Conditional Probability P(p/x) and Energy Variance at X for Quantum Bound States

If one defines a conditional probability P(p/x) = a(p) exp(ipx) / W(x) where W(x)=wavefunction then Sum over p P(p/x) = 1, but P(p/x) is complex and its real and imaginary parts may have positive and negative values at different x points. Nevertheless one may establish an average energy conservation equation with the same En (energy of the nth level) at each x which mimics classical mechanics for all n values. This equation is identical to the time-independent Schrodinger equation.

     If one treats P(p/x) as a distribution and obtains an average En that holds at each x point like temperature holds at each x for a statistical mechanical gas, one should be able to calculate the variance at x as well i.e.  Sum over p  P(p/x) [ pp/2m + V(x) -En] [ pp/2m +V(x) -En] = Var(E(x)). 

We examine this value for a particle in a box with infinite potential walls and for a harmonic oscillator. One may then average Var(En(x). We also consider implications on entropy.