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| 両方とも前のリビジョン前のリビジョン次のリビジョン | 前のリビジョン | ||
| research:memos:reduces_transition_probability [2020/04/01 22:47] – [Reduced matrix element of the electric operator] kobayash | research:memos:reduces_transition_probability [2020/04/01 23:25] (現在) – [Reduced matrix element of the electric operator for wave functions in a spherically symmetric potential] kobayash | ||
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| 行 68: | 行 68: | ||
| & \times\left[\Gamma\left(\frac{l_1+l_2+\lambda+3}{2}\right) | & \times\left[\Gamma\left(\frac{l_1+l_2+\lambda+3}{2}\right) | ||
| \sum_{k=0}^{\min(n_1-1, | \sum_{k=0}^{\min(n_1-1, | ||
| + | \end{align} | ||
| + | |||
| + | If $l_1+\lambda=l_2$, | ||
| + | |||
| + | \begin{align} | ||
| + | \langle{j_2}\left|r^\lambda\right|{j_1}\rangle | ||
| + | = (-1)^{n_1+n_2}\left(\frac{\hbar}{m\omega}\right)^{\frac{l_2-l_1}{2}}\sqrt{\frac{(n_1-1)!\Gamma(n_2+l_2+1/ | ||
| + | \binom{l_2-l_1}{n_1-n_2}. | ||
| + | \end{align} | ||
| + | |||
| + | Therefore, | ||
| + | |||
| + | \begin{align} | ||
| + | B(E\lambda; j_1 \rightarrow j_2) | ||
| + | =&\ e^2 \frac{2\lambda+1}{4\pi}\left\langle j_1 \frac{1}{2} \lambda 0 \left| j_2 \frac{1}{2}\right.\right\rangle^2 | ||
| + | \left[(-1)^{n_1+n_2}\left(\frac{\hbar}{m\omega}\right)^{\frac{l_2-l_1}{2}}\sqrt{\frac{(n_1-1)!\Gamma(n_2+l_2+1/ | ||
| + | \binom{l_2-l_1}{n_1-n_2}\right]^2\\ | ||
| + | =&\ e^2 \frac{2\lambda+1}{4\pi}\left\langle j_1 \frac{1}{2} \lambda 0 \left| j_2 \frac{1}{2}\right.\right\rangle^2 | ||
| + | \left(\frac{\hbar}{m\omega}\right)^{\lambda}\frac{(n_1-1)!\Gamma(n_2+l_2+1/ | ||
| + | \binom{\lambda}{n_1-n_2}^2. | ||
| + | \end{align} | ||
| + | |||
| + | |||
| + | If $l_1+\lambda=l_2$ and $\lambda=1$, | ||
| + | |||
| + | |||
| + | \begin{align} | ||
| + | B(E\lambda; j_1 \rightarrow j_2) | ||
| + | =&\ e^2 \frac{2+1}{4\pi}\left\langle j_1 \frac{1}{2} 1 0 \left| j_2 \frac{1}{2}\right.\right\rangle^2 | ||
| + | \frac{\hbar}{m\omega}\frac{(n_1-1)!\Gamma(n_2+l_2+1/ | ||
| + | \binom{1}{n_1-n_2}^2\\ | ||
| + | =&\ e^2 \frac{3}{4\pi}\left\langle j_1 \frac{1}{2} 1 0 \left| j_2 \frac{1}{2}\right.\right\rangle^2 | ||
| + | \frac{\hbar}{m\omega}\frac{(n_1-1)!\Gamma(n_2+l_2+1/ | ||
| \end{align} | \end{align} | ||
