The occupation of the centre-of-mass quantum states of a gas layer of non-interacting bosons is studied theoretically. The atomic gas is trapped in a gravito-optical trap in one dimension, moving freely in the two transverse dimensions. The eigenfunctions and eigenvalues of the trap are considered. Using the discrete spectrum, a quantum statistical effect is shown in the longitudinal density profile, obtained by tracing over the transverse Bose-Einstein momentum distribution.