Density Due to Collective versus Fluctuation Kinetic Energy in Classical, Classical Statistical and Quantum Mechanics

The purpose of this note is to consider spatial densities in the case of classical mechanics (1/v), quantum mechanics (W(x)*W(x) for bound states) and statistical mechanics exp(-V(x)/T) with respect to the effects of collective motion. In the classical mechanical case, with a density of 1/v, the entire motion is collective and seems to be modeled as a compressible gas with flux continuity. (It seems that this idea may even possibly lead to Newton´s second law.) In the case of quantum mechanics, high energy eigenvalue densities are supposed to approach the classical density near peak density points. We try to investigate why by considering root mean square velocities and what we think is fluctuation flux in a quantum system. For peak wavefunction values dW(x)/dx=0 and we try to show that these points represent places where there is no fluctuation flux. Thus, at such points, there is only collective motion and one would expect W(x1)W(x1) vave(x1) =W(x2)W(x2) vave(x2) between two such points x1 and x2, i.e. a continuity equation. For points in between, fluctuation flux mixes with collective velocity to create a more complicated pattern. In such high energy cases, cycling time 1/E from exp(iEt) is small, so the idea of a density at a point x at a time t with a small delta t seems to make sense. Actual results of the intersection of 1/v(x) and the quantum density are not actually at the peak, but the peak seems to be the approximating the match. (3) For low quantum energies, the cycling time 1/En is long compared to times required to traverse the system and fluctuation flux mixes with collective motion. For the quantum oscillator this leads to a Gaussian. We also examine the case of the classical statistical oscillator and argue that the fluctuation flux (the thermal kinetic energy) is mixed with collective kinetic energy which leads to higher interparticle spacing for places where the potential is higher