Q:

Are distributions of position and momentum assumed to be independent in quantum mechanics?

Given a wave-function of a single particle we can calculate probability density for positions. We can also calculate probability density for momenta. Are these probability densities assumed to be always independent?
Or, in other words, if we measure position and momentum of a particle (for example electron in hydrogen being in the ground state), should we expect that these two random quantities independent?

A:

To say that position and momentum are not independent random variables in QM really undersells the point. It is not that we cannot measure them independently, the Heisenberg uncertainty principle states that we cannot measure both quantities simultaneously at all.
What this means is that the distribution of momentum depends not only on the outcome of a measurement of position, it depends on whether we measured position at all. We cannot measure momentum and then ask what we would have measured if we had also tried to measure position, because that would change the outcome of the momentum measurement.

