On the Quantization of the Motion of Liquids

Abstract: For building up a theory of superfluid Helium-4, Lev Landau ingeniously unified the principles of quantum mechanics with the principles of hydrodynamics. By introducing a velocity operator he was able to derive a quantum analogue of the mass density flux equation and a quantum Euler equation. The article shows that it is possible to formulate a consistent quantum theory of the ideal liquid based on Landau's Quantum Hydrodynamics. This means that it is added the quantum analogue of the momentum density flux equation, the energy density flux equation, the entropy density flux equation, and a quantum vorticity equation. These are all manifestly in agreement with the classical limit of the flux equations of classical ideal liquids. It is also shown that the various aspects that have been criticised about Landau's Quantum Hydrodynamics - e.g. the application of the inverse of the mass density operator - are not justified. Such a consistent quantum theory of an ideal liquid may therefore provide a starting point of a complementary approach in the theoretical research of superfluid Helium 4.