Aproximación armónica#
(372)#\[ \left(\hat{T}_n + E_i(\{\vec{R}\})\right)\Psi_n^{\alpha}=E_{i,\alpha}\Psi_n^{\alpha}\]
(373)#\[\tilde{R}_{\alpha^\prime}=\sqrt{M_\alpha}R_{\alpha^\prime}\]
(374)#\[\hat{T}_n = \sum_{\alpha^\prime} \frac{\hbar^2}{2M_\alpha^\prime} \frac{d^2}{dR^2_{\alpha^\prime}} \Rightarrow \hat{T}_n = \sum_{\alpha^\prime} \frac{\hbar^2}{2} \frac{d^2}{d\tilde{R}^2_{\alpha^\prime}}\]
(375)#\[E_{i}(\{\tilde{R}\})\approx E_{i}(\tilde{R}_0)+\sum_{\alpha^\prime} 0 \tilde{R}_\alpha^\prime + \frac{1}{2}\sum_{\alpha^\prime,\beta^\prime} \left.\frac{d^2 E_i}{d\tilde{R}_{\alpha^\prime} d\tilde{R}_{\beta^\prime}} \right\vert_{R=R_0}\tilde{R}_{\alpha^\prime} \tilde{R}_{\beta^\prime} + \cdots\]
(376)#\[\mathcal{H}=\frac{1}{2} \left.\frac{d^2 E_i}{d\tilde{R}_{\alpha^\prime} d\tilde{R}_{\beta^\prime}} \right\vert_{R=R_0} \rightarrow \mathcal{H}Q_\alpha = K_\alpha Q_\alpha \rightarrow Q_\alpha = \sum_{\alpha^\prime} c_{\alpha \alpha^\prime} \tilde{R}_{\alpha^\prime}\]
Definición#
The divergence is an operator that is applied on a vector field.
(377)#\[\vec{\nabla}\cdot \vec{f}=\frac{\partial f_x}{\partial x}+\frac{\partial f_y}{\partial y}+\frac{\partial f_z}{\partial z}\]
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. 207 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.