Apologies if this is a little vague. It might not have a good answer.
Given the interpretation of $|\psi(x)|^2$ as a probability distribution it's unsurprising that a wave function that is concentrated around a point $x$ should behave at least a little like a classical particle at the point $x$.
Is there a similarly intuitive explanation for why a plane wave function $\exp(ik\cdot x)$ behaves somewhat like a classical particle with momentum $\hbar k$? I'm not looking for the standard explanation in terms of eigenstates of the momentum operator, but something that can be used pedagogically for people whose linear algebra isn't sophisticated enough for that.
For example it's not hard to see that a plane wave in n-dimensions has a direction associated with it but it's not intuitively obvious to me that a higher frequency wave should have a higher momentum (unless I reason via the Schrödinger equation which I don't want to do). It's also not surprising that a plane EM wave carries momentum, after all it can interact with charged matter via the Lorentz force and transfer momentum to it, but wave functions don't have such a straightforwardly interpretable interaction.
So how can we make it unsurprising that a plane wave function has a definite momentum?