A second-quantized cell-model Hamiltonian is derived to provide a model for quantum crystals and liquids. The boson crystal is treated in this article. The cells are divided into two sublattices: (i) regular, which are usually occupied, and (ii) interstitial, which are usually empty. The particle hard cores are simulated by assuming Fermi commutation relations for operators referring to a single cell; this allows a discussion in terms of a spin-analog Hamiltonian which is diagonalized in the spin-wave approximation. Because the Hamiltonian includes a term which allows tunneling between regular and interstitial sites, the ground state includes a description of zero-point motion and exchange via virtual intermediate interstitial occupation. Excited states include nonlocalized vacancies, interstitials, and vacancy-interstitial pairs. Phonon states are not included in this analysis. The model exhibits a phase transition to the Bose condensed state which is examined briefly.