Cross sections have been calculated for electron transfer into the $1s$, $2s$, and $2p$ states and for excitation into the $2s$ and $2p$ states in collisions between 1.5---15-keV protons and hydrogen atoms The triple-center, coupled-state method of Anderson, Antal, and McElroy has been used: Bound atomic states are centered on each nucleus and on a third center (the center of charge) in order to simulate the molecular character of slow collisions. Convergence of cross sections (and molecular potential-energy curves) with respect to the size of the basis has been studied; the present basis includes up to 36 states (16 gerade, 20 ungerade states). The $1s$ capture cross section is estimated to be converged, and the $2p$ and $2s$ cross sections are estimated to be converged to about 10% and 20%, respectively. Dips are observed in the $2s$ and $2p$ excitation cross sections at about 10 keV and in the $2p$ capture cross section at about 5 keV. Results have been compared with atomic-state, pseudostate, and molecular-state results, as well as with experimental results. The results agree fairly closely with the related AO+ pseudostate results of Fritsch and Lin at the lowest calculated energy, although there are differences at higher energies. There are differences from some of the many experimental results, but overall there is agreement within the experimental error limits.