The Madelung equations reformulate the Schrödinger equation—one of the foundations of quantum mechanics—into a hydrodynamic form that describes the flow of a nonclassical fluid. This overview revisits their derivation and key ideas.

This is a two-dimensional model of the simplest molecular ion, consisting of two protons and one electron. The electron, represented by a...

A quantum system in thermal equilibrium at finite temperature can be effectively described by a density matrix. In this overview, we derive a general expression for the equilibrium density matrix by extremizing the system’s von Neumann entropy.