Is it true that Schrödinger's equation can be derived from the heat diffusion equation?
We assume this is absolutely true.
Simply replace the dependent variable thermal energy U or T Kelvin with the quantum energy Ψ^2 and you get the time-independent Schrödinger equation for Ψ^2.
It is obvious that the SE solution for Ψ is the Sqrt of the solution for Ψ^2.
Note that time t and Ψ are here ∈(R).
Here is a brief introductory answer to shed some light on the question and its answer:
The main objection to the question and its answer is that the Schrödinger equation is an eigenvalue problem while the heat equation is not, which is not true.
The heat equation can be transformed into an eigenvalue problem thanks to the revolutionary hypothesis,
S(x,y,z,t)=Constant. Ψ^2(x,y,z,t). . . (1)
Where S is the source-sink term.
Equation (1) transforms the heat equation into an eigenvalue problem which in this case corresponds to the time-independent SE.
The physical meaning of equation (1) is that energy can be transformed into a quantum particle and vice versa for an isolated system in infinite free space.
A simple application of transforming the heat diffusion equation into SE via equation (1) is the numerical solution of a quantum particle in a 1D infinite potential well, as shown in Figures 1, 2, 3.
To be continued.
Fig 1 QM 103.png
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Fig.2 QM103.png
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Fig1.QM102.png
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Ismail Abbas
Yes, the Schrödinger equation can be derived from the equation of diffusion. The diffusion coefficient corresponds to the angular momentum operator in quantum theory.
Article Correlation between Diffusion Equation and Schrödinger Equation
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