通往量子少体问题解的最陡路径：面向量子少体问题的梯度变分方法
The Steepest Slope toward a Quantum Few-body Solution: Gradient Variational Methods for the Quantum Few-body Problem
AI总结：
针对量子少体问题中随机变分法随粒子数增加的优化与数值挑战，提出梯度变分方法族，实验证明其比随机变分法更高效有效，并分析了影响方法性能的关键因素。
摘要：
量子少体系统看似简单，实则不然。事实上，除少数特殊情况外，当粒子数超过两个时，其对应的薛定谔方程无法解析求解，人们必须借助近似方法来处理量子少体问题。其中，变分方法已被提出以简化数值计算并获得精确解，随机变分法（Stochastic Variational Method）便是其中一种，它采用随机搜索来确定用于构建波函数拟设的关联高斯基函数的数量与参数。然而，随着粒子数增加，随机方法面临数值计算与优化层面的挑战。\n我们提出了一类梯度变分方法，用梯度优化替代随机搜索。我们针对选定的少体问题，对比并实证评估了基准随机变分法、梯度变分方法族的若干实例以及部分混合方法的性能。结果表明，梯度方法与混合方法比随机变分法更高效、效果更优。我们还讨论了奇异性、振荡性以及梯度优化策略在各类方法性能中的作用。
英文摘要：
Quantum few-body systems are deceptively simple. Indeed, with the notable exception of a few special cases, their associated Schrodinger equation cannot be solved analytically for more than two particles. One has to resort to approximation methods to tackle quantum few-body problems. In particular, variational methods have been proposed to ease numerical calculations and obtain precise solutions. One such method is the Stochastic Variational Method, which employs a stochastic search to determine the number and parameters of correlated Gaussian basis functions used to construct an ansatz of the wave function. Stochastic methods, however, face numerical and optimization challenges as the number of particles increases. We introduce a family of gradient variational methods that replace stochastic search with gradient optimization. We comparatively and empirically evaluate the performance of the baseline Stochastic Variational Method, several instances of the gradient variational method family, and some hybrid methods for selected few-body problems. We show that gradient and hybrid methods can be more efficient and effective than the Stochastic Variational Method. We discuss the role of singularities, oscillations, and gradient optimization strategies in the performance of the respective methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
