Quantum Spatial Probability and Time Intervals in the High Energy Limit

 Linking quantum mechanics to classical mechanics seems to have been an early goal in quantum theory as represented by the correspondence principle. More recently there have been attempts to link the entropy of both a single bound quantum state and a thermal statistical system of quantum states to classical entropies (1),(2). For the former, the entropy is proportional to Integral dx density(x) ln[density(x)] where density(x)= delta t = 1 / v(x) with v(x) being velocity. Thus, density, which is a probability in Shannon’s entropy equation, is a  measure of the change in the value of delta t(x) in this classical entropy. Bound state quantum mechanics is said to be time-independent, but by the correspondence principle (1) the probability of the envelope of the wavefunction i.e. Envelope(W(x))* Envelope(W(x)) = delta t(x) so quantum probability (albeit linked to the envelope) is related to classical time intervals which adjust with x to yield the proper ration v(x)=delta x / delta t(x) with delta x being constant. 

    In this note we argue that the expectation value:  Integral W(x) d/dx dW/dx = Integral WW KEclassical(x) for a bound state is, in the high energy limit Integral dt KEclassical(x)  i.e. a portion of the action, just as in the classical case (i.e. the correspondence principle). This leads to the association of the higher energy limit of the envelope of W times itself with delta t(x).