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
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 $\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(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}.
The existence of ontologically different but empirically indistinguishable versions of non-relativistic quantum mechanics is one of the most serious and widely discussed cases of metaphysical underdetermination. I argue that this problem is unlikely to disappear when we turn to the next more fundamental theory, namely relativistic quantum field theory. The problem is merely overshadowed by yet another interpretational controversy, but it persists regardless of the outcome of this latter debate. I further argue that one should not be too optimistic about transferring ontological lessons learned in the context of one physical theory to the context of a more fundamental theory.