一种通用PSTD方法：求解任意形式局域势在菲涅尔区和远场区的量子散射
A General PSTD Method to Solve Quantum Scattering in the Fresnel and Far-field regions by A Localized Potential of Arbitrary Form
AI总结：
本文将FDTD/PSTD计算电动力学技术扩展到量子领域，结合总场/散射场方案和量子表面等效定理，构建了求解任意有限范围势量子散射的通用时域数值方法，可同时获得近场、菲涅尔区和远场波函数，并通过偏波法验证了精度。
摘要：
我们提出了一种时域方法，用于求解有限范围内任意势的量子散射。该方法可以获得全空间的散射波函数，包括近场、中场（即菲涅尔区）和远场。这是通过将FDTD计算电动力学的若干技术扩展到量子领域来实现的。总场/散射场方案自然地纳入了入射源条件。内部模型中的波函数，包括相互作用区和近场区域，通过PSTD/FDTD迭代直接计算。我们证明了表面等效定理的量子版本，并将外部自由空间中的波函数与内部模型中的PSTD/FDTD解联系起来。简要讨论了基于重叠区域分解和局部傅里叶基上FFT的PSTD并行实现。这些构建模块组合成一个数值系统，为势散射问题提供了一个通用、稳健的求解器。其精度通过已建立的偏波法进行验证，比较了两者在中心方势散射上的预测结果。进一步的研究表明，远场解不足以模拟菲涅尔区效应。
英文摘要：
We present a time domain method to solve quantum scattering by an arbitrary potential of finite range. The scattering wave function in full space can be obtained, including the near field, the mid field (i.e. Fresnel region) and the far field. This is achieved by extending several techniques of FDTD computational electrodynamics into the quantum realm. The total-field/scattered-field scheme naturally incorporates the incidence source condition. The wave function in the internal model, including the interaction region and the close near field, is directly computed through PSTD/FDTD iterations. The quantum version of surface equivalence theorem is proven and links the wave function in the external free space to the PSTD/FDTD solution in the internal model. Parallel implementation of PSTD based on overlapping domain decomposition and FFT on local Fourier-basis is briefly discussed. These building blocks unite into a numerical system that provides a general, robust solver to potential scattering problems. Its accuracy is verified by the established partial wave method, by comparing the predictions of both on the central square potential scattering. Further investigations show the far-field solution is inadequate for simulating Fresnel-region effects.
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