Free particle trapped in an infinite quantum well examined through the discrete calculus model

We use the discrete approach to solve the Schrödinger as well as the Bloch equations for a free particle and the quantum gas of free particles embedded in an infinite quantum well with the finite width. We obtain the expressions of energy eigenvalues, the eigenfunctions as well as the density matrix and partition function for the discrete case. By applying the continuous quantum mechanics limit we recover the corresponding results in the continuous-variable quantum mechanics.