Schrodingers Conditional Probability Density

Before proceeding with emplo5rment of the conditional probability (1.34) to further developments, giving its complexity, worth checking it for Schrodinger equation fulfillment, however for 1-dimensional case written in the quantum amplitude form  

Equation (1.36) is asked to have the formal (path integral) solution, i.e., factorizing the thermal amplitude, see Volumes 1 and II of the present five volume set (Putz, 2016b-c) 

Quantum Nanochemistry—Volume IV Quantum Solids and Orderability 

Along the spatial derivative obtained by the first derivation 

All together, Eqs. (1.38) and (1.41) in Eq. (1.36) provides the actual Schrodinger equation for the conditional probability quantum correction to be  

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Before the effective calculation of the conditioned probability density, worth exposing several essential characteristics of the Fokker-Planck equation as well as of the connection with the Schrodinger equation. To this aim one will be starting with the effective deduction of the Fokker-Planck equation. It can be successively write for the conditioned probability density (Feynman Hibbs, 1965 Balescu, 1975) ... 

In addition to initial conditions, solutions to the Schrodinger equation must obey eertain other eonstraints in form. They must be eontinuous funetions of all of their spatial eoordinates and must be single valued these properties allow T T to be interpreted as a probability density (i.e., the probability of finding a partiele at some position ean not be multivalued nor ean it be jerky or diseontinuous). The derivative of the wavefunetion must also be eontinuous exeept at points where the potential funetion undergoes an infinite jump (e.g., at the wall of an infinitely high and steep potential barrier). This eondition relates to the faet that the momentum must be eontinuous exeept at infinitely steep potential barriers where the momentum undergoes a sudden reversal. 

The probability density for a particle at a location is proportional to the square of the wavefunction at that point the wavefunction is found by solving the Schrodinger equation for the particle. When the equation is solved subject to the appropriate boundary conditions, it is found that the particle can possess only certain discrete energies. 

For the variational principle to hold, the chosen wavefunction must satisfy the same conditions we have considered necessary in interpreting the square of a true wave-function as a probability density. These are the smoothness and continuity and that the wavefunction be normalizable. The function and its first derivative must be continuous in all spatial variables of the system over the entire range of those variables the function must be single-valued and the integral of the square of the function over all space must be a finite value. Wavefunctions obtained from analytical solution of a Schrodinger equation and wavefunctions selected for variational treatment should satisfy these conditions. 

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