What if space itself were made of discrete points?

Polymer quantization, a technique borrowed from loop quantum gravity, replaces continuous space with a discrete topology, producing a quantum theory genuinely different from standard Schrödinger mechanics. Applied here to compact systems — a particle on a ring and a particle in a box — the approach yields exact energy levels and wavefunctions defined on finite lattices of points. As the lattice spacing shrinks, the solutions smoothly recover the familiar continuum results, offering a concrete test case for how discrete-space quantum theories can connect back to ordinary quantum mechanics.