Most physicists are familiar with how nonrelativistic quantum mechanics governs the behavior of particles in a Coulomb potential or a simple harmonic oscillator potential. But what happens when the particles become highly relativistic? Such questions used to lie mainly in the domain of high and intermediate energy physics. With the discovery that electrons in graphene behave like ultra-relativistic fermions, these questions are now accessible to the tools of condensed matter physics. In this talk, I will describe experiments in which a scanning tunneling microscope (STM) was employed to manipulate charged impurities to construct Coulomb and simple harmonic oscillator potentials in gate-tunable graphene devices. STM measurements were used to spectroscopically probe and image the resulting electronic wavefunctions of Dirac fermions in these potentials. For a Coulomb potential, we find that graphene’s massless electrons reorganize into “atomic collapse” states that are analogous to the bound states predicted for nuclei having atomic number greater than Z = 170. For a simple harmonic oscillator potential, we observe quantum interference patterns that can be identified with different principal and angular momentum quantum numbers.