We suppose,
i- the starting point is to replace the complex PDE for the complex wave function Ψ (probability amplitude) by a real PDE for Ψ^2.
The PDE for Ψ^2 which is the probability density of finding the quantum particle in the volume element x-t dx dy dz dt is assumed exactly equal to the energy density of the quantum particles.
ii- Consequently, the solution of the Bmatrix chain follows the same procedure for solving the heat diffusion equation explained above in the last question:
How to invert a 2D and 3D Laplacian matrix without using MATLAB iteration or any other conventional mathematical method?
iii-The third and final step is to take the square root of the solution for Ψ^2 as the solution for Ψ itself.
As simple as that!
Note that particle physics is different from wave mechanics because they both have a different physical nature.
The nature of particles is described or modeled as being small and confined to a point in space, while a particle wave is something that propagates and extends almost into infinite free space. Additionally, particles collide and scatter while matter waves refract, diffract, and interfere in a process called superposition. They add or cancel each other in superpositions.
The question arises how can Bmatrix chains describe the seemingly opposite nature via the same matrix?
The answer lies in the specific definition of the boundary conditions in both cases.
Assuming that the boundary conditions in Bmatrix chains are Dirichlet or similar, then they describe a microscopic or macroscopic particle. On the other hand, assuming that the boundary condition extends to infinity, the chains of the B matrix describe the mechanics of wave particles.
Dear Ismail Abbas,
You wrote:
"The third and final step is to take the square root of the solution for Ψ^2 as the solution for Ψ itself...
Note that particle physics is different from wave mechanics because they both have different physical nature"
Well I don't think they have different physical nature as if we take energy as the real fundamental issue, that can manifest itself as a wave or as "particle", as was shown by Erwin Schrödinger in 1926, when he presented as a postulate its famous wave equation, that in fact can be deduced with Basic Systemic Unit concept based on Euler' Relation, I mean complex numbers, with which Time and Space, are in fact, two different orders of reality, not reducible the one to the other, as you can see in the small paper in the attachment, where as a matter of fact, the square root of Ψ, is deduced too.
My Best Regards
Edgar Paternina
Retired electrical engineer.