质量三角形的运动学几何以及三体系统薛定谔方程简化为仅在三角形参数上定义的偏微分方程

Kinematic geometry of mass-triangles and reduction of Schrödinger's equation of three-body systems to partial differential equations solely defined on triangular parameters.

作者信息

Hsiang W Y

机构信息

Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA.

出版信息

Proc Natl Acad Sci U S A. 1997 Aug 19;94(17):8936-8. doi: 10.1073/pnas.94.17.8936.

DOI:10.1073/pnas.94.17.8936

PMID:11038564

原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC22968/

Abstract

Schrödinger's equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, R9, naturally equipped with Jacobi's kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger's equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger's equation to corresponding PDEs solely defined on triangular parameters--i.e., at the level of R6/SO(3)--has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),R6) which enable us to obtain such a reduction of Schrödinger's equation of three-body systems to PDEs solely defined on triangular parameters.

摘要

三体系统的薛定谔方程是一个定义在九维构型空间(R^9)上的线性偏微分方程（PDE），该空间自然配备了雅可比运动度量以及平移和旋转对称性。薛定谔方程关于平移对称性的自然不变性使我们能够将构型空间简化为六维空间，而旋转对称性则提供了角动量守恒的量子力学版本。然而，关于如何最大化利用旋转不变性，以便能够将薛定谔方程简化为仅在三角形参数上定义的相应偏微分方程（即，在(R^6/SO(3))层面）的问题，从未得到充分探讨。本文描述了关于((SO(3),R^6))的轨道几何和调和分析的结果，这些结果使我们能够将三体系统的薛定谔方程简化为仅在三角形参数上定义的偏微分方程。