In description of motion in a uniform strong magnetic field in the presence of external potentials the velocities of the cyclotron rotation and the guiding center coordinates are respectively fast and slow variables. We developed a quantum adiabatic expansion for this situation. In the adiabatic limit we find the effective action of the slow variables in which the energy `surfaces` are local Landau levels depending on the coordinates of the guiding centers. The levels are not equidistant and may exhibit avoided crossings at which the transitions between the levels are concentrated. The action includes additional terms arising from Berry phase. These terms modify the simplectic form of the dynamics of the guiding centers and effect the WKB quantization. As a result it has a form of flux quantization through closed classical orbit of an effective magnetic field which in addition to the original field includes inhomogenous terms arising from the Berry phase and depending on the external potential. We present explicit calculations for the potentials which are weakly inhomogenous on the scale of the magnetic length. The standard strong field projection on a single Landau level is the first term of our expansion. (author).