Differential and total displacement and helium production rates are calculated for copper irradiated by spallation neutrons and 760 MeV protons at the Clinton P. Anderson Los Alamos Meson Physics Facility (LAMPF). The calculations are performed using the SPECTER and VNMTC computer codes, the latter being specially designed for spallation radiation damage calculations. For comparison, similar SPECTER calculations are also described for irradiation of copper in EBR-II and RTNS-II. The results indicate substantial contributions to the displacement and helium production rates due to neutrons in the high-energy tail (above 20 MeV) of the LAMPF spallation neutron spectrum. Still higher production rates are calculated for irradiations in the direct proton beam. These results will provide useful background information for research to be conducted at a new irradiation facility at LAMPF.