Dmitry Zhdanov

Quantum mechanics as an effective fluid theory. Fundamentals and chemical applications

In quantum mechanics, we cannot define trajectory of a particle. However, there exist consistent ways to describe the quantum evolution as a flow of effective compressible probability fluid in phase space. In this picture (called Husimi representation), the normalized fluid distribution replaces conventional quantum wavefunction or density matrix, and the Schrodinger equation is substituted by the continuity equation for fluid density. The quantum dynamics, in principle, can be fully characterized in terms of trajectories of elementary parcels of this Husimi fluid. Interestingly, this possibility was de facto and unconsciously used by quantum chemists for decades in their semiclassical trajectory-based approaches to solve multidimensional Schrodinger equation. At the same time, the detailed closed-form expressions characterizing multidimensional Husimi fluid were elaborated only for a special class of systems, and their generalizations are available only for one-dimensional case.

In my talk, I will present systematic development of hydrodynamic analogy for quantum mechanics in phase-space representation. First, the continuity equations for Wigner and Husimi functions of a generic multidimensional closed quantum system will be derived. Second, Bohmian mechanics will be reinterpreted as a singular limit of these equation in a special gauge. Based on these two results, I will propose a new numerical method for solving multidimensional time-dependent Schrodinger equation in the basis of squeezed coherent states evolving with the phase space flow.