- #1
sergiokapone
- 302
- 17
Is any way to get Rodrigues' rotation formula from matrix exponential
\begin{equation}
e^{i\phi (\star\vec{n}) } = e^{i\phi (\vec{n}\cdot\hat{\vec{S}}) } = \hat{I} + (\star\vec{n})\sin\phi + (\star\vec{n})^2( 1 - \cos\phi ).
\end{equation}
using SO(3) groups comutators properties ONLY like
\begin{equation}\hat S_k\hat S_j-\hat S_j\hat S_k=i\varepsilon_{kjl}\hat S_l,\qquad \hat S^2=2\hat I ?\end{equation}
where ##\vec{n} = (n_x,n_y,n_z)^{\top}##, ##\vec{n}^2 = 1##, and
\begin{equation}\label{}
(\vec{n}\cdot\hat{\vec{S}}) = \star\vec{n} =
\begin{pmatrix}
0 & -n_z & n_y \\
n_z & 0 & -n_x \\
-n_y & n_x & 0 \\
\end{pmatrix},
\end{equation}
\begin{multline}\label{}
\hat{\vec{S}} =
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & i \\
0 & -i & 0 \\
\end{pmatrix}
\vec{e}_x
+
\begin{pmatrix}
0 & 0 & -i \\
0 & 0 & 0 \\
i & 0 & 0 \\
\end{pmatrix}
\vec{e}_y
+
\begin{pmatrix}
0 & i & 0 \\
-i & 0 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}
\vec{e}_z =
\hat{S}_x \vec{e}_x +
\hat{S}_y \vec{e}_y +
\hat{S}_z \vec{e}_z
.
\end{multline}
References: https://en.wikipedia.org/wiki/Axis–angle_representation#Exponential_map_from_so(3)_to_SO(3)
\begin{equation}
e^{i\phi (\star\vec{n}) } = e^{i\phi (\vec{n}\cdot\hat{\vec{S}}) } = \hat{I} + (\star\vec{n})\sin\phi + (\star\vec{n})^2( 1 - \cos\phi ).
\end{equation}
using SO(3) groups comutators properties ONLY like
\begin{equation}\hat S_k\hat S_j-\hat S_j\hat S_k=i\varepsilon_{kjl}\hat S_l,\qquad \hat S^2=2\hat I ?\end{equation}
where ##\vec{n} = (n_x,n_y,n_z)^{\top}##, ##\vec{n}^2 = 1##, and
\begin{equation}\label{}
(\vec{n}\cdot\hat{\vec{S}}) = \star\vec{n} =
\begin{pmatrix}
0 & -n_z & n_y \\
n_z & 0 & -n_x \\
-n_y & n_x & 0 \\
\end{pmatrix},
\end{equation}
\begin{multline}\label{}
\hat{\vec{S}} =
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & i \\
0 & -i & 0 \\
\end{pmatrix}
\vec{e}_x
+
\begin{pmatrix}
0 & 0 & -i \\
0 & 0 & 0 \\
i & 0 & 0 \\
\end{pmatrix}
\vec{e}_y
+
\begin{pmatrix}
0 & i & 0 \\
-i & 0 & 0 \\
0 & 0 & 0 \\
\end{pmatrix}
\vec{e}_z =
\hat{S}_x \vec{e}_x +
\hat{S}_y \vec{e}_y +
\hat{S}_z \vec{e}_z
.
\end{multline}
References: https://en.wikipedia.org/wiki/Axis–angle_representation#Exponential_map_from_so(3)_to_SO(3)
Last edited: