We study an N particle, two mode Bose-Hubbard system, modelling a Bose-Einstein condensate in a double-well potential. Besides the full many particle description one often introduces a mean field approximation, which can be interpreted as a classical limit of the many particle system. In this talk the correspondence between both descriptions is investigated. We introduce a semiclassical approximation based on the classical dynamics of the mean field Gross-Pitaevskii equation, which is expected to be valid for large N. By using a WKB-type quantization condition we can reconstruct the quantum properties of the N particle system approximately from the mean field model. The resulting eigenvalues and eigenstates are found to be in very good agreement with the exact ones, even for small values of N.
In addition we consider Landau-Zener type dynamics for the full many particle system as well as in the mean field approximation. In the mean field system the emergence of novel nonlinear eigenstates leads to a breakdown of adiabaticity. It is shown that this phenomenon corresponds to quasi-degenerate avoided crossings of the many particle levels. By solving the many particle problem within an independent crossings approximation, we derive an explicit generalized Landau-Zener formula. A comparison with numerical results for the many particle system and the mean field approximation shows excellent agreement.