Fisher Information, Minimization and the Schrodinger Equation

In classical mechanics, it is often argued that the Maxwell-Boltzmann distribution may be obtained by maximizing entropy subject to a constraint. A historical derivation of this distribution from the Boltzmann transport equation, however, does not seem to make use of entropy extremization, but rather uses two particle scattering, equal probability for time reversal scattering and conservation of kinetic energy. I.e. use is made of F(p1)F(p2)=F(p3)F(p4) ((1)) where F(p) is the probability distribution for momentum p. Furthermore, the Boltzmann transport equation is not necessary for obtaining ((1)), it follows from an “and” condition of probability,independence and time reversal ideas. Recently, there have been a number of papers in the literature describing how the extremization of Fisher information subject to a constraint may lead to the derivation of the Schrodinger equation. The constraint appears to be related to the potential energy. This, however, begs the question: Why does one need a constraint on potential energy and not use some other constraint? The potential energy V(x) is a function of x so F(x)V(x) provides geometric information about F(x), the spatial probability. If geometric information is the only consideration, it seems any function of x could provide similar information; there would be no reason to choose V(x) over others. Thus, it seems the idea of potential energy or energy in general is of fundamental importance. In this note, we examine conditions used in one source in the literature to extremize Fisher information to obtain the time-independent Schrodinger equation and then consider this process in relation to the ground state of the oscillator which mimics classical statistical mechanics. We find that in such a case, no extremization is necessary and that the integral being maximized equals the constraint. For cases other than the oscillator ground state, however, the Fisher information extremization seems interesting except for the fact that an integral: Integral dx V(x) F(x) seems to be forced into the problem. We try to argue that one may formulate that Schrodinger equation in terms of a plane wave average for kinetic energy which together with conservation of total energy at each point x yields the time-independent Schrodinger equation. In this equation, W(x), the wavefunction, appears as a normalization constant. This equation appears as an operator B(x)W(x) = EW(x) which we argue is of the form of the result of an extremization equation. Thus, given the Schrodinger equation, one may find a function, which when minimized or maximized yields the time-independent Schrodinger equation. That does not necessarily mean one must use extremization in order to obtain the equation, although it should give some insight into the problem