We perform spin-polarized density functional theory calculations for a hydrogen atom interacting exohedrally with a (5,5) single-walled carbon nanotube (SWNT). We also perform full three-dimensional (3D) quantum dynamics calculations of the H atom bound rovibrational states. We initially focus on the four sites of highest symmetry, along which we compute potential energy surface (PES) values at 33 separate, nonuniformly spaced radial values. These 132 geometries are sufficient to define the primary potential interaction regions. We find a weak physisorptive region between 2.5 and 3.5 Å from the SWNT wall, with a maximum well depth of 51 meV, relative to the desorption limit. We also find a chemisorptive region, extending from about 1.0 out to 1.5 Å from the SWNT wall. The maximum well depth of 0.755 eV occurs at 1.15 Å from the SWNT wall, nearly directly above a carbon atom. A small barrier of 54 meV lies between these two binding regions. There are also two types of transition states that lie between adjacent chemisorption wells. In addition to the high-symmetry sites, a detailed and accurate characterization of the PES requires density functional theory calculations along a large number of interstitial sites-18 in all. Using these 18×33 geometries, and exploiting the full D10h symmetry of the system, we fit a global analytical PES, using a Fourier basis in the cylindrical coordinates, with radially dependent expansion coefficients (rms error 3.8 meV). We then perform a mixed spectral basis/phase-space optimized discrete variable representation calculation of all bound rovibrational H atom eigenfunctions and energy levels. We also discuss ramifications for the possible use of SWNTs as hydrogen storage devices.