The discretized Schr ̈ odinger equation for the finite square well and its relationship to solid-state physics

The discretized Schrödinger equation is most often used to solve onedimensional quantum mechanics problems numerically. While it has been recognized for some time that this equation is equivalent to a simple tightbinding model and that the discretization imposes an underlying bandstructure unlike free-space quantum mechanics on the problem, the physical implications of this equivalence largely have been unappreciated and the pedagogical advantages accruing from presenting the problem as one of solid-state physics (and not numerics) remain generally unexplored. This is especially true for the analytically solvable discretized finite square well presented here. There are profound differences in the physics of this model and its continuous-space counterpart which are direct consequences of the imposed bandstructure. For example, in the discrete model the number of bound states plus transmission resonances equals the number of atoms in the quantum well.