  We define a deformed kinetic energy operator for a discrete position space
with a finite number of points. The structure may be either periodic or
nonperiodic with well-defined end points. It is shown that for the nonperiodic
case the translation operator becomes nonunitary due to the end points. This
uniquely defines an algebra which has the desired unique representation. Energy
eigenvalues and energy wave functions for both cases are found. As expected, in
the continuum limit the solution for the nonperiodic case becomes the same as
the solution of an infinite one dimensional square well and the periodic case
solution becomes the same as the solution of a particle in a box with periodic
boundary conditions.
