With the aim to construct a dynamical model with quantum group symmetry, the
q-deformed Schrödinger 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 Schrödinger equation and eigenvalues. We also present an alternative way to solve the Schrödinger equation which is based on the
q-analysis. We represent the Schrödinger 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.
The Schrodinger equation with position-dependent mass (PDM) becomes one of interesting subjects in the study of quantum systems, because of its wide applications in many physical problems. Meanwhile, harmonic oscillator becomes important model in most physical problems. In this paper, analytical solutions of the PDM Schrodinger equation, i.e. energy eigenvalues En and eigenfunctions , of a one dimensional harmonic oscillator system using transformation method has been studied. The results showed that there was no difference between the energy eigenvalues, En of the harmonic oscillator systems with and without PDM, whereas it is not for the eigenfunctions, of these two harmonic oscillator systems. This difference was due to the presence of dimensionless mass function and the changes of x becomes variable for the eigenfunctions of the harmonic oscillator system with PDM. Moreover, the differences in the eigenfunctions will affect the dynamical properties of the quantum systems. It can be concluded that the harmonic oscillator system with PDM is the generalization of the harmonic oscillator system with constant mass.