We propose an experiment to determine the density operator for the lon
gitudinal quantum state of an atomic beam. The method is based on tomo
graphic reconstruction of the Wigner function via a set of measured pr
obability distributions for the phase-space rotated position operator.
The time evolution of the Wigner function in free space effectively p
erforms the required phase-space rotation. We propose a state-selectiv
e time-dependent technique for measuring longitudinal probability dist
ributions.