Studies of quantum fields and gravity suggest the existence of a minimal
length, such as Planck length \cite{Floratos,Kempf}. It is natural to ask how
the existence of a minimal length may modify the results in elementary quantum
mechanics (QM) problems familiar to us \cite{Gasiorowicz}. In this paper we
address a simple problem from elementary non-relativistic quantum mechanics,
called "particle in a box", where the usual continuum (1+1)-space-time is
supplanted by a space-time lattice. Our lattice consists of a grid of
λ0​×τ0​ rectangles, where λ0​, the lattice
parameter, is a fundamental length (say Planck length) and, we take τ0​ to
be equal to λ0​/c. The corresponding Schrodinger equation becomes a
difference equation, the solution of which yields the q-eigenfunctions and
q-eigenvalues of the energy operator as a function of λ0​. The
q-eigenfunctions form an orthonormal set and both q-eigenfunctions and
q-eigenvalues reduce to continuum solutions as λ0​→0.
The corrections to eigenvalues because of the assumed lattice is shown to be
O(λ02​). We then compute the uncertainties in position and momentum,
Δx,Δp for the box problem and study the consequent modification
of Heisenberg uncertainty relation due to the assumption of space-time lattice,
in contrast to modifications suggested by other investigations such as
\cite{Floratos}