Over the past decade this has become possible with deep inelastic neutron scattering.
Calculation of the probability distribution functions using path integral techniques with potentials from Density Functional Theory (DFT) is considered in
Tunneling and delocalization effects in hydrogen bonded systems: A study in position and momentum space
by Joseph Morrone, Lin Lin, and Roberto Car
They consider three different situations, O-H...O, shown below, and corresponding to average proton donor-acceptor (oxygen-oxygen) distances of 2.53, 2.45, and 2.31 Angstroms respectively (top to bottom). They also correspond to three different phases of ice. I discussed Ice X earlier.
The three corresponding probability distributions for the proton along the oxygen-oxygen distance are shown below. System 1, 2, and 3 correspond to Ice X, VII, and VIII, respectively.
Notice the clear qualitative differences between the distributions in real space.
Now here is the surprising (and disappointing) thing.
The momentum space probability distributions are not that different, as seen below.
[I wish they had plotted system 1 as well].
Hence, one may worry whether one can really discern qualitative differences by finding small quantitative differences.
But the experimentalists claim they can.
Below are the results from a PRL
Anomalous Behavior of Proton Zero Point Motion in Water Confined in Carbon Nanotubes
n.b. they seem to be claiming that the oxygen-oxygen distance is only 2.1 A for the water inside the nanotube, considerably less than in bulk water or even ice X. Is this realistic or understandable?
When the potential has a barrier [but not necessarily tunneling] the momentum distribution has a node at high momentum. Morrone, Lin, and Car reproduce this is a simple model calculation but find that their path integral simulations cannot resolve it.