Wigner d矩阵的精确对角化高精度计算
High-precision evaluation of Wigner's d-matrix by exact diagonalization
摘要：
维格纳d矩阵的精确计算在各个研究领域都有重要意义。由于存在大量的数字，使用维格纳公式直接计算矩阵会损失精度。我们提出了一个简单的方法来避免这个问题，展开的d-矩阵到一个复杂的傅立叶级数和计算傅立叶系数的角动量算子$J_{y}$在$J_{z}$的本征基精确对角化。这种方法使我们能够计算d矩阵及其各种衍生物的自旋高达几千。我们的方法的d-矩阵的精度约为$10^{-14}$的自旋高达$100$。
英文摘要：
The precise calculations of the Wigner's d-matrix are important in various research fields. Due to the presence of large numbers, direct calculations of the matrix using the Wigner's formula suffer from loss of precision. We present a simple method to avoid this problem by expanding the d-matrix into a complex Fourier series and calculate the Fourier coefficients by exactly diagonalizing the angular-momentum operator $J_{y}$ in the eigenbasis of $J_{z}$. This method allows us to compute the d-matrix and its various derivatives for spins up to a few thousand. The precision of the d-matrix from our method is about $10^{-14}$ for spins up to $100$.
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