Suppose we have a world with one massive free particle and one observer. If the observer changes their velocity with respect to the particle, will this have the same effect on $\lvert\Psi(r,t)\rvert^2$ as if $\langle p\rangle$ was changed locally (for example, using compton scattering if we added another particle)?
Part of me says, yes: the "wavelength of $\Psi$" is related to momentum by the de Broglie relation, and momentum is relative (assuming we have just a Galilean transform here).
Part of me says, no: due to the uncertainty principle, the probability of finding a free particle anywhere would seem to be uniform and negligible over all space. From this argument, the wave function shouldn't change, because for a free particle, it wouldn't have a well-defined wavelength to begin with.
I know I'm missing some conceptual connection, because these can't both be true.
I am currently studying an intro QM course, so please take my undergraduate level of understanding into account!