宇称投影

1. 宇称投影算符

宇称投影算符\(\hat{P}\)作用在任意函数\(f(x)\)得到

\[\hat{P} f(x) = f(-x), \]

三维空间中则为

\[\hat{P} f(\vec{r}) = f(-\vec{r}). \]

如果有 \(\hat{P} f(\vec{r}) = f(-\vec{r}) = \pm f(\vec{r})\)，则称 \(f(\vec{r})\) 有偶、奇宇称。
宇称算符与反对称化算符是可交换的，所以它作用在多体波函数上，即作用在其中每个单粒子基上。

2. 多体波函数

任意多体波函数 \(\Psi\) 都可以分解为偶宇称、奇宇称两部分，

\[\Psi = \Psi_{even} + \Psi_{odd}. \]

若想从中投影出偶/奇宇称，则可以这样做：

\[\Psi_{even} = \frac{1}{2} ( 1 + \hat{P}) \Psi, ~~~ \Psi_{odd} = \frac{1}{2} ( 1 - \hat{P}) \Psi. \]

3. 角动量、宇称投影

在前面的笔记中，已经记了一些角动量投影相关的内容，角动量投影中最重要的是所谓“kernal”的计算：

\[\langle \Psi_1 | \hat{R} |\Psi_2 \rangle, \langle \Psi_1 | \hat{H} \hat{R} | \Psi_2 \rangle. \]

那么如果要进行偶宇称的计算，即需要关心

\[\langle \Psi^{(1)}_{even} |\hat{R}| \Psi^{(2)}_{even} \rangle, ~~~ \langle \Psi^{(1)}_{even} | \hat{H} \hat{R} | \Psi^{(2)}_{even} \rangle \]

因为 \(\hat{P}^\dagger = \hat{P}, [ \hat{P}, \hat{H}] = 0, [\hat{P}, \hat{R}] = 0\)，容易推得

\[\langle \Psi^{(1)}_{even} | \hat{R} | \Psi^{(2)}_{even} \rangle = \frac{1}{2} ( \langle \Psi_1 | \hat{R} | \Psi_2 \rangle + \langle \Psi_1 | \hat{R} \hat{P} | \Psi_2 \rangle ), \]

\[\langle \Psi^{(1)}_{even} |\hat{H} \hat{R} | \Psi^{(2)}_{even} \rangle = \frac{1}{2} ( \langle \Psi_1 |\hat{H} \hat{R} | \Psi_2 \rangle + \langle \Psi_1 | \hat{H} \hat{R} \hat{P} | \Psi_2 \rangle ), \]

\[\langle \Psi^{(1)}_{odd} | \hat{R} | \Psi^{(2)}_{odd} \rangle = \frac{1}{2} ( \langle \Psi_1 | \hat{R} | \Psi_2 \rangle - \langle \Psi_1 | \hat{R} \hat{P} | \Psi_2 \rangle ), \]

\[\langle \Psi^{(1)}_{odd} | \hat{H} \hat{R} | \Psi^{(2)}_{odd} \rangle = \frac{1}{2} ( \langle \Psi_1 | \hat{H} \hat{R} | \Psi_2 \rangle - \langle \Psi_1 | \hat{H} \hat{R} \hat{P} | \Psi_2 \rangle ). \]

所以计算量增大一倍即可。

4. 投影波函数上的跃迁

此处为雷杨添加。
因为 \(\hat{P}^{\dagger}=\hat{P},[\hat{P}, \hat{Q^+}]=0,\{\hat{P}, \hat{Q^-}\}=0, [\hat{P}, \hat{R}]=0\) ，容易推得

\[\left\langle\Psi_{\text {even }}^{(1)}|\hat Q^+\hat{R}| \Psi_{\text {even }}^{(2)}\right\rangle =\frac{1}{2}\left(\left\langle\Psi_1|\hat Q^+\hat{R}| \Psi_2\right\rangle+\left\langle\Psi_1|\hat Q^+\hat{R} \hat{P}| \Psi_2\right\rangle\right) \\ \left\langle\Psi_{\text {odd }}^{(1)}|\hat{Q}^+ \hat{R}| \Psi_{\text {odd }}^{(2)}\right\rangle =\frac{1}{2}\left(\left\langle\Psi_1|\hat{Q}^+ \hat{R}| \Psi_2\right\rangle-\left\langle\Psi_1|\hat{Q}^+ \hat{R} \hat{P}| \Psi_2\right\rangle\right) \\ \left\langle\Psi_{\text {even }}^{(1)}|\hat Q^- \hat{R}| \Psi_{\text {odd }}^{(2)}\right\rangle =\frac{1}{2}\left(\left\langle\Psi_1|\hat Q^-\hat{R}| \Psi_2\right\rangle-\left\langle\Psi_1|\hat Q^-\hat{R} \hat{P}| \Psi_2\right\rangle\right) \\ \left\langle\Psi_{\text {odd }}^{(1)}|\hat{Q}^- \hat{R}| \Psi_{\text {even }}^{(2)}\right\rangle =\frac{1}{2}\left(\left\langle\Psi_1|\hat{Q}^- \hat{R}| \Psi_2\right\rangle+\left\langle\Psi_1|\hat{Q}^- \hat{R} \hat{P}| \Psi_2\right\rangle\right) \]