Nonrelativistic quantum mechanics is obtained from nonrelativistic classical Hamiltonian mechanics by replacing the particle's spatial location by a complex-valued wave-function location distribution, and by replacing the particle's deterministic mechanics in universal time by an operator analog which acts on its wave function. But in relativistic classical mechanics, time isn't universal. One important time is given by a clock at rest with respect to the observer; in conjunction with the particle's observed spatial location it comprises the particle's observed space-time location. The Lorentz-invariant time of the particle's dynamics is its proper time, given by a clock at rest with respect to it. The wave function of relativistic quantum mechanics depends on the particle's observed space-time location, and time derivatives of the quantum operators are with respect to particle proper time. In the particle's zero-mass limit, however, the free-particle Schrödinger equation is independent of the particle's proper time, is second-order in observer time, and is real-valued. The relativistic correction to the hydrogen atom's Hamiltonian without spin is a very weak, short-range complement to the Coulomb potential.