Abstract: In order to derive a convenient solution to a one-dimensional quantum system, a method is presented to solve the one-dimensional quantum chain and quantum ring. The eigen-energy spectra of the two structures are compared by a geometric approach so as to further clarify the characteristics of the eigen-energy spectra. In addition, the changes in the eigen-energies are discussed in the presence of the fluctuated onsite energy, and the conditions are given under which the surface state in the quantum chain structure appears. On the other hand, no surface state appears in the quantum ring despite the fluctuation of the onsite energy.