Quantum Mechanical, Statistical Mechanical and Classical Spatial Probability Distributions

Quantum mechanical spatial probability densities may be obtained by solving the time independent Schrodinger equation. Statistical mechanical spatial densities may be obtained by maximizing Shannon´s entropy d(r)ln[d(r)] (where d(r) is the spatial density) subject to appropriate constraints. A classical spatial probability density may be obtained from 1/v(x) where v(x) is the velocity at position x obtained from kinetic energy + potential energy (x) =total energy.   Recently, there have been attempts to link these various spatial probability densities. For example for the ground state wavefunction of a harmonic oscillator, the Schrodinger equation can be obtained from varying Shannon´s entropy with constraints because V(x) W*(x) W(x) (where W(x) is the wavefunction) can be written in the form of Shannon´s entropy times a constant [3]. Furthermore, Casal et al  have maximized Tsalis´ entropy to obtain good approximations to the ground state wavefunctions of oscillators and Coulomb potentials. In [1], the classical density 1/v(x) has been calculated using energy levels from quantum mechanics, namely E=hbar w (n+.5). This density has in turn been used to calculate entropy and compared with results from wavefunctions of the Schrodinger equation with good results even for low n values even though for n=0, the wavefunction W(x)=1/C exp(-ax2) and 1/v(x) are not at all similar. The purpose of this note, is to attempt to examine the relationship between these different spatial densities.