Kullback-Leibler Entropy and Quantum Average Momentum

 The Kullback-Leibler entropy is defined as: - Integral dx P(x) ln { P(x)/R(x)} where P(x) and R(x) are two classical probability distributions i.e. real, positive functions with values between 0 and 1. In (1), it is shown that the Kullback-Leibler entropy is equivalent to Fisher’s information if R(x) = P(x+delta x) where delta x is a tiny constant. We argue that in this case, the Kullback-Leibler entropy represents either average momentum multiplied by spatial density in a quantum bound state, i.e.  Integral dx  W(x)W(x) {dW/dx / W} which is 0 because overall the particle does not move in one direction (odd function). This suggests forwards and backwards internal motion which yields 0 overall translation, similar to fluctuations (in x not t) . In addition, it represents the “fluctuations” or average momentum for half the classical probability distribution associated with a quantum free particle exp(ipx), namely  C cos(px) cos(px). Again the Kullback-Leibler entropy for this distribution (or sin(px)sin(px)) is 0. (This is equivalent to the result of a particle in an infinite potential well if one considers pave in cos(pave x), the wavefunction.)

  Thus a statistical quantity, namely Kullback-Leibler entropy, seems to be associated with quantum mechanical motion (average momentum). This motion is linked with zero Kullback-Leibler entropy where P(x) is a distribution and R(x)=P(x+delta x). Furthermore, this seems to show, we argue, that d/dx in Fisher’s information is linked with physical momentum.