We assume this to be true in a manner similar to the statistical modeling approach called Cairo techniques. Here, real time t exists in intervals quantized as a dimensionless integer 1,2 3 , . . .N and has been successfully used to solve time-dependent PDEs in 4D x-t unit space. some examples are heat diffusion versus time, Laplace and Poisson PDEs, sound volume and reverberation time in audio rooms, digital integration and differentiation, etc.
These classical physics solutions can be called statistical equivalence of the time-dependent diffusion problem.
[some examples are given by 1,2,3]
The Schrödinger PDE (SE) itself is no exception and the statistical equivalence of the SE exists.
Surprisingly, this SESE is more revealing and more comprehensive than the SE itself.
The route is quite long, how can we cross it?
1-A numerical statistical solution to the partial differential equations of Laplace and Poisson, Researchgate, IJISRT journal.
2-a statistical numerical solution for the time-dependent 3D heat diffusion problem without the need for the PD thermal equation or its FDM techniques, Researchgate, IJISRT journal.
3-Theory and design of audio rooms-A statistical view, Researchgate, IJISRT journal.
This is an introductory answer to shed some light on the question and part of its answer.
We assume that quantum physics and the Schrödinger equation itself constitute a subset of modern physics.
By modern physics we mean all universal laws of physics such as thermodynamics, electromagnetic waves, special and general relativity, Newtonian mechanics,..etc, as well as the modern definition of 4D binary transition probability introduced in 2020 via the transition matrix B and the Cairo technique [1,2,3].
The hypothetical definition of 4D x-t space via a proper transition matrix is essential and shows how mother nature works in x-t space, but is absent from both classical physics and quantum mechanics[3].
In order not to worry too much about the details of the theory, let's move on to the numerical solution of the Schrödinger equation first through pure mathematics and then via the statistical transition matrix B.
Somewhat surprisingly, the statistical matrix solution B of SE is more complete and more revealing than the pure mathematical solution, as the following analysis shows:
Let's start by discretizing the differential operator in SE as follows,
(d2/dx2)ψ≈[ψ(x−d)−2ψ(x)+ψ(x+d)]/d^2≡[ψn−1−2ψn+ψn+1]/d2 . . . (1)
where d is the distance between two successive points on the x-axis and
ψn is the value of the nth wave function indicate.
The Schrödinger equation can now be rewritten in this notation:
−ℏ^2/2m[(ψn−1)−2ψn+(ψn+1)]/d2+Vnψn=Eψn . . . (2)
Note that equation 2 is an eigenvalue matrix equation, namely:
Aψ = Eψ. . . (2)
more explicitly, the matrix A is in fact the super position of two matrices
M1 and M2.
M1 is a tridiagonal matrix, and is given by:
−ℏ^2/2md^2x
−2 1 0 0 . . . . . . . . . . . . . . . . . . . . . . 0
1 -2 1 0 0 0 . . . . . . . . . .. . . . . . . . . . 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0 0 0 . . . . . . . . . . . . . . . . . . . 0 0 1 -2
Call it Matrix M1.
And the diagonal matrix,
V1 0 0 0 . . . . . . . 0 0
0 V2 0 ...0 0 . .. . 0 0
. . . . . . . . . . . . . . . .
0 0 0 . . . . . . . 0 0 Vn
Call it Matrix M2.
And the time-independent numerical solution of Schrodinger equation is given in the matrix form,
[ψ1 ψ2 . . . . ψN−1 ψN]T = (M1+M2) .[V1 V2 . . . . . . .VN−1 VN)T . . . . (3)
Obviously, the integer N must be large enough to guarantee more precision.
In the present literature, the solution of the eigenvalues and eigenvectors of equation 3 is obtained via ready-to-use algorithms in Matlab or Python. .etc.
It should be mentioned that the definition of boundary conditions is as follows:
ψ1=ψn=0
which usually works.
Let us now move on to the statistical equivalence of the time-dependent 1D Schrödinger equation which is a simple extension of the 1D matrix chains B with the only difference that the applied potential V(x) is the source/sink term[1,2, 3].
In the statistical solution, the Schrödinger equation and its derivatives are completely ignored as if they never existed.
The most important innovation here is that the important ratio V(X)/E where E is the total energy is expressed by the numerical value of the diagonal element RO of the transition matrix B.
We have shown in previous articles [1,2,3] that:
The 1-D statistical transition matrix B is expressed as follows:
B=
RO 1/2-RO 0 0 0 0 0... . . . . . .0
1/2-RO RO 0.5-RO 0 0 0 . . . . 0
0 0.5-RO RO 1/2-RO 0 0 0 . . .0
. . . . . . . . . . . . . . . . . . . . . .
0 0 0 . . . . . . . . . . . . .1/2-RO RO
Eq (4)
and the statistical solution of SE is given by:
U(x,t)=(B^0+B+B^2 . . . +B^N) . (b +S). . . . .(5)
Where b is the boundary conditions vector arranged in a suitable order and S is the source/sink term vector which is replaced by the potential V(x).
Note that equation 5 for sufficiently large N, the infinite series
B^0+B+B^2 . . . +B^N + . .
is reduced to the inverse of I-B.
In other words,
B^0+B+B^2 . . . +B^N=[1/(I-B)] . . (6)
where B^0=I, the unitary matrix.
Equation 5 for U(x,t) introduces a transient transfer function D(x,N) which is equivalent to the sum of the two matrices M1+M2.
The statistical solution is given by,
U(x,t)=D(x,N ). V(x)
It is worth mentioning that equations 4.5 replace and completely ignore SE.
FEATURES OF THE STATISTICAL SOLUTION:
Equations 4,5 impose far-reaching predictions in the field of quantum physics.
They define five distinct criteria for the statistical equivalence of the Schrödinger equation and therefore for SE itself.
These five distinct criteria are defined according to the value of the main diagonal R which represents the potential energy/total energy ratio.
1-: RO=0.
This means that the potential energy is zero.
Here, the statistical solution is reduced to solving the Laplace and Poisson PDE for the energy emw in a box, that is to say in a limited medium [1].
2- RO is an element of the interval ]0.1/2[
Here, the statistical solution is reduced to solving a time-dependent PDE for heat diffusion in a material of given geometric shape [2,4].
This shows that the Schrödinger equation and the thermal diffusion equation of classical physics are somehow the same thing.
3- RO=1/2.
Here, the statistical transition matrix B and consequently the transfer matrix D reduce to the unitary matrix I.
The solution is reduced to a stationary heat flow in a perfect thermal insulator.
4: RO element of the interval ]1/2,1[.
Here we find the important oscillatory or wave statistical solution of the transition matrix B, as illustrated in the next section.
5: RO=1.
This means that the kinetic energy is zero.
Here, the elements or inputs of matrices D diverge to infinity and therefore the solution does not exist. There is no solution for SE when KE is zero.
However, in the current literature, most mathematicians and theoretical physicists assume that the quantum minimum KE corresponds to h/2 and not zero without giving any explanation.
ILLUSTRATIVE EXAMPLE
The following is an illustrative example of calculating the statistical solution of the Schrödinger equation for a quantum particle in a 1D infinite potential well.
The transfer matrix D for 15 nodes resulting from equation 5 for N=20 terms is calculated using equation 5 and the numerical results are as follows:
2.6766 -0.7136 0.1881 -4.8344E-2 1.1889E-2 -2.7363E-3 0 0 0 0 0 0 0 00-
0.7136 2.8647 -0.7619 0.2 -5.108E-2 1.2466E-2 -2.8457E-3 0 0 0 0 0 0 00.1881 -0.7619 2.8766 -0.7647 0.2005 -5.1189E-2 1.2484E-2 -2.8484E-3 0 0 0 0 0 0 0-4.8344E-2 0.2 -0.7647 2.8771 -0.7648 0.2006 -5.1192E-2 1.2485E-2 -2.8484E-3 0 0 0 0 0 01.1889E-2 -5.108E-2 0.2005 -0.7648 2.8772 -0.7648 0.2006 -5.1192E-2 1.2485E-2 -2.8484E-3 0 0 0 0-2.7363E-03 1.2466E-2 -5.1189E-2 0.2006 -0.7648 2.8772 -0.7648 0.2006 -5.1192E-2 1.2485E-2 -2.8484E-3 0 0 0 00 -2.8457E-3 1.2484E-2 -5.1192E-2 0.2006 -0.7648 2.8772 -0.7648 0.2006 -5.1192E-2 1.2485E-2 -2.8484E -3 0 0 00 0 -2.8484E-3 1.2485E-2 -5.1192E-2 0.2006 -0.7648 2.8772 -0.7648 0.2006 -5.1192E-2 1.2485E-2 -2, 8484E-3 0 00 0 0 -2.8484E-3 1.2485E-2 -5.1192E-2 0.2006 -0.7648 2.8772 -0.7648 0.2006 -5.1192E-2 1.2484E-2 -2.8457E-3 00 0 0 0 -2.8484E-3 1.2485E-2 -5.1192E-2 0.2006 -0.7648 2.8772 -0.7648 0.2006 -5.1189E-2 1.2466E-2 -2.7363E-30 0 0 0 0 -2.8484E-3 1.2485E-2 -5.1192E-2 0.2006 -0.7648 2.8772 -0.7648 0.2005 -5.108E-2 1.1889E-20 0 0 0 0 0 -2.8484E-3 1.2485E-2 -5.1192E-2 0.2006 -0.7648 2.8772 -0.7647 0.20 -4.8344E-20 0 0 0 0 0 0 -2.8484E-3 1.2484E-2 -5.1189E-2 0.2005 -0.7647 2.8766 -0.7619 0.18810 0 0 0 0 0 0 0 -2.8457E-3 1.2466E-2 -5.108E-2 0.20 -0.7619 2.8647 -0.71360 0 0 0 0 0 0 0 0 -2.7363E-3 1.1889E-2 -4.8344E-2 0.1881 -0.7136 2.6766
Call it Matrix M7.
It is clear that the tridiagonal matrix 7 has the same eigenvalues and the same eigenvectors as the tridiagonal matrix M1+M2 for the presupposed potential distribution V(x).
Furthermore, it can be shown that the multiplication D(x).V(x) is quite valid for different types of V(x) other than the flat potential well.
1-A numerical statistical solution to the partial differential equations of Laplace and Poisson, Researchgate and IJISRT review ,Nov 2020.
2-a statistical numerical solution for the time-dependent 3D heat diffusion problem without the need for the PD thermal equation or its FDM techniques, Researchgate and IJISRT review ,July 2020.
3-What is missing in mathematics and theoretical physics, Researcchgate and IJISRT,Mars 2023.
4-A rigorous experimental technique to measure the thermal diffusivity of metals in different 3D forms, Researchgate and IJISRT review ,Oct 2022.
Answer II-continued
This is only a brief answer, on the one hand to clarify the question and its answer, and on the other hand to thank our mathematician colleagues who usually formulate their questions/answers in a purely mathematical language that we, physicists, do not we don't use. . not very often. .
Mathematics without physics is meaningless and physics without mathematics is blind.
It is worth mentioning that I. Newton, who was the first to discover the methods of differential calculus and iterative calculus, was a physicist working on double-slit interference experiments.
The “statistical chain”? (or a “transition” matrix?) exist and currently we know of two:
1-Markov transition matrix and statistical Markov chain.
2-The transition matrix B "which is more physical" and its statistical chain.
Returning to the matrices, we clarify the following:
i-Square matrices are a subset of mathematical matrices.
II-physical square matrices that have physical meaning (such as the transition matrix B) are a subset of square matrices.
iii- Not all matrix equations are eigenvalue equations.
For example, the matrix of the numerical solution of the heat diffusion equation results in a system of non-homogeneous first-order linear algebraic equations while the matrix of the numerical solution of the Schrödinger equation is homogeneous and results in a Eigenvalue problem with multiple eigenvalues. and their corresponding eigenvectors.
iii- Statistical transition chains do not operate in the Hamiltonian space "PE+KE=constant which is suitable for closed systems" and have their own space where PE/E =constant.
iv-Generally speaking, in the statistical transition matrix the “eigenvalue” is the dominant eigenvalue equal to 1.
I hope this answer can help ,
To be continued.
Here is a more complete and coherent answer published on Researchgate and the IJISRT journal:
A statistical numerical solution for the time-independent Schrödinger equation, November 2023.
Your comments on the article are welcome.
Please go to,
A statistical numerical solution for the time-independent Schrödinger equation -II
It is more complete and more understandable.
Another way to describe quantum particle dynamics is to use statistical transition matrices that completely ignore the Schrödinger equation as if it never existed in the same way that one solves the heat diffusion equation without going through thermal PDE itself.
today we only know one physical transition matrix which is the transition matrix B resulting from the so-called Cairo technique.
Step 1
Construct the 2D statistical matrix B corresponding to Figure 1 which represents a quantum particle in a 2D infinite potential well.
📷
Fig 1. A quantum particle in a 2D infinite potential well
The basis for generating an eigen or proper matrix is the 2D matrix B with 9 equidistant free nodes as shown in Figure 2, nodes 1-9.
Note that RO=0 because PE is zero.
It is expressed by the same matrix M1 explained previously in the example of 2D thermal conduction.
step 2
Compose the proper or eigen matrix M2 as given by,
M2=M1+S(x,y)
Where S(x,y ) is a diagonal matrix and S=C1*V(x,y)
The resulting eigenmatrix M2 will be given by,
M2=
1/14 1/4 0 1/4 0 0 0 0 0
1/4 4/14 1/4 0 1/4 0 0 0 0
0 1/4 1/14 0 0 1/4 0 0 0
1/4 0 0 4/14 1/4 0 1/4 0 0
0 1/4 0 1/4 9/14 1/4 0 1/4 0
0 0 1/4 0 1/4 4/14 0 0 1/4
0 0 0 1/4 0 0 1/14 1/4 0
0 0 0 0 1/4 0 1/4 4/14 1/4
0 0 0 0 0 1/4 0 1/4 1/14
Where C1 is substituted for by the factor 1/14.
step 3
The energy eigenvector E(x,y) is equal to the principal diagonal of the matrix A which gives the following eigenvector equation,
2/14 1/4 0 1/4 0 0 0 0 0
1/4 4/14 1/4 0 1/4 0 0 0 0
0 1/4 2/14 0 0 1/4 0 0 0
1/4 0 0 4/14 1/4 0 1/4 0 0
0 1/4 0 1/4 9/14 1/4 0 1/4 0
0 0 1/4 0 1/4 4/14 0 0 1/4
0 0 0 1/4 0 0 2/14 1/4 0
0 0 0 0 1/4 0 1/4 4/14 1/4
0 0 0 0 0 1/4 0 1/4 2/14
*
[2/14 4/14 2/14 4/14 9/14 4/14 2/14 4/14 2/14]T
is equal to,
[8/49 123/392 8/49 123/392 137/196 123/392 8/49 123/392 8/49] T
Showing that the energy eigenvector is=
[2/14 4/14 2/14 4/14 9/14 4/14 2/14 4/14 2/14 ] T
with a dominant eigenvalue almost equal to 1.
The reason why we multiply the nodes 1,3,6 and 9 by the factor 2 is that these nodes are located at the four intersections of the two axes x and y where the rule E=Ex+Ey applies.
The x-oriented eigenvectors and the y-oriented eigenvectors are shown in Figure 2 in black and red lines.
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