Wave mechanics in quantum phase space: Harmonic oscillator

In this paper, we solve the rigorous solutions of the stationary Schrodinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems. and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems. and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.