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. 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. 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×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 Schroedinger 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→0. The corrections to eigenvalues because of the assumed lattice is shown to be O(l₀²)O(lambda_{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.