  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
$\lambda_0 \times \tau_0 $ rectangles, where $\lambda_0$, the lattice
parameter, is a fundamental length (say Planck length) and, we take $\tau_0$ to
be equal to $\lambda_0/c$. 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 $\lambda_0 $. The
$q$-eigenfunctions form an orthonormal set and both $q$-eigenfunctions and
$q$-eigenvalues reduce to continuum solutions as $ \lambda_0 \rightarrow 0 .$
The corrections to eigenvalues because of the assumed lattice is shown to be
$O(\lambda_0^2).$ We then compute the uncertainties in position and momentum,
$\Delta x, \Delta p$ 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}.
