On a Probability Density Equation

The stationary Schrödinger equation depending on spatial coordinates has been considered. The problem of obtaining a differential relationship for the wave function squared was posed. By extracting Schrödinger’s equation itself from this relationship a differential equation for a physically interpretable quantity, i.e. the probability density (wave function squared), has been formulated. As an example the one-dimensional case admitting a simple analytic solution was considered. The solution obtained is shown to be a solution squared of the corresponding nonlinear differential equation for the probability density. In the final section a more general non-stationary case was considered for the potential involving a time-dependent term, such potentials are found in the non-stationary perturbation theory. The constant in separating the variables remains real. Thus the procedure considered proves to be similar to that presented above for the stationary equation.

Schrödinger equation, stationary, probability density, real, nonlinear