I want to talk about the well-known analogy between the <PERSON> equation and the Navier-Stokes equation. It reveals the connection between the solution of the <PERSON> equation and Navier-Stokes. In this case, the <PERSON> equation describes the continuity equation. It is similar to the continuity equation with a density equal to $\rho=\psi^*\psi$. 346
We write the <PERSON> equation and transform it using the identity $\frac{\partial^2 \psi}{\partial x_l^2}=\psi[\frac{\partial^2 ln \psi}{\partial x_l^2}+\frac{1}{\psi^2}(\frac{\partial \psi}{\partial x_l})^2]$ $$i\hbar\frac{\partial \psi}{\partial t}=\frac{-\hbar^2}{2m}\sum _{l=1}^3 \frac{\partial^2 \psi}{\partial x_l^2}+U\psi=\frac{-\hbar^2}{2m}\psi[\frac{\partial^2 ln \psi}{\partial x_l^2}+\frac{1}{\psi^2}(\frac{\partial \psi}{\partial x_l})^2] +U\psi$$
Dividing by mass $ m \psi $, we obtain the equation $$\frac{i\hbar}{m}\frac{\partial ln \psi}{\partial t}+\frac{\hbar^2}{2m^2}(\frac{\partial ln \psi}{\partial x_l})^2=\frac{-\hbar^2}{2m^2}\sum_{l=1}^3 \frac{\partial^2 ln \psi}{\partial x_l^2} +U/m$$
We obtain the partial differential equation, taking the gradient from both sides of the equation, we introduce the real speed by the formula $$\vec V=-i\frac{\hbar}{m} \nabla ln\psi(1) $$. $$\frac{\partial \frac{ i\hbar }{m} \nabla ln \psi}{\partial t}+\frac{\hbar^2}{m^2}\frac{\partial ln \psi}{\partial x_l}\frac{\partial \nabla ln \psi}{\partial x_l}=\frac{i\hbar}{2m}\sum_{l=1}^3 \frac{\partial^2 i\frac{\hbar}{m} \nabla ln \psi}{\partial x_l^2} +\nabla U/m$$ Substituting the speed value into the transformed <PERSON> equation, we get$$\frac{\partial V_p}{\partial t}+\sum_{l=1}^3 V_l \frac{\partial V_p}{\partial x_l}=\nu \sum_{l=1}^3 \frac{\partial^2 V_p}{\partial x_l^2}-\frac{\partial U}{\partial x^p}/m,\nu=\frac{i\hbar}{2m}$$ We obtain the three-dimensional Navier - Stokes equation with pressure corresponding to the potential. The relationship between the two solutions allows us to conclude that in the microworld an imaginary solution is possible, which is obtained by substituting the real wave function in equation (1).
The imaginary kinematic viscosity of the vacuum is also awaiting an explanation. I have already raised this question in the topic "How to explain imaginary kinematic viscosity of a vacuum?", But not sufficiently argued. This topic can lead to the description of new properties of vacuum. 955