We assume that the conservation of probability does not exist in quantum mechanics as in the case of classical mechanics.
According to Shrodinger's equation, there is a continuous generation of probability density in subatomic systems.
No, that’s wrong, as the study of any textbook on qusntum mechanics explains.
Instead of making trivially wrong statements, it might be a better idea to take the time to study classical and quantum mechanics.
This is only a preliminary introductory response to shed light on the importance of this strategic issue.
In fact, the question itself leads to a more comprehensive question:
Is it possible to solve Schrödinger's equation via a statistical transition matrix without needing Schrödinger's equation itself?
We recall how the time-dependent heat equation is solved in its most general case without needing the PDE of heat itself.
However, there are few minorities of "scientists" who may judge the matter as trivial due to a lack of understanding.
The statistical transition matrix for the Schrödinger equation is not complicated but a bit long which will be the subject of the next answer.
**Answer continued part II.
This is just an introductory answer to explain the solution to a more comprehensive question:
Is it possible to solve Schrödinger's equation via a statistical transition matrix without needing Schrödinger's equation itself?
We recall how the time-dependent heat equation is solved in its most general case without needing the PDE of heat itself.
Brief:
The transition matrix B in 1D x-t space for n free nodes is given by,
B=
RO 1/2-RO/2 0 0 0 0 0
1/2-RO/2 1/2- 0 0 0 0 0
---
0 0 0 0 0 1/2-RO/2 RO
The corresponding statistical transition matrix for the Schrödinger equation (we call it Q-matrix) is given by,
Q=
2+2md^2 Y(x1)/h^2 -1 0 0 0 0 0
-1 2+2md^2 Y(x2)/h^2 -1 0 0 0 0
---------------
0 0 0 0 0 -1 2+2md^2 Y(xn)/h^2
The similarity of the classical B-matrix and the quantum Q-matrix is obvious.
Hint: the quantum diffusion coefficient β^ 2 = ℏ / (2 m) [1].
** In a following answer we provide detailed illustrative numerical solutions (in 1D, 2D and 3D unit spaces) of the calculations of the outputs of Schrödinger's equation in an infinite potential well via the transition matrix technique Q, which is also a product of Cairo Technique Theory.
. . . . . .
Ref: 1-
Schrödinger's equation as a diffusion equation
Mita, Katsunori (Department of Physics, St. Mary's College of Maryland, St. Mary's City, May 21, Maryland 20686).
*** Answer continued part III.
This question is not just about physics (classical/quantum) or mathematics, but a rigorous and comprehensive combination or incorporation of all three.
I'm sure many researchgate contributors can answer this question as well as I, if not better.
In fact, it is possible to solve the Schrödinger equation via a statistical transition matrix without needing the Schrödinger equation itself in the same way as solving the heat diffusion PDE, in his case the most general, without needing the heat equation itself.
The only requirement is the ability to imagine nature in 3D geometry (solid geometry) and then, in a later stage, in 4D x-t unit block space.
The energy field in 4D x-t unit space is resolved in B-matrix chains and can also be resolved via any other suitable statistical transition matrix.
My recommendation here is not to listen to those "scientists" who dismiss the whole matter as insignificant due to their own lack of imagination and understanding.
While waiting for your answers or suggestions, I am preparing my own answer based on a modified B-Transition matrix (we call it Q-matrix).
Therefore, in a following answer, we provide our own numerical answers (where I assume you can personally provide answers that are adequate or better than mine) calculations of the outputs of the Schrödinger equation in an infinite potential well via the technique of the Q-matrix for the quantum transition.
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