  We consider the $q$-deformed Schr\"odinger equation of the harmonic
oscillator on the $N$-dimensional quantum Euclidian space. The creation and
annihilation operator are found, which systematically produce all energy levels
and eigenfunctions of the Schr\"odinger equation. In order to get the
$q$-series representation of the eigenfunction, we also give an alternative way
to solve the Schr\"odinger equation which is based on the $q$-analysis. We
represent the Schr\"odinger equation by the $q$-difference equation and solve
it by using $q$-polynomials and $q$-exponential functions.
