  With the aim to construct a dynamical model with quantum group symmetry, the
$q$-deformed Schr\"odinger equation of the harmonic oscillator on the
$N$-dimensional quantum Euclidian space is investigated. After reviewing the
differential calculus on the $q$-Euclidian space, the $q$-analog of the
creation-annihilation operator is constructed. It is shown that it produces
systematically all eigenfunctions of the Schr\"odinger equation and
eigenvalues. We also present 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. The problem of the involution corresponding to
the reality condition is discussed.
