The quantum dynamics of a chaotic billiard with moving boundary is considered in this work. We found a shape parameter Hamiltonian expansion which enables us to obtain the spectrum of the deformed billiard for deformations so large as the characteristic wave length. Then, for a specified time dependent shape variation, the quantum dynamics of a particle inside the billiard is integrated directly. In particular, the dispersion of the energy is studied in the Bunimovich stadium billiard with oscillating boundary. The results showed that the distribution of energy spreads diffusively for the first oscillations of the boundary (${< Delta^2 E}> =2 D t$). We studied the diffusion contant $D$ as a function of the boundary velocity and found differences with theoretical predictions based on random matrix theory. By extracting highly phase space localized structures from the spectrum, previous differences were reduced significantly. This fact provides the first numerical evidence of the influence of phase space localization on the quantum diffusion of a chaotic system.
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