Edwards James P, González-Domínguez Víctor A, Huet Idrish, Trejo Maria Anabel
Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, C.P. 58040, Mexico.
Universidad Politécnica de Chiapas, Carretera Tuxtla-Villaflores, Km. 1+500, Las Brisas, 29150 Suchiapa, Chiapas, Mexico.
Phys Rev E. 2022 Jun;105(6-1):064132. doi: 10.1103/PhysRevE.105.064132.
We outline an approach to calculating the quantum mechanical propagator in the presence of geometrically nontrivial Dirichlet boundary conditions. The method is based on a generalization of an integral transform of the propagator studied in previous work (the so-called "hit function") and a convergent sequence of Padé approximants that exposes the limit of perfectly reflecting boundaries. In this paper the generalized hit function is defined as a many-point propagator, and we describe its relation to the sum over trajectories in the Feynman path integral. We then show how it can be used to calculate the Feynman propagator. We calculate analytically all such hit functions in D=1 and D=3 dimensions, giving recursion relations between them in the same or different dimensions and apply the results to the simple cases of propagation in the presence of perfectly conducting planar and spherical plates. We use these results to conjecture a general analytical formula for the propagator when Dirichlet boundary conditions are present in a given geometry, also explaining how it can be extended for application for more general, nonlocalized potentials. Our work has resonance with previous results obtained by Grosche in the study of path integrals in the presence of delta potentials. We indicate the eventual application in a relativistic context to determining Casimir energies using this technique.
我们概述了一种在存在几何非平凡狄利克雷边界条件的情况下计算量子力学传播子的方法。该方法基于对先前工作中研究的传播子的积分变换（所谓的“命中函数”）的推广以及揭示完全反射边界极限的帕德逼近的收敛序列。在本文中，广义命中函数被定义为多点传播子，我们描述了它与费曼路径积分中轨迹求和的关系。然后我们展示了如何用它来计算费曼传播子。我们解析地计算了一维和三维中的所有此类命中函数，给出了它们在相同或不同维度之间的递归关系，并将结果应用于存在理想导电平面和球面平板时的简单传播情况。我们利用这些结果推测了在给定几何中存在狄利克雷边界条件时传播子的一般解析公式，还解释了如何将其扩展以应用于更一般的非定域势。我们的工作与格罗舍在研究存在δ势的路径积分时获得的先前结果有共鸣。我们指出了该技术在相对论背景下最终用于确定卡西米尔能量的应用。
