Quantum mechanics of a free particle from properties of the Dirac delta function

Based on the assumption that the probability density of finding a free particle is independent of position, we infer the form of the eigenfunction for the free particle, $\bra{x} p > = \exp(ipx/\hbar)/\sqrt{2\pi\hbar}$. The canonical commutation relation between the momentum and position operators and the Ehrenfest theorem in the free particle case are derived solely from differentiation of the delta function and the form of $\bra{x} p >$.