3. Let $P$ and $Q$ be two points inside the square pyramid $S-ABCD$, with the lateral edge length being $\frac{\sqrt{3}}{2} a$ and the base edge length being $a$. Then the range of $\angle PSQ$ is $\qquad$
3. $\left(0, \pi-\arccos \frac{1}{3}\right)$.

As shown in Figure 5, it is clear that
$\angle P S Q \geqslant 0$, when $P$ and $Q$ coincide with the diagonal vertices of the base, $\angle P S Q$ reaches its maximum value, $B D=\sqrt{2} a, S B=$ $S D=\frac{\sqrt{3}}{2} a$; then
$$
\cos \angle B S D=\frac{S B^{2}+S D^{2}-B D^{2}}{2 S \bar{B} \cdot S D}=-\frac{1}{3},
$$

thus $\angle B S D=\pi-\arccos \frac{1}{3}$. 
Therefore, the range of $\angle P S Q$ is $\left(0, \pi-\operatorname{arccos} \frac{1}{3}\right)$.