Contrary to the opinion of the iron guards of SE who believe that this is the only unified field theory, the truth is that modern statistical mechanics of Cairo techniques and its B matrix chain products is the theory unified fields. Which means it can be the source of all SE solutions or any other time-dependent PDE.
There are too many examples of validation, but rather difficult to find.
The Schrödinger equation is a fundamental equation in quantum mechanics that describes the wave function of a particle, but it is not a part of statistical mechanics. Statistical mechanics is a separate branch of physics that aims to explain the macroscopic behavior of systems in terms of their microscopic constituents, using probability theory.
While quantum mechanics, including the Schrödinger equation, is important for understanding the behavior of individual particles and small systems, it is often impractical to apply it to large systems with many particles. Instead, statistical mechanics provides a framework for describing the average behavior of such systems without needing to solve the Schrödinger equation for all the particles.
In summary, the Schrödinger equation is a fundamental equation in quantum mechanics, but it is not a part of statistical mechanics. Statistical mechanics is a separate branch of physics that builds on quantum mechanics and classical mechanics to describe the macroscopic behavior of systems using probability theory.
This is just a brief response to shed some light on the question and its answer and to thank our fellow contributors for their helpful answers.
The response from our Argentinian friend “Neither one nor the other.
Different topics” is completely false.
Both can be defined as sets and have their own intersection and union (from an applications perspective) as two sets would.
The problem of solving a quantum particle in a one-dimensional well is an intersection application that exists as an element in both the Schrödinger equation and modern statistical mechanics.
The problem of solving the 1D, 2D and 3D dimensional thermal equation is another intersection application that exists as an application element in both sets “Schrödinger Equation and Modern Statistical Mechanics”.
Another angry response from our Argentinian friend.
His answer is interesting but totally flew and resembles the old iron guards of the Schrödinger equation.
1. Mother nature does not have two faces, classical and quantum, but only one face to show what modern statistical mechanics is.
Look at the question title.
A striking example is Einstein's proof of Planck's BB radiation law without any use of the quantum numbers 1,2,3, etc.
2. Any subject explicitly containing the word quantum must be understood as a deviated subset of modern statistical mechanics.
This is what the Cairo theory of techniques implies.
I hope you'll take a look.
3.Your statement “You would need to explore the quantum world
Statistical mechanics for more sense." makes no sense since probability and statistics (both macroscopic and microscopic) are a missing part of mathematics and classical theoretical physics [1].
They must be redefined in a 4D x-t unit space.
-----
1-What is missing in mathematics and theoretical physics,
Researchgate, IJISRT journal, March 2023.
For a more detailed answer,
Please see: An introductory framework for unified statistical field theory, Researchgate, IJISRT journal, May 2024.
Fundamentally, and this is what i find fascinating with the SE, it is that its solution contains discrete electron levels, the electron CANNOT be at any continuous distance from the nucleus, meaning the electron CAN ONLY occupy discrete or certain levels/distance from the nucleus, leading to the known periodic table of elements; so the SE predicts the existance of elements that are on the periodic table as per the standard model.
Im not angry
A SCHRODINGER eq. is normally for just
one electron not many as in stat mech
I understand why this may be confusing.
And yes, i defend the usual point of view.
Also stat mech has both clasical and quantum divisions.
Ever heard of Fermions and Bosons.?
The electron is a Fermion.
Libraries are full of or
probability or statistics.
Born interpretation in QM
Psi* Psi is probability density
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.
5. I told you stat mech and SE are diffdrent topics.
The statistics of electrons limits the
Ocupation of energy levels to 2 with
Oppozite spin. The exclusion principle.
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.
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