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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 αp 3 (the result of the maximum momentum assumption) and α 2 p 4 (the result of the minimum length assumption), where α ∼ 1/M Pl c 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 α 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 α0 Pl in GUP scenario.

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