diff options
Diffstat (limited to 'content/know/concept/selection-rules')
| -rw-r--r-- | content/know/concept/selection-rules/index.pdc | 9 |
1 files changed, 0 insertions, 9 deletions
diff --git a/content/know/concept/selection-rules/index.pdc b/content/know/concept/selection-rules/index.pdc index 22dfd64..9fda878 100644 --- a/content/know/concept/selection-rules/index.pdc +++ b/content/know/concept/selection-rules/index.pdc @@ -304,10 +304,6 @@ By inserting the expressions we found earlier for these commutators, we get: $$\begin{aligned} \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}} - %&= - 2 \hbar^2 \big( 2 (\hat{z} \hat{L}_x - \hat{x} \hat{L}_z - i \hbar \hat{y}) \hat{L}_z - %- 2 (\hat{x} \hat{L}_y - \hat{y} \hat{L}_x - i \hbar \hat{z}) \hat{L}_y - %- (\hat{L}^2 \hat{x} - \hat{x} \hat{L}^2) \big) - %\\ &= - 4 \hbar^2 \big( \hat{z} \hat{L}_x \hat{L}_z - \hat{x} \hat{L}_z^2 - i \hbar \hat{y} \hat{L}_z + \hat{y} \hat{L}_x \hat{L}_y - \hat{x} \hat{L}_y^2 + i \hbar \hat{z} \hat{L}_y \big) \\ &\qquad\qquad + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big) @@ -334,10 +330,6 @@ which we use to arrive at: $$\begin{aligned} \comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\hat{x}}} - %&= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2 - %- \hat{x} \hat{L}_x^2 - \hat{x} \hat{L}_y^2 - \hat{x} \hat{L}_z^2 \big) - %+ 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big) - %\\ &= - 4 \hbar^2 \big( \hat{z} \hat{L}_z \hat{L}_x + \hat{y} \hat{L}_y \hat{L}_x + \hat{x} \hat{L}_x^2 - \hat{x} \hat{L}^2 \big) + 2 \hbar^2 \big( \hat{L}^2 \hat{x} - \hat{x} \hat{L}^2 \big) \\ @@ -386,7 +378,6 @@ so the right-hand side becomes: $$\begin{aligned} 2 \hbar^2 \matrixel{f}{\vu{r} \hat{L}^2 \!\!+\! \hat{L}^2 \!\vu{r}}{i} - %\matrixel{\ell_f m_f}{\comm*{\hat{L}^2}{\comm*{\hat{L}^2}{\vu{r}}}}{\ell_i m_i} &= 2 \hbar^2 \big( \matrixel{f}{\vu{r} \hat{L}^2}{i} + \matrixel{f}{\hat{L}^2 \vu{r}}{i} \big) \\ &= 2 \hbar^2 \big( \hbar^2 \ell_i (\ell_i \!+\! 1) \matrixel{f}{\vu{r}}{i} |