The propagator for the step potential using the path decomposition expansion
James M. Yearsley
我们提出了一个直接的路径积分推导的传播子的存在下，一个步骤的潜力。推导利用了路径分解展开（PDX），以及传播子作为格路径的极限的定义。
We present a direct path integral derivation of the propagator in the presence of a step potential. The derivation makes use of the Path Decomposition Expansion (PDX), and also of the definition of the propagator as a limit of lattice paths.
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The propagator for the step potential and delta function potential using the path decomposition expansion
我们提出了一个推导的传播子的粒子在阶梯和δ函数的潜力。这些传播是已知的，但我们提出了一个直接的路径积分推导，基于路径分解展开和布朗运动的路径积分的定义。推导利用了卡塔兰数的性质，卡塔兰数列举了某些类别的格路径。
In designing quantum control, it is generally required to simulate the controlled system evolution with a classical computer. However, computing the time evolution operator can be quite resource-consuming since the total Hamiltonian is often hard to diagonalize. In this paper, we mitigate this issue by substituting the time evolution segments with their Trotter decompositions, which reduces the propagator into a combination of single-qubit operations and fixed-time system evolutions. The resulting procedure can provide substantial speed gain with acceptable costs in the propagator error. As a …
A Chebychev propagator for inhomogeneous Schrödinger equations
我们提出了一个传播计划的时间依赖性非均匀薛定谔方程，例如在最佳控制理论或反应散射计算中出现。一个正式的解决方案的基础上的多项式展开的非齐次项。它是由切比雪夫多项式的近似。不同的变种的非齐次传播的证明和应用到两个例子，从最优控制理论。分析了收敛性和数值效率。
Controllability of quantum mechanical systems by root space decomposition of su(N)
利用半单李代数的结构理论，研究了N能级量子力学系统的幺正传播子在单个控制场作用下的可控性。在一般配置以及在几个退化的情况下，提供了充分条件的向量场。
In this and subsequent paper arXiv:1011.5185 we develop a recursive approach for calculating the short-time expansion of the propagator for a general quantum system in a time-dependent potential to orders that have not yet been accessible before. To this end the propagator is expressed in terms of a discretized effective potential, for which we derive and analytically solve a set of efficient recursion relations. Such a discretized effective potential can be used to substantially speed up numerical Monte Carlo simulations for path integrals, or to set up various analytic approximation techniqu…
An Operator Derivation of the Path Decomposition Expansion
路径分解展开是一种路径积分技术，用于将配置空间中的路径上的和分解为不同空间区域中的路径上的和。它导致了配置空间传播子在配置空间中的任意表面上的分解。例如，它可以用于计算第一次穿越时间的分布。原来的证明在很大程度上依赖于位置表示，特别是对路径积分的属性。本文利用投影算子给出了路分解展开式的一个初等证明。这导致了一个版本的路径分解扩展更一般的配置空间形式比先前给出的。作为例子，给出了动量空间中的路径分解展开。
A simple fourth order propagator based on the Magnus expansion in the Liouville space: Application to a $Λ$-system and assessment of the rotating wave approximation
本文提出基于Magnus展开的四阶传播算子，应用于Λ系统并评估旋转波近似在不同场强下的性能。
One-dimensional Dunkl Quantum Mechanics: A Path Integral Approach
本文利用 Feynman 路径积分方法推导了一维 Wigner-Dunkl 量子力学中的传播子，并通过自由粒子与谐振子情形验证结果，进而从谱分解得到能谱和束缚态波函数。
The explicit criterion is derived in detail for the convergence of the Magnus expansion and the existence of the Magnus solution in the interaction picture, i.e., the exponential propagator in the weakly coupled spin (I=1/2) system SnAMX... (n=1,2,3,...) in which only spin group S is subjected to a time-dependent shaped selective radiofrequency pulse. The derivation is built on the same scheme as the Maricq's (J. Chem. Phys. 86 (1987) 5647). It is shown that the criterion depends only upon amplitude of the time-dependent field applied to the system, and the Magnus expansion converges and the M…
We present a recursive procedure, which is based on the small time expansion of the propagator, in order to generate a semi-classical expansion of the \textit{quantum action} for a quantum mechanical potential in arbitrary dimensions. In the method we use the spectral information emerges from the singularities of the propagator on the complex $t$ plane, which are handled by the $i\ve$ prescription and basic complex analysis. This feature allows for generalization to higher dimensions. We illustrate the procedure by providing simple examples in non-relativistic quantum mechanics.
Simple and general unitarity conserving numerical real time propagators of time dependent Schrödinger equation based on Magnus expansion
本文基于Magnus展开提出了一种简单且通用的守恒单位元实时间传播子方法，用于时间依赖的薛定谔方程，推导出不同阶次的近似表达式，适用于多种量子动力学应用。
We present an algorithmic method for the digital quantum simulation of many-body locally indivisible non-Markovian open quantum systems. It consists of two parts: firstly, a Suzuki-Lie-Trotter decomposition of the global system propagator into the product of subsystem propagators, which may not be quantum channels, and secondly, an algorithmic procedure for the implementation of the subsystem propagators through unitary operations and measurements on a dilated space. By providing rigorous error bounds for the relevant Suzuki-Lie-Trotter decomposition, we are able to analyze the efficiency of t…
Starting from a formulation of Correlated Worldline (CWL) theory in terms of functional integrals over paths, we define propagators for particles and matter fields in this theory. We show that the most natural formulation of CWL theory involves a rescaling of the generating functional for the theory; correlation functions then simplify, and all loops containing gravitons disappear from perturbative expansions. The spacetime metric obeys the Einstein equation, sourced by all of the interacting CWL paths. The matter paths are correlated by gravitation, thereby violating quantum mechanics for lar…
Nonconvergence of the Feynman-Dyson diagrammatic perturbation expansion of propagators
本研究通过数值计算揭示Feynman-Dyson图微扰展开的多种病态行为及不收敛根源，指出Padé近似可修正问题，同时质疑该理论与相关假设结合的有效性。
A recent proposal to connect the loop quantization with the spin foam model for cosmology via the path integral is hereby adapted to the case of mechanical systems within the framework of the so called polymer quantum mechanics. The mechanical models we consider are deparametrized and thus the group averaging technique is used to deal with the corresponding constraints. The transition amplitudes are written in a vertex expansion form used in the spin foam models, where here a vertex is actually a jump in position. Polymer propagators previously obtained by spectral methods for a nonrelativisti…
Improved convergence radius of the Fer expansion for Hermitian generators
本文针对含时哈密顿量的 Fer 展开，将其收敛半径从 2 改进约 30% 至约 2.6058，扩大了该幺正展开的适用条件。
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