There are many theories, but only two are serious.
i- In 1925, the Heisenberg-Born-Jordan matrix introduced the first consistent matrix formulation of quantum mechanics, albeit somewhat strange and incomplete.
It states that:
Ψ = √h
0 -i√1 0 0
i√1 0 -i√2 0 . .
0 i√2 0 -i√3 . . .
0 0 i√3 0 -i√4 . .
etc.
. . . Q1
It is clear that the Heisenberg matrix Q1 is the energy operator of the quantum mechanical system in the sense that:
E= ∭Ψ^2 (x,y,z,t) dx dy dz. where E is the total energy of the quantum mechanical system E (x,y,z,t) = ΨΨ* =Ψ^2.
Equation Q1 suggests that Heisenberg imaginary matrix describes the square root of the energy density of the mechanical system not the the energy density itself.
We suggest the following improvement which is physical and logical,
This improvement is the subject of this question.
We propose here two major modifications to improve the Q1 matrix above:
i- Transform the matrix into a real matrix by removing the complex unit i and replace Q by its square Q^2.
ii- Transform the matrix above into a sawtooth pattern, which is the most efficient for a quantum matrix.
This improvement actually constitutes a revival, or even the rise of matrix mechanics.
Then the new Q^2 matrix for 1D and 19 free nodes will be written as:
And its square Q^2 is expressed as,0 -1^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01^0.5 0 -2^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 2^0.5 0 -3^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 3^0.5 0 -4^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 4^0.5 0 -5^0.5 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 5^0.5 0 -6^0.5 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 6^0.5 0 -7^0.5 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 7^0.5 0 -8^0.5 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 8^0.5 0 -9^0.5 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 9^0.5 0 -8^0.5 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 8^0.5 0 -7^0.5 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 7^0.5 0 -6^0.5 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 6^0.5 0 -5^0.5 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 5^0.5 0 4^0.5 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 4^0.5 0 3^0.5 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 3^0.5 0 2^0.5 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2^0.5 0 1^0.5 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1^0.5 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1^0.5 0
Note that the new matrix Q brings a striking fact because it contains √2 √3 . . etc in addition to √10, accurate to 8 significant digits.-1 0 2^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 -3 0 6^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 02^0.5 0 -5 0 2*3^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 00 6^0.5 0 -7 0 2*5^0.5 0 0 0 0 0 0 0 0 0 0 0 0 00 0 2*3^0.5 0 -9 0 30^0.5 0 0 0 0 0 0 0 0 0 0 0 00 0 0 2*5^0.5 0 -11 0 42^0.5 0 0 0 0 0 0 0 0 0 0 00 0 0 0 30^0.5 0 -13 0 2*14^0.5 0 0 0 0 0 0 0 0 0 00 0 0 0 0 42^0.5 0 -15 0 6*2^0.5 0 0 0 0 0 0 0 0 00 0 0 0 0 0 2*14^0.5 0 -17 0 6*2^0.5 0 0 0 0 0 0 0 00 0 0 0 0 0 0 6*2^0.5 0 -17 0 2*14^0.5 0 0 0 0 0 0 00 0 0 0 0 0 0 0 6*2^0.5 0 -15 0 42^0.5 0 0 0 0 0 00 0 0 0 0 0 0 0 0 2*14^0.5 0 -13 0 30^0.5 0 0 0 0 00 0 0 0 0 0 0 0 0 0 42^0.5 0 -11 0 -2*5^0.5 0 0 0 00 0 0 0 0 0 0 0 0 0 0 30^0.5 0 -1 0 2*3^0.5 0 0 00 0 0 0 0 0 0 0 0 0 0 0 2*5^0.5 0 7 0 6^0.5 0 00 0 0 0 0 0 0 0 0 0 0 0 0 2*3^0.5 0 5 0 2^0.5 00 0 0 0 0 0 0 0 0 0 0 0 0 0 6^0.5 0 3 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2^0.5 0 1 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0
Note also that the new matrix Q gives another striking fact where the input sub-diagonals are [2,6,12,20,30,42,56,72,56,42,30,20,12,6,2]
which exactly matches the required sawtooth shape.
This improvement actually constitutes a revival, or even the rise of matrix mechanics
Again, for 1D 17 free nodes, we have,
.Q=
-1 0 2^0.5 0 0 0 0 0 0 0 0
0 -3 0 6^0.5 0 0 0 0 0 0 0
2^0.5 0 -5 0 2*3^0.5 0 0 0 0 0 0
0 6^0.5 0 -7 0 2*5^0.5 0 0 0 0 0
0 0 2*3^0.5 0 -9 0 30^0.5 0 0 0 0
0 0 0 2*5^0.5 0 -11 0 30^0.5 0 0 0
0 0 0 0 30^0.5 0 -11 0 2*5^0.5 0 0
0 0 0 0 0 30^0.5 0 -9 0 2*3^0.5 0
0 0 0 0 0 0 2*5^0.5 0 -7 0 6^0.5
0 0 0 0 0 0 0 2*3^0.5 0 -5 0
0 0 0 0 0 0 0 0 6^0.5 0 -2
Note that the new matrix Q contains √2, √3, √5, etc. in addition to √10, all with 8-digit precision.
We assume that neither E. Schrödinger nor N. Bohr understood how quantum mechanics works.
Cairo's AI techniques predict that QM operates according to the square of SE rather than SE itself.
Below, we present the solution for the QM particle in infinite 1D and 2D potential wells, and in a closed control volume box.
The first step is to completely neglect the PDE of SE, as if it never existed, and solve for its squared PDE, supplemented by the AB law proposed by the author:
S(x,y,z,t)=Const.V(x,y,z,t)
The AB law means that the spontaneous potential of SE can be transformed into quantum matter and vice versa.
Here, the PDE matrix of QM for 17 nodes is given by the eigenmatrix equation:
166/2187 244/2187 266/2187 140/2187 35/2187 0 0 0 0 0 0 0 0 0 0 0 0
427/4374 853/4374 434/2187 322/2187 245/4374 35/4374 0 0 0 0 0 0 0 0 0 0 0
133/1458 124/729 343/1458 2/9 205/1458 10/243 5/1458 0 0 0 0 0 0 0 0 0 0
175/4374 230/2187 5/27 373/1458 355/1458 280/2187 55/2187 5/4374 0 0 0 0 0 0 0 0 0
35/4374 70/2187 205/2187 142/729 211/729 194/729 227/2187 26/2187 1/4374 0 0 0 0 0 0 0 0
0 5/1458 5/243 56/729 97/486 505/1458 205/729 50/729 5/1458 0 0 0 0 0 0 0 0
0 0 5/4374 22/2187 227/4374 410/2187 1981/4374 584/2187 43/1458 0 0 0 0 0 0 0 0
0 0 0 1/4374 13/4374 50/2187 292/2187 2849/4374 827/4374 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 827/4374 2849/4374 292/2187 50/2187 13/4374 1/4374 0 0 0
0 0 0 0 0 0 0 0 43/1458 584/2187 1981/4374 410/2187 227/4374 22/2187 5/4374 0 0
0 0 0 0 0 0 0 0 5/1458 50/729 205/729 505/1458 97/486 56/729 5/243 5/1458 0
0 0 0 0 0 0 0 0 1/4374 26/2187 227/2187 194/729 211/729 142/729 205/2187 70/2187 35/4374
0 0 0 0 0 0 0 0 0 5/4374 55/2187 280/2187 355/1458 373/1458 5/27 230/2187 175/4374
0 0 0 0 0 0 0 0 0 0 5/1458 10/243 205/1458 2/9 343/1458 124/729 133/1458
0 0 0 0 0 0 0 0 0 0 0 35/4374 245/4374 322/2187 434/2187 853/4374 427/4374
0 0 0 0 0 0 0 0 0 0 0 0 35/2187 140/2187 266/2187 244/2187 166/2187
.
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 = 9
8 8
7 7
6 6
5 5
4 4
3 3
2 2
1 1
It is clear that the energy eigenfunction is given by,
[1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1]^T units
The starting point here is AB statistical theory:
It is true that Cairo intelligence techniques = natural intelligence = artificial intelligence in the strict sense = unified field theory.
The above rule is based on the assumption of the existence of an energy density transition matrix (called the B matrix):
U(x,y,z,t+dt)=B. U(x,y,z,t)
It should be noted that to date, we only know two transition matrices:
i) the mathematical statistical Markov matrix;
ii) the B transition matrix and its statistical transition chains.
We all know that the mathematical statistical Markov matrix does not admit boundary conditions or a source/sink term S, while the B transition matrix allows both.
The superiority of the B transition matrix over the Markov matrix is obvious. It is also worth noting that over the past four years, the B transition matrix has been used to efficiently solve:
1) time-dependent PDEs in classical physics, such as heat diffusion, sound intensity by electromagnetic scattering, and reverberation time in audio rooms in the most general case;
2) Time-dependent and time-independent PDEs in quantum physics.
3- Numerical mathematical integration and differentiation in 1D, 2D, and 3D.
4- Rigorous derivation of the static distribution formula, such as the Gaussian (normal) distribution.
Etc.
The only default regarding Cairo's theory of techniques is that it reveals the fatal fallacies of Nil Bohr in all fields of science.
Nil Bohr was almost completely outside the scope of the four-dimensional unitary x-t space of nature and imposed his misleading concept of R^4 space (three-dimensional geometry with real time as the external controller) on the entire scientific community for over a century. This raises the question: is the West losing the battle of classical/quantum physics, and even mathematics?
i- In 1925, the Heisenberg-Born-Jordan matrix introduced the first consistent matrix formulation of quantum mechanics, albeit somewhat strange and incomplete (probably related to Bohr's thinking).
It states that:
Ψ = √h
0 -i√1 0 0
i√1 0 -i√2 0 . .
0 i√2 0 -i√3 . . .
0 0 i√3 0 -i√4 . .
etc.
. . . Q1
It is clear that the Heisenberg matrix Q1 is the energy operator of the quantum system in the sense that:
E= ∭Ψ^2 (x,y,z,t) dx dy dz. where E is the total energy of the quantum mechanical system E (x,y,z,t) = ΨΨ* =Ψ^2.
Equation Q1 suggests that the imaginary Heisenberg matrix describes the square root of the energy density of the mechanical system, not the energy density itself.
We propose the following improvement, both physical and logical.
The theory of this improvement and its practical application to solving quantum mechanical systems are the subject of this question.
We propose two major modifications to improve the above matrix Q1:
i- Transform the matrix into a real matrix by removing the complex unit i and replacing Q with its square Q^2.
ii- Transform the above matrix into a sawtooth pattern, which is the most efficient for a quantum matrix, as shown in Figure 1.
This improvement actually constitutes a revival, or even the rise of matrix mechanics.
The new matrix Q^2 for 1D and 19 free nodes will then be written as:
0 -1^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1^0.5 0 -2^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 2^0.5 0 -3^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 3^0.5 0 -4^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 4^0.5 0 -5^0.5 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 5^0.5 0 -6^0.5 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 6^0.5 0 -7^0.5 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 7^0.5 0 -8^0.5 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 8^0.5 0 -9^0.5 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 9^0.5 0 -8^0.5 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 8^0.5 0 -7^0.5 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 7^0.5 0 -6^0.5 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 6^0.5 0 -5^0.5 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 5^0.5 0 4^0.5 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 4^0.5 0 3^0.5 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 3^0.5 0 2^0.5 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2^0.5 0 1^0.5 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1^0.5 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1^0.5 0
And its square Q^2 will be expressed as follows:
166/2187 244/2187 266/2187 140/2187 35/2187 0 0 0 0 0 0 0 0 0 0 0 0
427/4374 853/4374 434/2187 322/2187 245/4374 35/4374 0 0 0 0 0 0 0 0 0 0 0
133/1458 124/729 343/1458 2/9 205/1458 10/243 5/1458 0 0 0 0 0 0 0 0 0 0
175/4374 230/2187 5/27 373/1458 355/1458 280/2187 55/2187 5/4374 0 0 0 0 0 0 0 0 0
35/4374 70/2187 205/2187 142/729 211/729 194/729 227/2187 26/2187 1/4374 0 0 0 0 0 0 0 0
0 5/1458 5/243 56/729 97/486 505/1458 205/729 50/729 5/1458 0 0 0 0 0 0 0 0
0 0 5/4374 22/2187 227/4374 410/2187 1981/4374 584/2187 43/1458 0 0 0 0 0 0 0 0
0 0 0 1/4374 13/4374 50/2187 292/2187 2849/4374 827/4374 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 827/4374 2849/4374 292/2187 50/2187 13/4374 1/4374 0 0 0
0 0 0 0 0 0 0 0 43/1458 584/2187 1981/4374 410/2187 227/4374 22/2187 5/4374 0 0
0 0 0 0 0 0 0 0 5/1458 50/729 205/729 505/1458 97/486 56/729 5/243 5/1458 0
0 0 0 0 0 0 0 0 1/4374 26/2187 227/2187 194/729 211/729 142/729 205/2187 70/2187 35/4374
0 0 0 0 0 0 0 0 0 5/4374 55/2187 280/2187 355/1458 373/1458 5/27 230/2187 175/4374
0 0 0 0 0 0 0 0 0 0 5/1458 10/243 205/1458 2/9 343/1458 124/729 133/1458
0 0 0 0 0 0 0 0 0 0 0 35/4374 245/4374 322/2187 434/2187 853/4374 427/4374
0 0 0 0 0 0 0 0 0 0 0 0 35/2187 140/2187 266/2187 244/2187 166/2187
The Q^2 above is a kind of miracle:
1- the new matrix Q^2 brings a striking fact: the transition matrix in quantum mechanics should be composed of non-rational square root numbers such as √2 √3 ,√6, etc in addition to multiples of √10, accurate to 8 significant digits.
2-The sawtooth shape vector shown in Fig 1 is the eigenvector of the matrix Q^2 in the sense that,
Q^2 .. [1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1]T
=[1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1]T
This means that the sawtooth-shaped solution in Fig. 1 is exactly the steady-state solution for a 1D source-less QM system like that of a quantum particle in an infinite potential well.
3- If we multiply the quantum matrix Q^2 by the central field potential of the hydrogen atom, i.e.
[1 4 9 16 25 36 49 64 81 64 49 36 25 16 9 4 1] T
in arbitrary dimensionless units, we obtain:
[739/243 1630/243 2821/243 164/9 241/9 112/3 449/9 580/9 81 580/9 449/9 112/3 241/9 164/9 2821/243 1630/243 739/243] T
This corresponds to almost identical energy eigenvalues of hydrogen atom, with a relative error of less than 10%.
This means that the solution for the energy levels in the hydrogen atom does not require the Bohr quantization hypothesis,
mv 2 Pi r = n h.
--------
To be continued.
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