  A quantum wave function with localization on classical periodic orbits in a
mesoscopic elliptic billiard has been achieved by appropriately superposing
nearly degenerate eigenstates expressed as products of Mathieu functions. We
analyze and discuss the rotational and librational regimes of motion in the
elliptic billiard. Simplified line equations corresponding to the classical
trajectories can be extracted from the quantum state as an integral equation
involving angular Mathieu functions. The phase factors appearing in the
integrals are connected to the classical initial positions and velocity
components. We analyze the probability current density, the phase maps, and the
vortex distributions of the periodic orbit quantum states for both rotational
and librational motions; furthermore, they may represent traveling and standing
trajectories inside the elliptic billiard.
