投影模型中的单体跃迁

1. 投影框架下的本征态

如果变分得到的能量最低波函数为 \(|PC\rangle\),这里 PC 是 pair condensate 的缩写,表示我们最近做的对凝聚组态,但下面的讨论并不局限于这一种组态。
如果要得到角动量为 \((J,M)\) 的近似本征态,我们先从 \(|PC\rangle\) 中投影出来 \(2J+1\) 个角动量为 \((J,M)\) 的基矢:

\[\hat{P}^J_{MK} | PC \rangle, K = -J, -J+1, \cdots, J. \]

然后假定近似波函数为这些基矢的线性展开,

\[\psi^r_{JM} = \sum_K g^r_{JK} \hat{P}^J_{MK} | PC \rangle. \]

那么,要求解 \(g^r_{JK}\),需要构造 Hill-Wheeler 方程。

\[H^J_{K'K} = \langle \Phi | \hat{H} \hat{P}^J_{K'K} | \Phi \rangle, ~~~~ N^J_{K'K} = \langle \Phi | \hat{P}^J_{K'K} | \Phi \rangle, \]

Hill-Wheeler 方程为

\[\forall K', \sum_K H^J_{K'K} g^r_{JK} = \epsilon_{r,J} \sum_K N^J_{K'K} g^r_{JK}. \]

得到 \(g^r_{JK}\),即得近似本征波函数 \(\psi^r_{JM}\)

2. 投影框架下的单体跃迁

不妨把单体跃迁算符记作:

\[\hat{Q}^s_\sigma = \hat{Q}^{\pi s}_\sigma + \hat{Q}^{\nu s}_\sigma, \]

2.1 投影基矢上的约化矩阵元

这个单体算符在投影基矢上的约化矩阵元为 \(\langle (\hat{P}^{J'}_{*K'} PC) || Q^s || \hat{P}^J_{* K} PC \rangle\),则有

\[\langle PC | (\hat{P}^{J'}_{M'K'})^\dagger Q^s_\sigma \hat{P}^J_{MK} | PC \rangle = (JMs\sigma|J'M') \langle (\hat{P}^{J'}_{*K'} PC) || Q^s || \hat{P}^J_{* K} PC \rangle. \]

另外,由于 \((\hat{P}^{J'}_{M' K'})^\dagger = \hat{P}^{J'}_{K' M'}\),它的作用是将右侧 \((J',M')\) 张量挑出来并且旋转为 \((J',K')\),所以有

\[\langle PC | (\hat{P}^{J'}_{M'K'})^\dagger Q^s_\sigma \hat{P}^J_{MK} | PC \rangle = \langle PC | (\hat{P}^{J'}_{M'K'})^\dagger \sum_{J'' M''} (s \sigma J M | J'' M'')(Q^s \hat{P}^J_{*K}|PC\rangle)^{J''}_{M''}, = (s \sigma J M | J' M' ) \langle PC | (Q^s \hat{P}^J_{*K} )^{J'}_{K'} | PC \rangle. \]

对照上面两式,得到

\[\langle (\hat{P}^{J'}_{*K'} PC) || Q^s || \hat{P}^J_{* K} PC \rangle = (-1)^{s + J - J'} \langle PC | (Q^s \hat{P}^J_{*K} )^{J'}_{K'} | PC \rangle = \sum_{\sigma M} C^{J' K'}_{J M s \sigma} \langle PC | Q^s_\sigma \hat{P}^J_{MK} | PC \rangle. \]

2.2 投影波函数上的约化矩阵元

在上文的约定之下,假设初态波函数为

\[\psi^r_{JM} = \sum_k g^r_{JK} \hat{P}^J_{MK} | PC \rangle, \]

末态波函数为

\[\psi^{r'}_{J'M'} = \sum_{K'} g^{r'}_{J'K'} \hat{P}^{J'}_{M'K'} |PC \rangle, \]

那么,初末态之间,\(\hat{Q}^s\) 的约化矩阵元为

\[\langle \psi^{r'}_{J'} || \hat{Q}^s || \psi^r_J \rangle = \sum_{KK'} g^r_{JK} g^{r'}_{J'K'} \langle (\hat{P}^{J'}_{* K'} PC ) || \hat{Q}^s || ( \hat{P}^J_{* K} PC ) \rangle = \sum_{KK'} g^r_{JK} g^{r'}_{J'K'} \sum_{\sigma M} C^{J' K'}_{J M s \sigma} \langle PC | Q^s_\sigma \hat{P}^J_{MK} | PC \rangle. \]

2.3 约化跃迁概率

两个投影本征态之间的约化跃迁概率为

\[B(F:J_r \rightarrow J'_{r'} ) = \frac{2J' +1}{2J +1} | \langle \psi^{r'}_{J'} || \hat{Q}^s || \psi^r_J \rangle |^2 = \frac{2J' +1}{2J+1} \left| \sum_{KK'} g^r_{JK} g^{r'}_{J'K'} \langle (\hat{P}^{J'}_{* K'} PC ) || \hat{Q}^s || ( \hat{P}^J_{* K} PC ) \rangle \right|^2 \\ = \frac{2J' +1}{2J+1} \left|\sum_{KK'} g^r_{JK} g^{r'}_{J'K'} \sum_{\sigma M} C^{J' K'}_{J M s \sigma} \langle PC | Q^s_\sigma \hat{P}^J_{MK} | PC \rangle \right|^2 \\ = \frac{2J' +1}{2J+1} \left| \sum_{K K' \sigma } g^r_{JK} g^{r'}_{J'K'} C^{J', K'}_{J K'-\sigma ; s\sigma} \langle PC | Q^s_\sigma \hat{P}^J_{K'-\sigma, K} | PC \rangle \right|^2. \]

posted on 2021-06-10 15:45  luyi07  阅读(230)  评论(0编辑  收藏  举报

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