Doing a Google search brings up the statement (Schrödinger's equation and the duality diffusion equations are equivalent. In turn, Schrödinger's 1931 conjecture is solved.).
The above statement is absolutely false.
The heat diffusion equation lives and operates in real measurable space-time while the Schrödinger equation lives and operates in imaginary space-time.
Furthermore, the Schrödinger equation itself is incomplete because it ignores median symmetry, boundary conditions and the principle of least action. . etc.
The Schrödinger equation, like the diffusion equation, follows from first principles.
Preprint Chaotic dynamics of an electron
This is just an introductory answer to shed some light on the question and its answer.
The heat diffusion equation lives and operates in real measurable space-time while the Schrödinger equation lives and operates in imaginary space-time.
What is striking is that both are subsets of the numerical statistical method of Cairo techniques, which is the unified energy density field theory.
It predicts a connection between the solution of the heat diffusion equation and the Schrödinger PDE in 1D, 2D, 3D as follows [1]:
[Q]=[I-B]+C[I].
Where Q is the transition matrix for the Schrödinger equation and B is the transition matrix for the thermal diffusion equation.
This means that to solve the Schrödinger PDE for a given configuration, one only needs to solve the heat diffusion for the same configuration.
An illustrative 1D example is shown in Figure 1.
Note that the straight line solution of Ψ^2 is the same as that of the distribution of thermal energy (temperature T) in a long isolated bar under Drichlet boundary conditions.
To be continued.
1- Quantum buzzle, vacuum dynamics and Big Bang, Researchgate, IJISRT journal, June 2024.
Dear Ismail Abbas ,
Since high speed orbitals in atoms are not covered by Schrodinger equations (non lorentz Invariant) and classic Heat equation do not involve high speeds, we should avoid the term space-time to my understanding. That depends also on the fact that Schrodinger's equation cannot be changed to match SR requirements (to my opinion non realistic).
Yes one operates in space and time in real numbers the other in the space of complex numbers..
We have to remember though that the complex number use in Electromagnetism was introduced by Heaviside for the circuit analysis and in that case it involved the description of the status of the circuits...
There is a certain similarity in structure
and consequences, but physically address different things.
1 temperature
2 mass comcentration
3 probability density
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