  We investigate the scattering and two-body bound states of two ultracold
atoms in a quasi-two-dimensional (quasi-2D) confinement, with the confinement
potential being an infinite square well (box potential) in the transverse
($z$-) direction, and the motion of the atoms in the $x$-$y$ plane being free.
Specifically, we calculate the effective 2D scattering length and 2D effective
range of the low-energy scattering, as well as the energy and the
transverse-excited-mode probability of the bound states. Comparing these
results with those obtained under a harmonic transverse confinement potential,
we find that in most of the cases the 2D effective range for the box
confinement is approximately 0.28 of the one for the harmonic confinement.
Moreover, the transverse-excited-mode probability of the bound states for the
box confinement is also much lower than the one for the harmonic confinement.
These results suggest that the transverse excitation in the box confinement is
notably weaker than the one in a harmonic confinement. Therefore, achieving
quasi-2D ultracold gases well-described by pure-2D effective models,
particularly those with 2D contact interaction, is more feasible through box
confinement. Our results are helpful for the quantum simulation of 2D many-body
physics with ultracold atoms, e.g., the suppression of 2D effective range may
lead to an enhancement of quantum anomaly in two-dimensional Fermi Gases.
Additionally, our calculation method is applicable to the two-body problems of
ultracold atoms in other types of quasi-2D confinements.
