The Schrodinger Equation and Maximization of Entropy

Given that quantum mechanics is argued to be a statistical theory, there have been attempts to investigate if the Schrodinger equation follows from the maximization of Shannon’s entropy. For example in (1), Shannon’s entropy using P(x)=density is maximized subject to energy constraints and the Schrodinger equation appears in the zero temperature limit. Furthermore, traditionally it is argued that the free particle quantum wavefunction exp(ipx) has a modulus of 1 at all x points, which seems to be a statement of maximum entropy. If that is the case, how does exp(ipx) follow from the maximization of entropy ? In this note, we try to examine the role of maximum entropy in the Schrodinger equation and specifically argue that two principles are used. First, maximization of entropy using conditional probability not P(x) is applied using an xW (W=wavefunction) and not energy constraint. Secondly, Newton’s conservation of energy is applied to a quantum ensemble of free particle conditional probabilities obtained in the first step. Thus, statistical arguments are only part of the quantum picture.