The mechanical product of the B matrix of the Cairo techniques predicts that if the solution of the heat diffusion equation is forced by the boundary conditions and physical properties of the material involved, then the solution of the quantum mechanical system is spontaneous and function of the size of the quantum object as well as the external potential if applicable.
The question arises as to why the quantum mechanics solution and the classical physics solution emerge from the same transition matrix B?
Please take a look at the article,
A spontaneous statistical solution of Schrödinger's partial differential equation
tion
February 2024
Researchgate, IJISRT International Journal, February 2024.
We hope you find a detailed and useful answer.
The spontaneous statistical solution of Schrödinger's partial differential equation is very efficient and creative.
An important reason to replace the Schrödinger equation with the equivalent B-matrix mechanics is that you move from the domain of ES where many questions remain unanswered to the domain of statistical matrix equations where almost all questions have answers.
For example, the numerical results of the solution for a 1D infinite potential well are shown in the following Figures 1,2.
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Fig.1 Spatial distribution of the wave function ψ in a 1D infinite potential well. (maximum probability)
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Fig.2 Spatial distribution of the wave function ψ in a 1D infinite potential well. (near maximum probability)
Furthermore, the equivalent mechanics of matrix B can directly show that the solution is invariant and reversible over time and that it follows the rule of least action.
Unless you're talking about perturbation theory, the terms "spontaneous" and "stimulated" is not defined with respect to the Schrödinger equation, so we clearly have a case of mislabelling here.
This is a brief answer, first to shed some light on the question and its answer, and second to thank our sincere German colleague for his helpful answer:
Unless you're talking about perturbation theory, the terms "spontaneous" and "stimulated" are not defined with respect to the Schrödinger equation, so we clearly have a case of mislabeling here. »
*The B-matrix mechanic completely ignores the Schrödinger PDE and all its derivatives as if it had never existed[1].
An important reason to replace the Schrödinger equation with the equivalent B-matrix mechanics is that you move from the realm of the Schrödinger equation where many questions remain unanswered to the realm of statistical matrix equations where almost all questions have answers.
In extreme cases, it is like going from the darkness of night to the light of a sunny day.
Furthermore, the correspondence principle according to which in the limit of large quantum numbers the predictions of quantum physics become identical to the predictions of classical physics does not apply to classical physics problems such as heat diffusion and waves sound.
It is worth mentioning that the concept of the terms "spontaneous" and "stimulated" existed decades before Schrödinger's equation in 1927.
Furthermore, these terms or concepts of spontaneity and stimulation can be well defined via B-matrix mechanics outside of the overall SE context.
1- A spontaneous statistical solution of the Schrödinger equation, IJISRT journal, Researchgate, February 2024.
For interested contributors:
Please take a look at,
Cairo Technics Solution of Schrödinger's Partial Differential Equation - Time Dependence, Researchgate, IJISRT Journal, March 24.
We assume there is a detailed answer.
Many physicists and mathematicians assume that mother nature has two distinct languages, one for macroscopic objects in classical physics and the second for microscopic subatomic objects which is Schrödinger's PDE and its derivatives.
Moreover, the old iron guards of SE believe that these two languages reduce to a single one which is the solution of SE, assuming that eventually it can deal with macroscopic objects (Via SE has no scales and principle of Correspondence).
The opposite is true:
The Schrödinger equation is only a subset of modern probability classical mechanics.
SE itself is a probability conservation equation!
We assume that nature has only one language to speak to itself, namely the physical statistical chains of the B matrix capable of solving the classical heat diffusion equation and the quantum Schrödinger PDE in a configuration space 3D.
It is striking that the closed, empty configuration bounded in 1D, 2D and 3D has its own probability and statistics, even without any energy density field inside.
A striking example of the above statement appears in limited mathematical integrations:
I= ∫ y dx from x=a to x=b, . . . . (1),
I=∫∫ W(x,y) dx dy from x=a to x=b and y=b to y=c. . . . . (2)
..etc..
Equations 1,2 can be calculated numerically and precisely via the transition chains of the matrix B[1] by applying the so-called statistical weights,
I{1D,7 nodes}=6 h /77(6*Y1 +11* Y2 + 14* Y3+15* Y4 +14* Y5 + 11*Y6 + 6*Y7). . . . (1*)
I{2.9 nodes}= 9 h^2 /29.5
2.75 3.5 2.75
3.5 4.5 3.5
2.75 3.5 2.75
(Y11,Y12,Y13,Y21,Y22,Y23,Y31,Y32,Y3) .. . . (2*)
NOW, believe it or not, equations 1*,2* are exactly the numerical statistical solution for ψ(r)^2 in the time-independent Schrödinger PDE.
Showing that the solution of the time-dependent Schrödinger PDE is spontaneous rather than stimulated.
[1]Effective unconventional approach to statistical differentiation and statistical integration, Researchgate, IJISRT journal, November 2022.
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