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Many-body quantum-mechanical stationary states that have real valuedwavefunctions are shown to satisfy a classical conservation of energy equationwith a kinetic energy function. The terms in the equation depend on theprobability distribution, and, in addition, pressure and velocity functions,but these functions also depend on the probability distribution. There are twopossible directions of the velocity that satisfy the energy equation. A linearmomentum function is defined that integrates to zero, and this property isconsistent with the expectation value of the linear momentum for stationarystates with real-valued wave functions. The energy equation is integrated toobtain a version of the well known energy equation involving reduced densitymatrices, where the kinetic energy functional of the one-particle densitymatrix is replaced by a function of the electron density and a velocityfunction. Also, the noninteracting kinetic energy functional from theHohenberg--Kohn theorem is given as an explicit functional of the orbitaldensities. For the purpose of describing the behavior of particles in astationary state, a model based on the energy equation is constructed. Themodel is evaluated for the two different velocity directions using the groundsstate of the particle in a one-dimensional box and the hydrogen atom. For onevelocity direction, equations of motions with contradictory properties areobtained, and, in the other, an unstable system is found. A discussion is givenwith suggestions of additional elements that might improve the model.

A Quantum Mechanics Conservation of Energy Equation for Stationary States with Real Valued Wave Functions