Localization of a wave function with channels of a quartic oscillatorx²y²/2

Abstract

The famous system of x 2 y 2 /2 has four infinite channels along the axes. The classical motion of this system not only reveals chaotic behavior but also expels a particle from the central region through the channels. We quantitatively discuss the localized eigenstates with furthest extension along the channel by adiabatic approximation and numerical calculation. We measure the eigenstates of this system by Shannon information entropy, position uncertainty and Husimi distribution. We show that the localization of eigenstates is strongly affected by the infinite channel. We also reveal the obvious difference from a similar system with the potential 0.01(x 4 + y 4 )/4 + x 2 y 2 /2, which has channels with finite spatial extension.