We investigate the quantum states of hydrogen atom motion on Pd(111) surface and in its subsurface by calculating the wavefunctions and the eigenenergies for hydrogen atom motion within the framework of the variation method on an adiabatic potential energy surface (PES), obtained through first-principles calculations, for the hydrogen atom motion. The calculated results show that the ground-state wavefunction for the hydrogen atom motion localizes on the face-centered cubic (fcc) hollow site of the surface. The higher excited state wavefunctions are distributed between the first and second layers, and subsequently delocalized under the second atom layer. These suggest that an effective diffusion path of the hydrogen atom into the subsurface area passes through the fcc hollow site to the octahedral sites in the subsurface. Moreover, activation energies for diffusion of H and D atoms over the saddle point of the PES between the fcc hollow site and the first (second) octahedral site are estimated as 598 (882)meV and 646 (939)meV, respectively. Furthermore, the activation energies for diffusion of H and D atoms over the saddle point of the PES between the first (second) octahedral site and the fcc hollow site are estimated as 285 (483)meV and 323 (532)meV, respectively.