旋转算符
贡献者： addis
如果要将一个三维函数 $f( \boldsymbol{\mathbf{r}} )$ 绕 $z$ 轴旋转角 $\alpha$，我们可以使用极坐标 $f(r, \theta, \phi)$，并用平移算符 $ \exp\left(-\alpha \partial/\partial \phi \right) $ 对坐标 $\phi$ 进行 “平移”。那么球坐标中的算符 $ \partial/\partial \phi $ 在直角坐标中如何表示呢？令 $s = \sqrt{x^2 + y^2}$，该算符的意义是求函数在 $ \hat{\boldsymbol{\mathbf{\phi}}} $ 方向的方向导数乘以 $s$。$ \hat{\boldsymbol{\mathbf{\phi}}} = (-y/s, x/s, 0)$，所以
\begin{equation}
\frac{\partial f}{\partial \phi} = s \boldsymbol\nabla f \boldsymbol\cdot \hat{\boldsymbol{\mathbf{\phi}}} = x \frac{\partial f}{\partial y} - y \frac{\partial f}{\partial x} = \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) f~,
\end{equation}
于是我们就得到直角坐标系中绕 $z$ 轴逆时针旋转的算符为
\begin{equation}
R_z(\alpha) = \exp \left[-\alpha \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) \right] ~.
\end{equation}
同理，我们可以将极坐标的极轴指向 $x$ 或 $y$ 轴正方向，从而得出绕 $x$ 或 $y$ 轴逆时针旋转角 $\alpha$ 的算符分别为
\begin{equation}
\begin{aligned}
R_x(\alpha) &= \exp \left[-\alpha \left(y \frac{\partial}{\partial{z}} - z \frac{\partial}{\partial{y}} \right) \right] ~,\\
R_y(\alpha) &= \exp \left[-\alpha \left(z \frac{\partial}{\partial{x}} - x \frac{\partial}{\partial{z}} \right) \right] ~.
\end{aligned}
\end{equation}
例 1
要将 $f(x, y, z) = x$ 绕 $z$ 轴旋转 $\alpha$ 角，就计算
\begin{equation} \begin{aligned}
&\qquad\exp \left[-\alpha \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) \right] x\\
&= x -\alpha \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) x + \frac{1}{2!} \alpha^2 \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) ^2 x - \dots
\end{aligned} ~\end{equation}
其中
\begin{equation} \begin{aligned}
& \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) x = -y~,\\
& \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) ^2 x = \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) (-y) = -x~,\\
& \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) ^3 x = \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) (-x) = y~,\\
& \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) ^4 x = \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) y = x\\
&\dots
\end{aligned} ~\end{equation}
所以
\begin{equation} \begin{aligned}
&\qquad\exp \left[-\alpha \left(x \frac{\partial}{\partial{y}} - y \frac{\partial}{\partial{x}} \right) \right] x\\
&= \left(1 - \frac{1}{2!}\alpha^2 + \frac{1}{4!}\alpha^4 \dots \right) x + \left(\alpha - \frac{1}{3!}\alpha^3 + \frac{1}{5!}\alpha^5 \dots \right) y\\
&= x \cos\alpha + y \sin\alpha~.
\end{aligned} \end{equation}
显然这就是旋转后所得的函数。
在量子力学中，由直角坐标系中角动量算符 $L_x, L_y, L_z$ 的定义，三个旋转算符分别可以记为
\begin{equation}
\exp\left(- \mathrm{i} \frac{\alpha L_x}{\hbar}\right) ~,
\qquad
\exp\left(- \mathrm{i} \frac{\alpha L_y}{\hbar}\right) ~,
\qquad
\exp\left(- \mathrm{i} \frac{\alpha L_z}{\hbar}\right) ~.
\end{equation}