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Elementary Quantum Mechanics in a Space-time Lattice

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 $λ_0 \times τ_0 $ rectangles, where $λ_0$, the lattice parameter, is a fundamental length (say Planck length) and, we take $τ_0$ to be equal to $λ_0/c$. 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 $λ_0 $. The $q$-eigenfunctions form an orthonormal set and both $q$-eigenfunctions and $q$-eigenvalues reduce to continuum solutions as $ λ_0 \rightarrow 0 .$ The corrections to eigenvalues because of the assumed lattice is shown to be $O(λ_0^2).$ 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}.