• 主题
高级搜索  >
历史搜索 清空历史
首页> 外文会议>Mathematical methods for information science and economics >Kronecker Power Series in Quantum Mechanical Probabilistic Evolution Approach: Managing Arbitrariness in Spectral Issues of the Propagation Superoperator
【24h】

Kronecker Power Series in Quantum Mechanical Probabilistic Evolution Approach: Managing Arbitrariness in Spectral Issues of the Propagation Superoperator

机译：量子力学概率演化方法中的Kronecker幂级数：管理传播超级算子的频谱问题中的任意性

获取原文
获取原文并翻译 | 示例
页面导航
  • 摘要
  • 著录项
  • 相似文献
  • 相关主题

摘要

Recently we have started to use Kronecker power series instead of the multivariate Taylor series, in the formulation of the "Probabilistic Evolution Approach (PEA)" to ODEs, Quantum Expectation Value Dynamics, and, Classical Statistical Mechanics within the perspective of Liouville equation, density and partition functions. The basic idea has been to expand the unknown entities in terms of Kronecker powers of a vector describing the system under consideration. This system vector is either composed of the temporally varying unknowns in the case of ODEs or certain operators' expectation values varying in time for the other cases. The Kronecker powers of the state vector (or their expec tation values in the case of Quantum Mechanics or Classical Statistical Mechanics) have been considered as the basis set elements and certain ODEs for each of them have been constructed. The result in all cases (for ODEs, Quantum Expectation Values, Statistical Mechanical Expectation Values) was a first order linear homogeneous infinite vector ODE with generally initial value impositions, such that the coefficient matrix (we call "Evolution Matrix) for this infinite explicit vector ODE was a constant infinite square matrix. The formal analytical solution of this infinite vector ODE can be obtained and requires the evaluation of an exponential matrix varying in time whose proportionality coefficient is the Evolution Matrix. This evaluation is facilitated when the evolution matrix (which is in upper block Hessenberg form most generally) becomes block triangular because of certain limitations in the system. Triangularity makes the spectral analyses quite simple. Beyond that, the case of multinomiality where the Evolution Matrix has the main diagonal and its few upper neighbor diagonals as the nonvanishing substructures, enables to use the finite order block recursions to get solution to PEA equations. The case of conicality is the simplest form of the multinomiality and corresponds to two term block recursions whose solutions can be analytically constructed as infinite series of the initial values of the Kronecker powers of the state vector or its expectation values. In fact all multinomial cases can be converted to two term block recursions via appropriate order reductive manipulations. What we have told above is somehow the review of the last year developments of the "Probabilistic Evolution Theory" and it will be kept sufficiently comprehensive but, at the same time, sufficiently short during the presentation. The remaining part is completely new and based on recently developed issues. The Kronecker powers of the state vector(s) contain certain number of identicalities or linear dependences as the price of the brevity in the relevant multivariate representation. These can be in fact reflected to the Kronecker power series coefficients as certain level of arbitrarinesses. These arbitrarinesses can be expressed in terms of certain flexible parameters which can be determined as what we want to obtain, of course, within certain limitations. A special emphasis will be given on the commutator algebra over the state vector's Kronecker powers. The propagation superoperator acting on an operator to give the time variant exponential function image of the operator's commutator with the Hamiltonian. The construction of certain eigenoperators of the propagation superoperator will be explained in the perspective of the Kronecker power series and the management of the arbitrariness appearing there.
机译：最近，我们开始使用Kronecker幂级数代替多元泰勒级数，在针对ODE，量子期望值动力学和古典统计力学的Liouville方程，密度的“概率演化方法（PEA）”的公式化中和分区功能。基本思想是根据描述系统的向量的Kronecker幂来扩展未知实体。在ODE的情况下，此系统矢量由时间上变化的未知数组成，在其他情况下，某些操作员的期望值随时间变化。状态向量的Kronecker幂（或在量子力学或经典统计力学的情况下，其期望值）已被视为基础集元素，并且已为它们构造了某些ODE。在所有情况下（对于ODE，量子期望值，统计机械期望值），结果都是一阶线性均质无限矢量ODE，其一般具有初始值强制，因此该无限显式矢量的系数矩阵（我们称为“演化矩阵”） ODE是一个恒定的无限方矩阵，可以得到此无限矢量ODE的形式解析解，并且需要对随时间变化的指数矩阵进行评估，其比例系数为Evolution Matrix。由于系统中的某些限制，最上部的Hessenberg形式变为块三角形。三角性使频谱分析变得非常简单。此外，在多项式的情况下，Evolution矩阵具有主对角线，而很少的上邻对角线为不消失的子结构，可以使用有限阶块递归来得到PEA方程的解。圆锥形的情况是多项式的最简单形式，对应于两个项块递归，其解可以解析为状态向量的Kronecker幂的初始值或其期望值的无穷级数。实际上，所有多项式情况都可以通过适当的阶数归约操作转换为两个项块递归。我们上面所说的是对“概率演化理论”去年发展的回顾，它将保持足够全面，但同时在演讲过程中也要足够简短。其余部分是全新的，并且基于最近开发的问题。状态向量的Kronecker幂包含一定数量的相同性或线性相关性，作为相关多元表示中简短性的代价。实际上，这些可以反映为Kronecker幂级数系数，具有一定程度的任意性。这些任意性可以用某些灵活的参数表示，这些参数可以确定为我们想要获得的，当然是在一定的限制范围内。将特别着重于状态向量的Kronecker幂的换向器代数。传播超级算子作用于算子上，以给出具有哈密顿量的算子换向器的时变指数函数图像。传播超级算子的某些本征算子的构造将在克罗内克幂级数和其中出现的任意性管理的角度进行解释。

著录项

相似文献

  • 外文文献
  • 中文文献
  • 专利
获取原文

客服邮箱：kefu@zhangqiaokeyan.com

京公网安备：11010802029741号 ICP备案号：京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有

  • 客服微信

  • 服务号