Quantization of Space in the Presence of a Minimal Length

In this article, we apply the Generalized Uncertainty Principle (CUP), which is consistent with quantum gravity theories to an elementary particle in a finite potential well, and study the quantum behavior in this system. The generalized Hamiltonian contains two additional terms, which are proportional to alpha p(3) (the result of the maximum momentum assumption) and alpha(2)p(4) (the result of the minimum length assumption), where alpha similar to 1/M-PIc is the CUP parameter. On the basis of the work by Ali et al., we solve the generalized Schrodinger equation which is extended to include the alpha(2) correction term, and find that the length L of the finite potential well must be quantized. Then a generalization to the double-square-well potential is discussed. The result shows that all the measurable lengths especially the distance between the two potential wells are quantized in units of alpha(0)l(PI) in CUP scenario.