![]() Applications of the projection methods are presented for simple nuclear models, realistic calculations in relatively small configuration spaces, nuclear energy density functional (EDF) theory, as well as in other mesoscopic systems. In order to avoid the numerical complexity of exact projection techniques, various approximation methods available in the literature are examined. In this article, we discuss general group-theory techniques to restore the broken symmetries, and provide detailed expressions on the restoration of translational, rotational, spin, isospin, parity and gauge symmetries, where the latter corresponds to the restoration of the particle number. However, the mean-field approach suffers from the drawback that the corresponding wave functions do not have sharp quantum numbers and, therefore, many results cannot be compared directly with experimental data. In this way, additional correlations are subsumed in the system. This approximation, not only provides computationally very simple solutions even for systems with many particles, but due to the non-linearity, it also allows for obtaining solutions that break essential symmetries of the system, often connected with phase transitions. ![]() In this approach, the linear Schrödinger equation for the exact many-body wave function is mapped onto a non-linear one-body potential problem. Some examples are strongly interacting medium and heavy mass atomic nuclei and mesoscopic condensed matter systems. ![]() The mean-field approximation based on effective interactions or density functionals plays a pivotal role in the description of finite quantum many-body systems that are too large to be treated by ab initio methods. ![]()
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