Quantization of Space in the Presence of a Minimal Length

In this article, we apply the Generalized Uncertainty Principle (GUP), 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 αp³ (the result of the maximum momentum assumption) and α²p⁴ (the result of the minimum length assumption), where α ～ 1 / M_Plc is the GUP parameter. On the basis of the work by Ali et al., we solve the generalized Schrödinger equation which is extended to include the α² 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 α₀l_Pl in GUP scenario.