We numerically study quantum mechanical features of the Bunimovich stadium
billiard and the rational billiards which approach the former as the number of
their sides increases. The statistics of energy levels and eigenfunctions of the
rational billiards becomes indistinguishable from that of the Bunimovich
stadium billiard below a certain energy. This fact contradicts the classical
picture in which the Bunimovich stadium billiard is chaotic, but the rational
billiard is pseudointegrable. It is numerically confirmed that the wave functions
do not detect the fine structure, which is much smaller than the wavelength.