Uncertainty Relations in the Madelung Picture

Abstract

Madelung showed how the complex Schrödinger equation can be rewritten in terms of two real equations, one for the phase and one for the amplitude of the complex wave function, where both equations are not independent of each other, but coupled. Although these equations formally look like classical hydrodynamic equations, they contain all the information about the quantum system. Concerning the quantum mechanical uncertainties of position and momentum, however, this is not so obvious at first sight. We show how these uncertainties are related to the phase and amplitude of the wave function in position and momentum space and, particularly, that the contribution from the phase essentially depends on the position–momentum correlations. This will be illustrated explicitly using generalized coherent states as examples.

Louis de Broglie Double Solution Theory Confirms the Wave-Particle Duality Principle

“…Unfortunately, we must admit that Madelung's article went unnoticed by many physicists of that time. Now interest in the Madelung equations has increased due to the fact that the flow of an ideal fluid gives a fine explanation of complex-valued solutions of the Schrödinger equation -wave mechanics corresponds to the Eulerian picture [23][24][25][26].…”