Cairo Theory of Techniques expands to cover the theory and practice of almost all areas of classical physics and quantum mechanics in addition to pure mathematical problems.

In fact, the modern solution for the quantum energy density described by the Schrödinger equation and its derivatives is only a subset of the modern statistical theory of Cairo techniques (or any other adequate numerical statistical theory ).

We prefer to present the evidence that quantum theory is a subset of the Cairo techniques via a somewhat exceptional way of explaining this broad field.

This presentation consists of asking relevant questions Q and proposing their appropriate answers A.

To begin with, the following assumptions are assumed to be true: 1-The Schrödinger equation is only valid for isolating infinite free space.

2- The Dirichlet boundary condition and the Neumann condition exist together and lead to the same point,

a f + b df/dx=c, where f is the function of interest and a,b,c are 3 constants

It's clear that,

i-if a/b tends towards 0, we recover Neumann

ii-if b/a tends towards 0, we recover Dirichlet

iii-when c = 0, in the context of the diffusion equation

3- All solutions of quantum mechanics problems in the 1D, 2D, 3D domains are symmetrical with respect to the midpoint or center of mass of the domain.

4- The square of the wave function Ψ (x,y,z,,t) =Ψ (x,y,z,,t) Ψ* (x,y,z,,t) represents the probability of finding the quantum particle to the volume element and meanwhile the quantum energy density U.

The physical picture is now clear and the Schrodiger equation for Ψ^2 or U is similar to the thermal diffusion equation:

dU/dt)partial= D Nabla^2 U+S . . . (1)

Subject to the boundary conditions of Diriclet and Neumann.

TO BE CONTINUED.

Answer II – Continued

The Dirichlet boundary conditions of equation 1 are,

Ψ(-∞)=Ψ(+∞)=0

In the language of statistical mechanics,[1]

Ψ^2(-∞)=Ψ^2(+∞)=0

which is the condition of an isolated quantum system.

And the quantum transition matrix Q is related to the classical transition matrix B via the relation,

Q=I+CI-B

C is constant depending on the size and structure of the transition matrix B.

Furthermore, the provision of eigenvalues ​​occurs automatically or spontaneously via the spatio-temporal evolution of the energy density field U of the quantum system,

(I+CI-B). U = λ.U

U=V+KE where KE is Nabla^U and V is a potential energy assumed or known in advance.

For the eigenvalue λ =1 it is in a way the fastest root or the principle of least action.

For example, the solution for quantum particles in a one-dimensional infinite potential well follows from the theory explained in reference 1.

We present the results of benchmark 1 for 9 free nodes in the following table,

X = 0 1 2 3 4 5 4 3 2 1 0

Const.V(x)= 0 C 1 C 2C 3C 4 C . . .

Potential energy V(x) vs x for a particle in an infinite potential well. Note that V(x) is symmetrical about the midpoint and has a right triangular shape

It should be noted that B-matrix strings are still in their infancy but are capable of defeating the Schrödinger equation almost everywhere.

What will happen in the next ten years?

To be continued.

1-An introductory framework for unified statistical field theory, Researchgate, IJISRT journal, May 2024.

Answer III (continued)

Nature is beautiful and the 4D x-t B-matrix chain solution is equally beautiful.

The following example is a nice explanation of the resolving power of transition chains of matrix B in the 3D geometric domain (3D box) shown in Figure 1.

The isolated rectangular box is discretized into 27 equally spaced free nodes.

The limits of the quantum dot are simultaneously subject to the Dirichlet and Neumann boundary conditions as explained previously in answer 1.

We first introduce and explain some physical and mathematical facts about the numerical statistical methods of Cairo techniques and its B-matrix chains:

1-B matrix chains are the only transition matrix chains other than the well-known Markov chains.

2-If we compare the old image of partial differential equation techniques including Schrödinger's partial differential equation with the modern statistical mechanics of Cairo techniques, the superiority of the latter is confirmed and appears as the challenge of the future.

3-We emphasize once again that the Cairo technical procedure

has been successfully applied to solve time-dependent PDEs in classical physics, such as the heat diffusion equation and the spatio-temporal distribution of sound energy density in audio rooms.

4- Extending the B matrix chains to cover almost all energy density field situations QM is not

complicated but quite long and requires mastery of certain

prerequisite in matrix algebra and ability to imagine nature in a 4D x-t unit space.

5- The modern approach to statistical mechanics completely neglects Schrödinger's PDE as if it had never existed.

The quantum energy density U (x, t) is expressed by:

U(x, t)=D(N) . (b + S) + IC. B^N. . . . . ………………..(2)

Where S is the vector of the source/sink term and IC is

the vector of initial conditions. Equation 2 is used as

statistical equivalence matrix PDE as a function of time which

was used to solve classical physics problems

such as thermal conduction PDEs and it is now proposed to find

a solution to Schrödinger's 3D geometric equation

shapes.

6-S(x,y,z,t)=Constant C1. V(x,y,z,t) . . . [2]……………………… (3)

Where S is a source term.

The constant in equation 3 is found by trial and error.

Equation 7 is a breakthrough because it characterizes

the physical domain of validity of SE and suggests a

statistical solution to the 3D SE which bypasses the SE itself [2] (Eq.4).

i.e. ∂ψ/∂t = -ħ²/2m ∂²ψ/∂x² + V(x)ψ . . . (4)

7-We emphasize once again that the Cairo technical approach does not require iteration of the out-of-the-box MATLAB or PYTHON algorithm or any other conventional mathematical iteration method.]

To determine whether this wave function satisfies the Schrödinger equation, we need to substitute it into the time-dependent Schrödinger equation:

Show that the numerical method of the Cairo technique and its chains of B matrices are the only unified field theory and what's more, it's so beautiful.

Ref 2- A 3D numerical statistical solution for the time-independent Schrödinger equation, Researchgate, IJISRT journal,

December 2023

To be continued.

The Schrödinger equation is the most-cited equation of the twentieth century. It may not end up being the most-cited one of the twenty-first century, but it certainly is far from dying. It is not likely to die at all in the foreseeable future, because Newton's force law has not died either.

The situation of the two equations is very similar in some respects: both Newton's mechanics and Schrödinger's quantum mechanics are non-relativistic theories. The former has been superseded by special and general relativity, the latter by quantum field theory. Still, we use the old theories whenever the situation allows it (because neither relativistic speeds nor relativistic partices are involved), so they remain valid in their domain of application. Therefore, there is no reason to expect Schrödinger's equation to ever die, just as there is no reason to expect Newtonian mechanics to die...

Some of the restrictions that have been enumerated for the Schrödinger equation simply don't apply. The Schrödinger equation is not limited to infinite domains, for example, it can be perfectly well solved for particles in a finite box.

Moreover, quantum mechanics is not a subset of statistical mechanics, it rather is an underlying fundamental theory allowing a description of microscopic systems. Statistical mechanics is valid only for macroscopic systems, so it cannot ever replace quantum mechanics (but needs it to be able to define entropy consistently, moreover, magnetization of macroscopic objects as well as black-body radiation are phenomena not explicable within classical statistical mechanics, you need the quantum version, which again may be based on the validity of the Schrödinger equation). The Schrödinger equation may be used in founding statistical mechanics but not vice versa.

Both the relativistic Schrödinger equation which is invariant under Lorentz transformations and the non-relativistic Schrödinger equation which is not invariant under Galilean transformations are for eternity.

Answer IV (continued)

Nature is beautiful and capable of solving any math or physics problem and B 4D x-t matrix string solution is equally beautiful and wireframe.

In particular probabilities and statistics which are well defined in the 4D unit space of matrix chains B but which belong to a missing part of mathematics and classical theoretical physics.

The following example is a nice explanation of the resolving power of the transition chains of the matrix B in 1D, 2D, 3D numerical integration which obviously exceeds the capacity of the Schrödinger equation or one of its derivatives.

1D statistical integration:

7 free nodes

In short, we arrive at,

The statistical integration formula for 7 nodes is given by,

I= 6 h/77(6.Y1 +11. Y2 + 14. Y3+15. Y4 +14. Y5 +

11.Y6 + 6.Y7). . . (1)

which is the statistical equivalence of Simpson's rule for 7

nodes.

Consider the case I=∫ y dx from x=2 to x=8 where y=X^2.

It is,

X = 2 3 4 5 6 7 8

Yes= 4 9 16 25 36 49 64

Trapezoidal ruler,

I =h*(Y1/2+Y2+Y3+Y4 +Y5+Y6+Y7/2)

=h*(2+9+16+25+36+49+32=169 square units.

Analytical integration expression,

I=X^3/3=(512-8)/3=168 square units.

Finally, the statistical integration formula for 7 nodes (Formula 1) is

given by,

I=6h/77(6*4 +11* 9 + 14* 16+15* 25 +14* 36 + 11*49 +

6*64)

I=167.455 square units.

This means that statistical integration is more precise than other numerical techniques.

Let's now move on to a more complex relationship y=x^3 on x=1 to

x=7 which gives,

X = 1 2 3 4 5 6 7

Y= 1 8 27 64 125 216 343

Trapezoidal ruler,

I =h*(Y1/2+Y2+Y3+Y4 +Y5+Y6+Y7/2)=612 square units

Analytical integration expression,

I=X^4/4=(2401-1)/4=600 square units.

Simpson's Rule: -----

Finally, the statistical integration formula for 7 nodes is

given by,

I=6h/77(6*1 +11* 8 + 14* 27+15* 64 +14* 125 + 11*216

+ 6*343)=598.13 square units.

Again, statistical integration is more precise than other numerical techniques.

2-Volume 7, number 11, November – 2022 International journal of innovative research sciences and technologies

ISSN Number: -2456-2165,

Is there a free math solution for

Digital integration

To be continued.

Answer V (continued)

Once again, nature is beautiful and powerful.

It is capable of solving all its mathematics or physics problems in the simplest and fastest ways, and the 4D x-t Matrix B series solution is equally beautiful and wireframed.

In particular, probabilities and statistics that are well defined in the four-dimensional unit space of matrix strings B but belong to a missing part of classical theoretical mathematics and physics are the underlying reason for the superiority of the former.

The following example is a nice explanation of the power of solving the transition series of matrix B to find the sum of infinite algebraic series, which clearly exceeds the power of the Schrödinger equation or one of its derivatives.

Here we consider the statistical-physical solution to a purely mathematical formula.

given by,

Consider the physical statistical solution to a purely mathematical formula.

The question  given by:

Using matrix algebra,[3] how can we show that the series of infinite integers

[(1+x)/2]^N is equal to (1+x)/(1-x), ∀x∈[0,1 [?

We assume that all mathematicians and physicists know that mathematics is the language of physics. However, not all mathematicians and physicists admit that modern physics (classical physics supplemented by B-matrix strings) can be the language of mathematics and replace it in certain areas/situations, as in the case of this question.

A numerical example for validation, the sum of the entire series, 0.99 + 0.99^2 + 0.99^3 + . . . . +0.99^Ntends to 190 as N tends to infinity. The proof of this question is based on the transfer matrix D :

plus the following rule (principle 1) [14].[For positive symmetric physical power matrices, the sum of their eigenvalues ​​is equal to the eigenvalue of their sum of power series] The question arises whether a classical mathematical proof can also be found?

We assume that the proposed axiom or mathematical principle:

[For the real statistical transition matrices that follow,

D(N)= B+B^2+B^3 + . +B^N

the sum of their eigenvalues ​​(λ +λ^ 2 + λ^3+ . . ., where λ∈[0,1[) is equal to the eigenvalue of their sum of integer series (i.e. λ of D(N) ] .. ..Principle (1)

It is obvious that λ is the eigenvalue of the transition matrix B and λ^2 is the eigenvalue of the matrix B^2. . etc.

Note that principle (1) is true and can be validated by numerical calculation.

However, to our knowledge, there is no rigorous mathematical proof of principle (1).

Additionally, using matrix algebra and principle 1, you can prove some mathematical formulas such as:

i- the infinite power series [(1+x)/2]^N is equal to (1+x)/(1-x), ∀x∈[0,1[.

ii- Moreover, the infinite integer series [(1 + Mx) / (1 + M)] ^ N is equal to (1 + Mx) / M (1-x), ∀x element of [0, 1[ and M is a positive integer?

The five examples above show beyond doubt that Schrödinger PDE is a subset of statistical matrix mechanics and not the other way around.

Statistical matrix mechanics is concerned with the nature of quantum particles and their associated energy density, but Schrödinger's PDE describes the probability distribution of the energy density of a quantum system over possible energy levels.

This means that Schrödinger PDE and statistical matrix mechanics are apparently two different subjects. However, Schrödinger's PDE remains a subset of statistical matrix mechanics, in the sense that any solution for the energy density in a quantum system found by Schrödinger's PDE can also be obtained via appropriate statistical matrix mechanics. And more via many missing physical and mathematical solutions in Schrödinger's PDE are well defined and explained in statistical matrix mechanics.

3-Using matrix algebra, how to show that the infinite integer series [(1+x)/2]^N is equal to (1+x)/(1-x), ∀x∈[0,1[ ? , Researchgate, IJISRT journal, November 2023.

To be continued.