Geometric potential of surfaces with physical applications in Euclidean spaces

Abstract

In this study, we considered skew curvatures of the surfaces to generate their geometric potentials. The method depends essentially on the mean and Gaussian curvatures and their principal curvatures. In quantum mechanics in the study of the dynamics of massive particle with mass m constrained to move on a surface. In such a case, the difference function of the squared mean curvature with the Gaussian curvature induces a geometric (scalar) potential. This potential appears in the Schröndiger-type equations. Considering the skew curvatures of the rotational surfaces, some results on the meridian curves are obtained. Furthermore, the geometric potentials of level surfaces and generalized helicoidal surfaces are calculated. Finally, we discuss the some applications of these types of surfaces in quantum mechanics.