diff options
Diffstat (limited to 'content/know/concept/einstein-coefficients')
| -rw-r--r-- | content/know/concept/einstein-coefficients/index.pdc | 16 |
1 files changed, 9 insertions, 7 deletions
diff --git a/content/know/concept/einstein-coefficients/index.pdc b/content/know/concept/einstein-coefficients/index.pdc index 80707c6..9feaf8c 100644 --- a/content/know/concept/einstein-coefficients/index.pdc +++ b/content/know/concept/einstein-coefficients/index.pdc @@ -170,19 +170,21 @@ $$\begin{aligned} = \frac{\big|\!\matrixel{a}{H_1}{b}\!\big|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_{ba} - \omega) t / 2 \big)}{(\omega_{ba} - \omega)^2} \end{aligned}$$ -If the location of the nucleus of the atom has $z = 0$, +If the nucleus is at $z = 0$, then generally $\ket{1}$ and $\ket{2}$ will be even or odd functions of $z$, -such that $\matrixel{1}{z}{1} = \matrixel{2}{z}{2} = 0$, leading to: +meaning that $\matrixel{1}{z}{1} = \matrixel{2}{z}{2} = 0$ +(see also [Laporte's selection rule](/know/concept/selection-rules/)), +leading to: $$\begin{gathered} - \matrixel{1}{H_1}{2} = - E_0 d + \matrixel{1}{H_1}{2} = - E_0 d^* \qquad - \matrixel{2}{H_1}{1} = - E_0 d^* + \matrixel{2}{H_1}{1} = - E_0 d \\ \matrixel{1}{H_1}{1} = \matrixel{2}{H_1}{2} = 0 \end{gathered}$$ -Where $d \equiv q \matrixel{1}{z}{2}$ is a constant, +Where $d \equiv q \matrixel{2}{z}{1}$ is a constant, namely the $z$-component of the **transition dipole moment**. The chance of an upward jump (i.e. absorption) is: @@ -284,12 +286,12 @@ Let us return to the matrix elements of the perturbation $\hat{H}_1$, and define the polarization unit vector $\vec{n}$: $$\begin{aligned} - \matrixel{1}{\hat{H}_1}{2} + \matrixel{2}{\hat{H}_1}{1} = - \vec{d} \cdot \vec{E}_0 = - E_0 (\vec{d} \cdot \vec{n}) \end{aligned}$$ -Where $\vec{d} \equiv q \matrixel{1}{\vec{r}}{2}$ is +Where $\vec{d} \equiv q \matrixel{2}{\vec{r}}{1}$ is the full **transition dipole moment** vector, which is usually complex. The goal is to calculate the average of $|\vec{d} \cdot \vec{n}|^2$. |