Abstract: It is shown, by an example of a single nucleon moving in a triaxial deformed potential, that the symmetry of the eigen-functions of the particle is determined by the form of the potential, and the potential is in turn dependent on the shape of the nucleus which produces it. When the symmetry of the potential varies with the deformation parameter, the change of the symmetry for the eigen-function can be expressed through the representation transformation of the quantum mechanics calculation or by the representation transformation of the eigen-function. Although the problem comes from nuclear structure theory, the idea of this work has the common significance for an independent particle moving in a mean field such as in quantum systems of atoms, molecules and clusters.

Key words: triaxial deformation parameter, diagonalization of matrix, representation transformation