Interacción luz-materia#
(347)#\[\begin{split}\vec{E}=\vec{E}_0 e^{i(kz-\omega t)}\vec{u}_x\\
\vec{B}=\frac{\vec{E}_0}{c} e^{i(kz-\omega t)}\vec{u}_y\end{split}\]
(348)#\[\begin{split} \left.
\begin{array}{l}
\text{Coulomb gauge} \\
\vec{\nabla}\vec{A}=0, v=0
\end{array}
\right\}\rightarrow\frac{1}{2m}\left[\vec{p}+e\vec{A}\right]^2=
\frac{1}{2m}\vec{p}^2+\frac{e}{2m}\left[\vec{p}\vec{A}+\vec{A}\vec{p}\right]+e^2\vert\vec{A}\vert^2\end{split}\]
(349)#\[h_\text{int}=\frac{e}{2m}\left[\vec{p}\vec{A}+\vec{A}\vec{p}\right]\]
(350)#\[\vec{A}=\vec{A}_0 \left[e^{i\left(\vec{q}\vec{r}-\omega t\right)}-e^{-i\left(\vec{q}\vec{r}-\omega t\right)}\right]\approx \vec{A}_0 \left(e^{-i\omega t}-e^{i\omega t}\right)~~~~q=2\pi/\lambda\]
(351)#\[v=e\vec{E}\vec{r}\]
(352)#\[\mathcal{P}_{i\rightarrow f} = \frac{2\pi}{\hbar}\left\vert \left\langle \psi_i \right\vert \hat{H}_\text{int} \left\vert \psi_f \right\rangle \right\vert^2 \]
(353)#\[\vec{\mu}_{if}=\left\langle \psi_i \right\vert \vec{r} \left\vert \psi_f \right\rangle\]
(354)#\[\mathcal{P}_{i\rightarrow f}=\frac{2\pi}{\hbar}\left\vert \left\langle \psi_i \right\vert -e\vec{E}\cdot\vec{r} \left\vert \psi_f \right\rangle \right\vert^2 = \frac{2\pi e^2}{\hbar}\left\vert \vec{E}\cdot\vec{\mu}_{if} \right\vert^2 \]
(355)#\[\mathcal{P}_{i\rightarrow f}=\frac{2\pi}{\hbar}\left\vert \vec{E}\cdot\vec{\mu}_{if} \right\vert^2 e^2\]
Reglas de transición#
(356)#\[\left\langle \psi_g \right\vert \vec{r} \left\vert \psi_u \right\rangle \neq0; ~~ \left\langle \psi_u \right\vert \vec{r} \left\vert \psi_g \right\rangle \neq 0; ~~ \left\langle \psi_g \right\vert \vec{r} \left\vert \psi_g \right\rangle = \left\langle \psi_u \right\vert \vec{r} \left\vert \psi_u \right\rangle=0\]
(357)#\[P_i = \frac{N_i}{N} = \frac{d_i}{Z} e^{-E_i/KT}\]
(358)#\[Z = \sum_i d_i e^{-E_i/KT}\]
(359)#\[\vec{\mu}_{ij}=\left\langle A_{a} \right\vert \vec{r} \left\vert B_{b} \right\rangle\]
(360)#\[\Delta \nu = \pm 1\]
(361)#\[\vec{\mu}_{ij}=\left\langle ev J^\prime M_{J^\prime} \right\vert \vec{\mu} \left\vert ev J M_{J} \right\rangle = \left\langle Y_{J^\prime}^{M_{J^\prime}} \right\vert \vec{\mu}_0 \left\vert Y_J^{M_{J}} \right\rangle\]
(362)#\[\vec{\mu}_0=\left\langle ev \right\vert \vec{\mu} \left\vert ev\right\rangle\]
(363)#\[\begin{split}\left\langle Y_{J^\prime}^{M_{J^\prime}} \right\vert \vec{\mu}_0 \left\vert Y_J^{M_{J}} \right\rangle\neq 0
\rightarrow
\vec{\mu}_0\rightarrow
\begin{cases}
J=1 \\
\text{paridad=-1}
\end{cases}
\Longrightarrow \Delta J = \pm 1 (+1 ~\text{absorción}, -1 ~\text{emisión})\end{split}\]
(364)#\[\Delta M_J =0, \pm 1\]
(365)#\[F(J)=B^\prime J(J+1)\]
(366)#\[J\rightarrow J+1: F(J+1)-F(J)=2B^\prime \left(J+1\right)\]
(367)#\[\vert\vec{\mu}_{J,J+1}\vert^2=\vert\vec{\mu}_{0}\vert^2 \frac{J+1}{2J+1} \rightarrow \frac{\vert\vec{\mu}_{0}\vert^2}{2} \text{para} J\gg1\]
(368)#\[P_{J}=(2J+1)e^{-\frac{hcB'J(J+1)}{kT}}\]
(369)#\[\frac{dP_J}{dJ}=0 \rightarrow J_{\text{max}}=\left(\frac{kT}{2hcB^\prime}\right)^{1/2}-\frac{1}{2}\]
Definición#
The divergence is an operator that is applied on a vector field.
(370)#\[\vec{\nabla}\cdot \vec{f}=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}\]
As can be seen, the result of the applying the divergence is a scalar field.
Interpretation#
The divergence measures the difference between the number of field lines that enter and leave from/to a particular point.
Ejemplo
Fig. 205 The divergence measures the number of lines entering/exiting a point.#
A positive divergence is associated with a majority of lines coming out (leaving) from a point. This point is a source of field lines. On the other hand, a point with a negative divergence is seen to come from a majority of lines entering into a point. This point is called a sink of field line.