Simplify: $\;$ $\left(\dfrac{\sqrt{a}}{2} - \dfrac{1}{2 \sqrt{a}}\right)^2 \left(\dfrac{\sqrt{a} - 1}{\sqrt{a} + 1} - \dfrac{\sqrt{a} + 1}{\sqrt{a} - 1}\right)$
$\left(\dfrac{\sqrt{a}}{2} - \dfrac{1}{2 \sqrt{a}}\right)^2 \left(\dfrac{\sqrt{a} - 1}{\sqrt{a} + 1} - \dfrac{\sqrt{a} + 1}{\sqrt{a} - 1}\right)$
$= \left(\dfrac{2a - 2}{4 \sqrt{a}}\right)^2 \left[\dfrac{\left(\sqrt{a} - 1\right)^2 - \left(\sqrt{a} + 1\right)^2}{\left(\sqrt{a} + 1\right) \left(\sqrt{a} - 1\right)}\right]$
$= \left(\dfrac{a - 1}{2 \sqrt{a}}\right)^2 \left[\dfrac{a - 2 \sqrt{a} + 1 - \left(a + 2 \sqrt{a} + 1\right)}{a - 1}\right]$
$= \dfrac{a - 1}{4a} \times \left(- 4 \sqrt{a}\right)$
$= \dfrac{\left(1 - a\right) \sqrt{a}}{a}$
$= \dfrac{\left(1 - a\right) a}{a \sqrt{a}}$
$= \dfrac{1 - a}{\sqrt{a}}$