Wigner D-matrix

Table of Contents

1. Wigner D-matrix

\begin{align} R(\alpha, \beta, \gamma) = e^{-\mathrm{i}\alpha J_{z}} e^{-\mathrm{i}\beta J_{y}} e^{-\mathrm{i}g J_{z}} \end{align} \begin{align} D^l_{m' m} (\alpha, \beta, \gamma) \langle l, m'| R(\alpha, \beta, \gamma) |l, m\rangle \end{align}

\(D^l_{m' m}(R)\) 是旋转操作 \(R\) 在一组基底 \(|l, m\rangle\) 下的表示的矩阵元( \(l\) 是固定 的, 也就是说 \(1=\sum_m |l, m\rangle\langle l, m|\) ).

\(D^l(R)\) 是 \(SO(3)\) 群元 \(R\) 在不可约表示 \(l\) 中的表示矩阵.

2. Relation to spherical harmonics

\begin{align} Y_{l, m}^{*} (\theta, \phi) = \langle l, m | \theta, \phi \rangle =& \langle l, m | R(\phi, \theta, \gamma) | \theta = 0, \phi = 0 \rangle \\ =&\sum_{m'} \langle l, m | R(\phi, \theta, \gamma) |l, m'\rangle\langle l, m' | \theta = 0, \phi = 0 \rangle \\ =&\sum_{m'} \langle l, m | R(\phi, \theta, \gamma) |l, m'\rangle \delta_{m', 0}\sqrt{\frac{2l + 1}{4\pi}} \\ =&\langle l, m | R(\phi, \theta, \gamma) |l, 0\rangle \sqrt{\frac{2l + 1}{4\pi}} \\ =& D^l_{m, 0}(\phi, \theta, \gamma) \sqrt{\frac{2l + 1}{4\pi}} \end{align}

3. Reference

Date: 2022-06-07 Tue 00:00

Created: 2022-06-07 Tue 15:06

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