  In early 90's Mandel and coworkers performed an experiment \cite{mandel} to
examine the significance of quantum phase operators by measuring the phase
between two optical fields. We show that this type of quantum mechanical phase
measurement is possible for matter-waves of ultracold atoms in a double well.
In the limit of low number of atoms quantum and classical phases are
drastically different. However, in the large particle number limit, they are
quite similar. We assert that the matter-wave counterpart of the experiment
\cite{mandel} is realizable with the evolving technology of atom optics.
