  We formulate a non-relativistic Hamiltonian in order to describe the
interaction between a moving dielectric membrane and radiation pressure. Such a
Hamiltonian is derived without making use of the single-mode adiabatic
approximation, and hence it enables us to incorporate multi-mode effects in
cavity optomechanics. By performing second quantization, we show how a set of
generalized Fock states can be constructed to represent quantum states of the
membrane and cavity field. In addition, we discuss examples showing how photon
scattering among different cavity modes would modify the interaction strengths
and the mechanical frequency of the membrane.
