The wave equation contains the entire state of a particle.
And a single vector can be represented in many different ways in different basis, and two of those ways happen to be the position and the momentum representations.
When you represent a wave equation as a function, you have to say what the variable of the function means. And depending on weather you say "it means position" or "it means momentum", the position and momentum operators will be written differently.
This is well shown at: Video 54. ""Visualization of Quantum Physics (Quantum Mechanics)" by udiprod (2017)".
Then the uncertainty principle follows immediately from a general property of the Fourier transform: https://en.wikipedia.org/w/index.php?title=Fourier_transform&oldid=961707157#Uncertainty_principle
- https://www.youtube.com/watch?v=bIIjIZBKgtI&list=PL54DF0652B30D99A4&index=59 "K2. Heisenberg Uncertainty Relation" published by doctorphys