We assume that V(x,y,z,t) is the external potential applied to the quantum particle enclosed in a closed system.
What is quite surprising is that there exists another spontaneous component for V which comes from the energy density of the system itself expressed by,
V(x, y, z, t)=Cons U(x, y ,z ,t) . . . . (1)
Eq 1 is a revolutionary breakthrough.
Equation 1 means that quantum energy can be transformed into quantum particles and vice versa.
Additionally, Equation 1 (predicted by the B-matrix chains of the Cairo Statistical Numerical Method) eliminates any confusion about whether the Schrödinger PDE is a wave equation or a diffusion equation and provides a definitive answer:
she could be both.
Usually no t. It is just some electrostatic or other potential.
For example between atoms a locally parabolic potential,
Or a 1/r coulomb potential
Surprisingly, the solution to Schrödinger's PDE and its physical interpretation is the least understood topic among mathematicians and theoretical physicists.
The short answer to this question is that the quantum particle is subject to the applied potential V(x,y,z,t).
This scalar potential V is composed of two components V1 and V2.
We can safely assume that,
V = C1. V1 + C2.V2. . . (1)
Where V1 is the external applied voltage and V2 is the spontaneous or self-applied voltage.
V2(x,y,z,t)=Cons. U(x,y,z,t) . . . (2)
Where U is the quantum energy density.
Obviously, C1 + C2 = 1.
Therefore, the Schrödinger PDE is part diffusion equation and part wave equation.
It is worth mentioning that equations 1,2 are not suggestions but rather certainty equations applied with extraordinary precision in solving SE for central field situations as well as for infinite 1D, 2D potential in addition to that of the isolated quantum dot.
To be continued.
Its well understood in physics.
First solve some typical conservation
Of energy problems in rational mechanics.
Answer II-Continued
The death of critical thinking will kill us long before AI. (Joan Westenberg)
This is just a brief response to shed some light on the question and its answer and to thank our Argentinian friend for his helpful response.
Schrödinger's PDE,
i h dΨ/dt)partial=h^2 . Nabla^2 Ψ/2m + V Ψ . . . . (1)
is precise but incomplete.
Now think about solving SE for Ψ^2 and not Ψ.
Equation 1 transforms to,
dΨ^2/dt)partial=C1. Nabla^2 Ψ/2m + C2 .V . . . . (2)
With the following hypotheses proven numerically,
i-Ψ^2=Ψ . Ψ*
ii-Ψ^2 is exactly equal to the quantum energy density of the quantum particle.
iii-Ψ^2 is exactly equal to the probability of finding the quantum particle in the 4D unit volume element x-t "dx dy dz dt"
iv-Real time t is completely lost and replaced by the dimensionless integer N dt.
Where N is the number of iterations or repetitions and dt is the time jump.
Equation 2 belongs to and is solved by matrix mechanics.
Equation 2 does not need any PDE or FDM method to be solved.
What is quite surprising is that equation 2 is more informative than equation 1.
To be continued.
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