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We perform a theoretical investigation into the classical and quantumdynamics of an optical field in a cavity containing a moving membrane("membrane-in-the-middle" set-up). Our approach is based on the Maxwell waveequation, and complements previous studies based on an effective Hamiltonian.The analysis shows that for slowly moving and weakly reflective membranes thedynamics can be approximated by unitary, first-order-in-time evolution given byan effective Schr\"{o}dinger-like equation with a Hamiltonian that does notdepend on the membrane speed. This approximate theory is the one typicallyadopted in cavity optomechanics and we develop a criterion for its validity.However, in more general situations the full second-order wave equationpredicts light dynamics which do not conserve energy, giving rise to parametricamplification (or reduction) that is forbidden under first order dynamics andcan be considered to be the classical counterpart of the dynamical Casimireffect. The case of a membrane moving at constant velocity can be mapped ontothe Landau-Zener problem but with additional terms responsible for fieldamplification. Furthermore, the nature of the adiabatic regime is ratherdifferent from the ordinary Schr\"odinger case, since mode amplitudes need notbe constant even when there are no transitions between them. The Landau-Zenerproblem for a field is therefore richer than in the standard single-particlecase. We use the work-energy theorem applied to the radiation pressure on themembrane as a self-consistency check for our solutions of the wave equation andas a tool to gain an intuitive understanding of energy pumped into/out of thelight field by the motion of the membrane.

Parametric amplification of light in a cavity with a moving dielectric membrane: Landau-Zener problem for the Maxwell field