This is just an introductory answer aimed at shedding more light on the question and its answer and on the other hand thanking our fellow American contributor for his brilliant, almost exact answer.

It is one of the best concerning Schrödinger PDE

iℏ(dψ/dt)= Ĥψ. . . (1) but not its square,

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

Where U=Ψ^2=Ψ . Ψ*

and S is the source/sink term (extrinsic or intrinsic).

Note that equation 2 is more complete and more eligible to constitute a unified field theory. Because,

Equation 2 is capable of solving classical physics situations such as the heat diffusion equation and quantum mechanical situations such as quantum particles in a central field or an infinite potential well.

Additionally, Equation 2 can find numerical statistical solutions to pure mathematical problems such as numerical differentiation and numerical integration as well as the sum of infinite integer series.

But how and why?

The question really comes down to a question of interpretation.

This is where the dispute lies.

Ψ^2 can be interpreted as the probability density in 4D x-t unit space.

Ψ^2 can be interpreted as the probability density in 4D x-t unit space.

In other words, equation 2 describes the evolution of the probability density which is mother nature itself, whether microscopic or macroscopic.

To be continued. @Preston Guynn

The Schrödinger equation is an approximation [1] useful for masssive particles (or, more correctly, low-momentum ones); I wouldn't want to use it for a unified theory.

See e.g. https://arxiv.org/abs/1309.2484

I find the question and answers very amusing. Why unifying, as the point of departure? They describe different things

What is quite striking is that when the Schrödinger wave equation remains inactive in the face of many physical and mathematical situations, its square PDE steps forward and does the work.

Contrary to what many mathematicians and physicists think, the proposed equation for Ψ^2 is unified field theory and not that of Ψ itself.

Numerical examples are countless but we present the following:

In the 3D + t space of the Schrödinger equation we must assume, without rigorous reasoning, that the separation of variables expresses f(t) as,

The probability density function of an exponential distribution is

𝑓(𝑥;𝜆)=(𝜆[𝑒−𝜆𝑥]) . . . (1) for 𝑥 ≥ 0

,

𝑓(𝑥;𝜆)= 0

for 𝑥< 0.

Here λ > 0 is the parameter of the distribution, often called the rate parameter.

and the,

Cumulative distribution function exponential distribution like,

𝐹(𝑥;𝜆)=(1−[𝑒−𝜆𝑥])

for 𝑥≥ 0 . . . . (2)

,

𝐹(𝑥;𝜆)=0

for 𝑥 < 0.

NOTE THAT expressions above 1,2 already appear [1,2] via B matrix strings in an alternative parameter:

𝑓(t;T)=(𝜆[𝑒−𝑥/T]) . . . (3) for 𝑥 ≥ 0

called the cooling curve.

𝐹(𝑥;T)= T max(1−[𝑒−𝑥/T]) . . .(4),

𝐹(𝑥;T)=0

for 𝑥 < 0.

called heating curve.

α = Log (0.5+0.5 OR). . . . . . . . . .(5)

α =[0,log1/2]

α Max = log1/2 Conductor

α Min = 0 Insulator

1/2+1/2 RO=Exp(α)

RO= 2*Exp(α)-1

RO Max = 2*1/2-1=0

Note that the necessary and sufficient conditions for deriving equations 3,4 are that the transition matrix satisfies:

B ^ N .I is exponential, which has been shown to be true [1].

To be continued.

[1] A rigorous experimental technique for measuring the thermal diffusivity of metals, Researchgate, IJISRT journal,

August 2022

The Schrödinger equation with the Bohr/Copenhagen interpretation is the central philosophy of modern quantum mechanics describing the wave function Ψ.

However, this is only valid for subatomic particles in infinite free space.

It can well describe the quantified energy levels in the hydrogen atom and the infinite potential wells.

Obviously, this is not enough to solve the energy density distribution function for material media such as the heat diffusion equation and sound intensity in audio rooms.

On the other hand, the square of the wave function Ψ^2=Ψ.Ψ*=U (energy density in 4D unit space) has its own statistical transition matrix (B-matrix for classical physics such as heat diffusion, sound intensity, EMW fields etc. and Q-matrix for 1D, 2D, 3D quantum mechanics situations in vacuum).

Additionally, Q matrix chains explain the formation of the Big Bang in a vacuum over millions or billions of years.

The properties of the chains of the matrix B and those of the chains of the matrix Q are remarkable and justify the hypothesis that they constitute the theory of unified fields.

The properties of matrix B valid for material supports are:

1-B^N tends to zero as N tends to infinity.

2-Solution vector U(x,y,z,t)=b .(B+B^2+B^3 + . . . B^N)

Where b is the Dirichlet boundary conditions.

3-B+B^2+B^3 + . . . B^N = 1/(I-B)

For N sufficiently large.

II-Properties of the Q-Matrix valid for the infinite vacuum are:

1-Q=[(1+Constant ).I -B]

I is the unitary matrix.

2-Norm Q^N=Norm Q. . for all N

3-The Dirichlet boundary conditions for an isolated quantum system in infinite free space are Ψ^2(-∞)=Ψ^2(+∞)=0

To be continued.

The starting point for obtaining the modified Schrödinger PDE describing ψ^2(x,y,z,t) cannot be the Schrödinger PDE itself because it is not complete.

The starting point is the following PDE,

dU/dt)partial=D.Nabla^2 U +S(x,y,z,t) . . . (1)

Where U=ψ^2=ψψ*.

Equation 1, just like Schrödinger's classic PDE, applies to infinite free space and must be supplemented by rules of vacuum dynamics, the most important of which is:

S(x,y,z,t) =Constant * V(x,y,z,t) . . . . (2)

In addition to other commonly accepted rules of vacuum dynamics.

In this case, in simple terms, ∫E∙dt can also be dimensionless.

In such a context, if we consider that time is discretized and woven into the 4D unit space, this would quite indicate a quantized, complete and simultaneous quantum state of the system.

Additionally, the principle of least action is inherent to the solution, which is not always the case for the classic Schrödinger equation.

To be continued.

The starting point for obtaining the modified Schrödinger PDE describing ψ^2(x,y,z,t) cannot be the Schrödinger PDE itself because it is not complete.

The starting point is the following PDE,

dU/dt)partial=D.Nabla^2 U +S(x,y,z,t) . . . (1)

Where U=ψ^2=ψψ*.

Equation 1 is proposed as a unified field theory capable of solving classical physics problems such as the heat diffusion equation, sound intensity in audio rooms, Poisson and Laplace PDE as well as situations of quantum mechanics such as subatomic particles in the central field and infinite potential well.

Equation 1, just like Schrödinger's classic PDE, applies to infinite free space and must be supplemented by rules of vacuum dynamics, the most important of which is:

S(x,y,z,t) =Constant * V(x,y,z,t) . . . . (2)

In addition to other commonly accepted rules of vacuum dynamics.

In this case, in simple terms, ∫ E∙dt can also be dimensionless.

In such a context, if we consider that time is discretized and woven into the 4D unit space, this would quite indicate a quantized, complete and simultaneous quantum state of the system.

Additionally, the principle of least action is inherent to the solution, which is not always the case for the classic Schrödinger equation.

To be continued.

It is easy to show that unified field theory belongs to the square of the Schrödinger wave equation rather than to the classical Schrödinger wave equation itself.

You first transform the Schrödinger PDE describing the wave function Ψ into another describing its square Ψ^2=Ψ. Ψ*. Strikingly, the modified Schrödinger PDE describing Ψ^2, when supplemented by the natural laws of vacuum dynamics, is more complete than the classical Schrödinger PDE itself.

The reformulated Schrödinger partial differential equation lives and operates in the 4D unit space and therefore belongs to the chains of the B matrix which is a product of the statistical method called Cairo techniques.

You can take a particular approach to show that unified field theory belongs to the square of the Schrödinger wave equation rather than the classical Schrödinger wave equation itself.

This particular approach focuses on applying B-matrix chain techniques to answer three fundamental questions that continually challenge the classic Schrödinger equation:

A-Can we replace Schrödinger's partial differential equation with a kind of statistical matrix mechanics capable of describing the formation and explosion of the Big Bang?

B-Can we deduce an expression for the reverberation time in audio rooms?

C-Is it possible to calculate a finite numerical integration via matrix mechanics?

Mathematicians and theoretical physicists with the greatest skills in the classical Schrödinger equation cannot answer these particular questions even in their dreams.

Finally, we present a correct and exact answer to the three questions A, B and C [1] proving that the unification of field theory is the square of the Schrödinger wave equation sublimated by the rules of vacuum dynamics .

1-Is unified field theory Schrödinger's wave equation or its square? A unitary spatio-temporal vision, ResearchGate, IJISRT journal, July 2024.