When looking at the wave function of a particle, I usually prefer to write
$$ \Psi(x,t) = A \exp(i(kx - \omega t)) $$
since it reminds me of classical waves for which I have an intuition ($k$ tells me how it moves through space ($x$) and $\omega$ tells me how it moves through time ($t$), roughly speaking). However, I noticed that you can translate this from the $(k,\omega)$ into the $(p,E)$ space by extracting $\hbar$:
$$ \Psi(x,t) = A \exp(i(px - Et)/\hbar) $$
When looking at this representation, I couldn't help but be remembered of the uncertainty relations
$$ \Delta p \Delta x \geq \hbar\\ \Delta E \Delta t \geq \hbar $$
This cannot be a coincidence, but the derivation of these relations (as described e.g. on Wikipedia) are simply over my head and my textbook merely motivates them with an argument based on wave packets.
What is the connection here? Is there an intuitive explanation for the recurrence of both uncertainty relations in the wave function of a matter particle?