Quantum hydrodynamics, Wigner transforms and the classical limit. (English) Zbl 0877.76087

Summary: We analyse the classical limit of the quantum hydrodynamic equations as the Planck constant tends to zero. The equations have the form of an Euler system with a constant pressure and a dispersive regularisation term, which (formally) tends to zero in the classical limit. The main tool of the analysis is the exploitation of a kinetic equation, which lies behind the quantum hydrodynamic system. The presented analysis can also be interpreted as an alternative approach to the geometrical optics (WKB)-analysis of the Schrödinger equation.


76Y05 Quantum hydrodynamics and relativistic hydrodynamics
76A02 Foundations of fluid mechanics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory