For polarization and angular momentum, rotating the basis corresponds to a very straightforward physical transformation, namely, the physical rotation of an experimental apparatus about an axis in space. But what is the corresponding physical transformation corresponding to a rotation of basis for linear position-momentum? For example, consider a two-slit apparatus. Rotating the basis cannot correspond to rotating the slits. That just transforms the measurement of position along one axis to position along a different axis. It is not a transformation of measuring position to measuring momentum. To measure momentum you have to measure wavelength, which means transforming the slits into a diffraction grating or something like that, but that's as far as I've gotten in figuring this out myself.
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Its worth looking at https://physics.stackexchange.com/questions/312834/experimentally-measure-velocity-momentum-of-a-particle-in-quantum-mechanics. These answers remind us that measurement in quantum physics, just like classical physics, is a messy business involving modelling of how your particular measurement device works. – isometry Nov 13 '18 at 06:03