The quantum states of a hydrogen atom (or hydrogen-like ion) located in a closed or opened vacuum space near a partially penetrable surface are investigated. The electronic potential energy beyond the surface is assumed to be constant. So the electron has a finite possibility to penetrate the surface. It is found that in the case while the electronic potential energy V0 beyond the surface is negative the number of electron bound states in the atom may be reduced. While V0 is quite low, the atom has no bound states at all. If V0 is close to the eigenenergy of atom, the eigenenergy reaches a minimum at some distance between the atom's nucleus and the surface. If V0 is positive, the energy of the hydrogen atom is smaller near a projection of the surface at the same nearest distance from the atom's nucleus to the surface. If V0 is negative, the hydrogen atom energy may be smaller near a pit (vacancy) or a channel.The electric dipole moment, the diamagnetic screening constant, the polarizability, and the hyperfine splitting of the atom are calculated for the ground states.