  A cavity quantum electrodynamical scenario is proposed for implementing a
Schr\"odinger microscope capable of amplifying differences between non
orthogonal atomic quantum states. The scheme involves an ensemble of
identically prepared two-level atoms interacting pairwise with a single mode of
the radiation field as described by the Tavis-Cummings model. By repeated
measurements of the cavity field and of one atom within each pair a
measurement-induced nonlinear quantum transformation of the relevant atomic
states can be realized. The intricate dynamical properties of this nonlinear
quantum transformation, which exhibits measurement-induced chaos, allows
approximate orthogonalization of atomic states by purification after a few
iterations of the protocol, and thus the application of the scheme for quantum
state discrimination.
