Semi-classical Limit

In this chapter, we present a semi-classical approachSemi-classical approximation based on the Wigner transform of the quantum density matrix [WIG]. We prefer this approach to the more famous WKB approximation [WKB] for two main reasons. Firstly, it is more suitable for studying the relationship between classical and quantum mechanics, in particular the correspondence between Poisson brackets and the commutation brackets, which are the basic tools of conservation theorems and the equations of motion. Secondly, the Wigner transform of the density is the natural extension of the particle distribution in the phase space and so it is suitable for the semi-classical treatment of quantum fluid dynamics.