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A mig redactar…
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$$ \mathcal{H}{\vec{R}}=-\frac{\hbar^2}{2M}\vec{\nabla}^2{\vec{R}} \qquad\qquad \mathcal{H}{\vec{r}}=-\frac{\hbar^2}{2\mu}\vec{\nabla}^2{\vec{r}}+V(\vec{r}) $$
A partir d’ara quan parlem de $\mathcal{H}$ ens estarem referint a $\mathcal{H}_{\vec{r}}$.
$$
\begin{gathered} \mathcal{H}{f s}=\mathcal{H}{P}+\mathcal{H}{r e l}+\mathcal{H}{S O}+\mathcal{H}{D} \\ \mathcal{H}{\mathrm{P}}=-\frac{\hbar^{2}}{2 \mathrm{~m}{\mathrm{e}}} \nabla^{2}+V(r) \quad \mathcal{H}{\mathrm{rel}}=-\frac{1}{8 \mathrm{~m}{\mathrm{e}}^{3} c^{2}} p^{4} \\ \mathcal{H}{\mathrm{SO}}=\frac{\hbar^{2}}{2 \mathrm{~m}{\mathrm{e}}^{2} c^{2}} \frac{1}{r} \frac{\mathrm{~d} V}{\mathrm{~d} r} \mathbf{L} \cdot \mathbf{S} \quad \mathcal{H}{\mathrm{D}}=\frac{\hbar^{2}}{8 \mathrm{~m}{\mathrm{e}}^{2} c^{2}}\left(\nabla^{2} V\right) \\ \left\langle\mathcal{H}{s o}\right\rangle=-E_{n} \frac{(\alpha Z)^{2}}{2 n} \times \begin{cases}\frac{1}{\left(l+\frac{1}{2}\right)(l+1)} & \text { si } j=l+\frac{1}{2} \\ \frac{-1}{l\left(l+\frac{1}{2}\right)} & \text { si } j=l-\frac{1}{2} \\ 0 & \text { si } l=0\end{cases} \\ \left\langle\mathcal{H}{r e l}\right\rangle=-E{n}\left(\frac{\alpha Z}{n}\right)^{2}\left(\frac{3}{4}-\frac{n}{l+\frac{1}{2}}\right) \leq 0 \\ \left\langle\mathcal{H}{\mathrm{D}}\right\rangle=-E{n} \frac{(\alpha Z)^{2}}{n} \delta_{l 0} \geq 0 \end{gathered} $$
$$ \vec{r}1 \cdot \vec{r}2 = r_1 r_2 \cos\theta{12} = r_1 r_2 \left( \frac{4\pi}{3} \sum{q=-1}^{+1} Y_{1q}^*(\hat{\mathbf{r}}1) Y{1q}(\hat{\mathbf{r}}_2) \right), $$