Classical Probability Theory and Quantum Wavefunctions

 In (1), we noted that a Gaussian wavefunction may be a solution to the time-independent Schrodinger equation (e.g. the ground state solution of the quantum oscillator) and simultaneously a solution of maximizing  spatial Shannon’s entropy  - density(x) ln(density(x)) subject to the constraint  V(x) xx constant, where density(x) =W*(x)W(x) with W(x) being the wavefunction.

      In (2), which deals strictly with classical probabilities and the Maxwell-Boltzmann factor n(ei) = exp(-ei/T), we argued that this function is equivalent to:   ln(n(ei)) = -ei/T. This in turn is equivalent to the maximization of  ln(n(ei)!) with the constraint ei n(ei) which is part of:

ln( N!/ Product over i n(ei)!) with the full constraint Sum over i ei n(ei)=Eave. Here n(ei)=N p(ei) where p(ei) is a probability and N a very large number. Thus a function of exp( something) may indicate that “something” * density(x) is a constraint with  -density ln(density) being Shannon’s entropy. 

    Quantum mechanics for a bound system yields a real wavefunction W(x) such that density(x)=W(x)W(x)=probability(x). Thus maximization of Shannon’s entropy - density(x) ln (density) subject to a constraint:  density(x)B(x)  should correspond to a solution for density of exp( B(x)). B(x) as part of the constraint, however, should represent a physical constraint. Thus if there is a solution to the Schrodinger equation of the form exp(.5B(x)) and B(x)density(x) represents the physical constraint in the system, classical maximization of probability solves the quantum problem in such a case. 

     The above ideas may be applied to the quantum oscillator ground state and essentially repeat what is done in (1). We, however, try to generalize them to see if they may apply to a free quantum particle. In this case the probability used is exp(ipx) which is complex. It corresponds to P(x)=exp(ipx)exp(-ipx)=1 which is a uniform distribution which has maximum lack of information.