The quantum mechanical problem of the hydrogen atom is treated by use of a finite difference equation in place of Schrödinger's differential equation. The exact solution leads to a wave vector energy expression that is readily converted to the Bohr-Rydberg formula. (The calculations here reported are limited to spherically symmetric states.) The wave vectors reduce to the familiar solutions of Schrödinger's equation as c → ∞. The internal consistency and limiting behavior provide support for the view that the equations employed could well constitute an approach to a relativistic formulation of wave mechanics.