  The proton momentum distribution, accessible by deep inelastic neutron
scattering, is a very sensitive probe of the potential of mean force
experienced by the protons in hydrogen-bonded systems. In this work we
introduce a novel estimator for the end to end distribution of the Feynman
paths, i.e. the Fourier transform of the momentum distribution. In this
formulation, free particle and environmental contributions factorize. Moreover,
the environmental contribution has a natural analogy to a free energy surface
in statistical mechanics, facilitating the interpretation of experiments. The
new formulation is not only conceptually but also computationally advantageous.
We illustrate the method with applications to an empirical water model,
ab-initio ice, and one dimensional model systems.
