Technically you're correct, but you seem to be shortchanging the import of the uncertainty principle being a consequence of the Fourier transform.
That "trivial" property of the Fourier transform fundamentally limits our ability to measure both position and momentum, or time and energy of a particle. Without that aspect of QM we could get effectively unlimited certainty of those various properties. It deeply implies the universe is non-local in nature.
That "trivial" property of the Fourier transform fundamentally limits our ability to measure both position and momentum, or time and energy of a particle. Without that aspect of QM we could get effectively unlimited certainty of those various properties. It deeply implies the universe is non-local in nature.