of finite heights. The result justifies the proposed scattering phase correction which paves the way for correcting other semiclassical methods based on Green functions, like Gutzwiller trace formula, dynamical zeta functions, and Landauer-Büttiker formula.
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, dynamical 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.
The scattering properties of any complex scattering potential, , can be obtained from the dynamics of a particular non-unitary two-level quantum system . The application of the adiabatic approximation to yields a semiclassical treatment of the scattering problem. We examine the adiabatic series expansion for the evolution operator of and use it to obtain corrections of arbitrary order to the semiclassical formula for the transfer matrix of v. This results in a high-energy approximation scheme that unlike the semiclassical approximation can be applied for potentials with large derivatives.
当为几个一个维的潜力的散布阶段能确切被导出时，更少在多维的量系统被知道。这个工作提供一个方法扩大一个维的阶段知识到多维的量子化规则。扩展在 Bogomolny 在井由不同高度的步潜力围住的二量使用的转移操作符方法的例子被说明。这个概括半古典作品方法精确地决定系统的精力光谱，它显示建议阶段修正的实质的角色。理论上，结果能被扩大到另外的半古典作品方法，例如 Gutzwiller 踪迹公式，动态希腊语的第六个字母功能，和半古典作品 LandauerB 映牯椠 t ??? 胴吗？？
Sequential stationary phase, i.e., the replacement of a multidimensional stationary phase evaluation by an ordered set of lower-dimensional stationary phase integrations, is applied to uniformly asymptotic path integral forms for the semiclassical propagator. The results are useful formulas for computing trajectory indices (generalized Maslov indices) in any quantum representation for general types of time-dependent Hamiltonians. Index connecting relations that relate the indices for the different representations are also obtained. We also demonstrate a general canonical structure for the semiclassical phase indices that arises naturally through the application of sequential stationary phase. \textcopyright{} 1996 The American Physical Society.