This is a brief answer, essentially aimed at establishing a comparative link between the two theories of quantum mechanics mentioned in this question.

The last century was marked by the advent of the first theory, based on the original Schrödinger equation of 1925, which was doomed to failure and destined to disappear sooner or later.

We recommend our distinguished contributors and readers to join the second theory before it is too late.

The first theory of QM, based on the original Schrödinger equation of 1925-1927, cannot describe energy discontinuities and therefore cannot explore the characteristics of the Big Bang explosion or the formation of black holes, unlike the second quantum theory of 2020.

We also add that one of the most important problems on which the second theory of QM surpasses the first is the ability to describe the vacuum in infinite free space, where QM itself lives and operates.

@This leads to the definition of external dark space and makes it possible to know the temperature of the vacuum and therefore dark energy and dark matter thanks to the statistical artificial intelligence used in the second theory of the SE squared equation.

Now we all know that there are two distinct theories of quantum mechanics: i-The quantum mechanics theory based on the original equation of Schrodinger and supplimented with the principle of superposition with Bohr interpretation in 1927. ii-The quantum mechanics theoy based on the square of original  Schrodinger equation supplimented with the artificial intelligence of the statistical theory of Cairo techniques prosedposed in 2020 We recall once again that this second theory of quantum mechancs  deals with and even improves all quantum properties of the original SE such as entanglement, superposition and quantized energy solutions. However, the main objective of this Q/A is to examine the possibilty of combining the  theory of quantum mechancs with the theory of general relativity (gravitation) to arrive at a better understanding of the darkenergy and dark matter in the outer deep space. Answer of this qustion that prove that   electromagnetic field energy can be transformed into gravitational field energy and vice versa via the second theory of quantum mechanics offers a positive answer.

The question arises [1]:

In this regard, the theory of relativity requires continuity, strict causality, and locality, while quantum theory requires non-continuity, non-causality, and non-locality.

Thus, the fundamental concepts of quantum theory and general relativity are in direct contradiction.

This is why these two theories have never been coherently unified.

We believe that the answer to this brilliant question is inherent and operates smoothly within the matrix mechanics of the numerical statistical theory of Cairo techniques and its resulting transition B-matrix chains as follows:

Recall that the theories of quantum mechanics and general relativity evolve and operate in infinite free space.

The difference is that quantum mechanical energy variations occur at very small increments of space and time, whereas general relativity requires the opposite.

The weaknesses of quantum mechanics and general relativity can be highlighted by studying the squared Schrödinger equation, supplemented by equation AB (proposed by the author):

S(x,y,z,t)=C1 V1 + C2 V2 . .. . (1) Equation AB.

If V1 represents the externally applied voltage and V2 the inherent or spontaneous internal voltage, then Equation 1 solves the problem of causality and locality.

The squared SE equation plus equation 1 can be solved in depth using the CONCRETE artificial intelligence proposed by the author.

In other words, the problem of causality and locality is solved by the second term in Equation 1.

Furthermore, the solution to the Schrödinger equation admits neither singularities nor discontinuities, unlike the solution of matrix mechanics. The existence of singularities is essential to the formation and explosion of the Big Bang, as well as to the formation and maintenance of black holes.

Ultimately, once again,

It is true that Cairo's intelligence techniques = natural intelligence = artificial intelligence in the strict sense = unified field theory.

It should be noted that the numerical statistical theory of Cairo techniques is defined in a 4D unitary x-t space where time is embedded in the 3D geometry and expressed as a dimensionless real integer.

Therefore, there is no need to express time as a complex quantity with two components: complex and imaginary.

To be continued.

1-Pierre Bayle, Professor of Physics, UPS Toulouse 1, France,

Personal communication.

Here are few questions on quantum physics of the Schrödinger equation and classical statistical physics of its square (energy diffusion), to clarify this question and the topic as a whole.

1-Q1-

Briefly explain the Schrödinger wave equation and its square.

A1-

Schrödinger partial differential equation:

i h dΨ/dt)partial=h^2 . Nabla^2 Ψ/2m + V Ψ . . . . (1)

With the Bohr-Copenhagen interpretation introducing entanglement and superposition Ψ.

The Schrödinger partial differential equation is precise but incomplete because it operates in an incomplete D^4 space (3D+t as an external controller). Now imagine solving the Schrödinger partial differential equation for Ψ^2 and not Ψ.

Equation 1 transforms into:

dΨ^2/dt) partial =C1.Nabla^2 Ψ/2m + C2 .V . . . . (2)

With the following statistically proven assumptions,

i-Ψ^2=Ψ . Ψ*

ii-Ψ^2 is exactly equal to the energy density of the quantum particle(s).

iii-Ψ^2 is the probability of finding the quantum particle in the 4D unit volume element x-t "dx dy dz dt"

iv- The actual time t is completely lost and replaced by the dimensionless integer N dt. In this 4D unit x-t space, the dimensionless time N is integrated into the 3D Cartesian space.

N is the number of iterations or repetitions and dt is the time jump.

Equation 2 is derived from and solved by the advanced artificial intelligence of modern transition matrix statistics.

Equation 2 is solved via matrix mechanics and does not require any PDE or FDM techniques to be solved.

Surprisingly, equation 2 is more informative than equation 1.

Q2

Is the quantum wavefunction Ψ a scalar, a vector, or neither?

A2

It is very likely that Ψ is none of these.

The quantum function Ψ^2 is an nxn square matrix (second-order tensor).

The question arises: is the answer to this question "shut up and calculate"?

Q3-

Is the Schrödinger equation an eigenvalue problem?

Is the heat diffusion equation an eigenvalue problem?

A3

The answer is yes in both cases.

Solving the heat diffusion equation using advanced AI for matrix chains B is an eigenvalue problem.

The most important thing is the preliminary selection of the principal diagonal elements RO (entries) of the statistical transition matrix B.

The solution is expressed by the transfer function of the heat diffusion equation D(N):

U(x,y,z,t)=D(N).[b+S] +B^N IC

IC is the vector of initial conditions (U(x,y,z,0))

The eigenvalue of the solution implies:

the exponent of the solution:

k*= log [(1 + RO) / (1-RO)]

Note that [(1 + RO) / (1-RO)] is equal to the eigenvalue 1 for B and [(1 + RO) / (1-RO)]^2 to the eigenvalue 2 for B^2, etc.

with a maximum of log 2 and a minimum of zero. [RO] is the diagonal vector of matrix B.

Vector RO = ( RO11 , RO22, RO33 , . . . ,RO n n)

The classical approach to the Schrödinger equation is an eigenvalue equation:

By definition, an operator acting on a function produces another function. However, a special case arises when the generated function is proportional to the original function.

A^ψ∝ψ . . . (1)

[This is a special case]

This case can be expressed as an equality by introducing a proportionality constant k.

A^ψ = k ψ . . . . . (2) [This solution applies to the special case of PDEs]

Not all functions solve an equation like those in equations 1 and 2.

ψ = (φ1+φ2)/√2

TO BE CONTINUED.

The statistical theory of Cairo techniques allows us to explain this complex situation in more detail:

The time-dependent diffusion equation (like the Laplacian operator equation) represents or models the natural binary collision relaxation process in a closed-surface control volume subject to Dirichlet boundary conditions.

1- In both time-dependent and time-independent processes, the information waves are always contained within the same control volume.

2- This means that the diffusion process is associated with and controlled by the information waves filling the control volume, both in macroscopic classical physics, such as the macroscopic heat diffusion equation, and in microscopic subatomic quantum physics.

Information waves are the driving force: no diffusion without permission.

Note that information waves are transmitted at the speed of light C.

3- This means that the Schrödinger partial differential equation is not a pure wave equation, but a partially diffusion and partially wave equation. 4- When we attempt to extract information from a quantum system through measurement, the information wave diminishes and the system transforms into a pure scattering system, a system of colliding particles.

This is sometimes called quantum wave collapse.

To be continued.