An infinite square-well potential as a limiting case of a square-well potential in a minimal-length scenario

One of the most widely problem studied in quantum mechanics is of an infinite square-well potential. In a minimal-length scenario its study requires additional care because the boundary conditions at the walls of the well are not well fixed. In order to avoid this we solve the finite square-well potential whose the boundary conditions are well fixed, even in a minimal-length scenario, and then we take the limit of the potential going to infinity to find the eigenfunctions and the energy equation for the infinite square-well potential. Although the first correction for the energy eigenvalues is the same as one found in the literature, our result shows that the eigenfunctions have the first derivative continuous at the square-well walls what is in disagreement with those previous work. That is because in the literature the authors have neglected the hyperbolic solutions and have assumed the discontinuity of the first derivative of the eigenfunctions at the walls of the infinite square-well which is not correct. As we show, the continuity of the first derivative of the eigenfunctions at the square-well walls guarantees the continuity of the probability current density and the unitarity of the time evolution operator.