This is only a preliminary answer intended to shed some light on the question and the answer sought.

*Rarely mentioned, matrix quantum mechanics is a formulation of quantum mechanical theory created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925 that succeeded in resolving the spectral lines of the hydrogen atom.

Their original theory predates Schrödinger's SE equation of

two years

.

It was the first conceptually self-contained and logically consistent formulation of quantum mechanics.

Their description of quantum leaps supplanted the electronic orbits of Bohr's model.

**To do this, it interprets the physical properties of the particles as matrices evolving over time. It is the equivalent of Schrödinger's wave formulation in quantum mechanics, as evidenced by Dirac's bra-ket notation.

Unlike the wave formulation, it produces operator spectra (mainly energy) by purely algebraic ladder operator methods. Based on these methods, they derived the spectrum of the hydrogen atom in 1926, before the development of wave mechanics.

The transition probabilities were not entirely classical quantities, because the only frequencies that appear in the Fourier series should be those seen in quantum jumps, not the fictitious ones from precise classical orbits analyzed by Fourier. They replaced the classical Fourier series with a matrix of coefficients,

In 1926, John von Neumann became David Hilbert's assistant (Catastrophic Incident in the Way of Matrix Quantum Mechanics) and he would coin the term Hilbert space to describe the algebra and analysis used in the development of quantum mechanics .

***It should be mentioned that the proposed statistical theory of Q-transition statistical chains is fundamentally and physically different from that of Heisenberg, Born and Jordan.

Now, what's new in the proposed statistical theory of Q-matrix transition chains?

To be continued.

The quantum wave function (often represented as Ψ) and the differentiation between possible and impossible wave functions, particularly in connection with the Schrödinger partial differential equation (PDE). Breaking down your question into parts:

1. Statistical Matrix Definition of Quantum Wave Function (Ψ): In quantum mechanics, the wave function Ψ describes the state of a quantum system. It contains information about the probabilities of various outcomes when measurements are made on the system. The square of the absolute value of the wave function, |Ψ|^2, represents the probability density of finding a particle in a particular state.

2. Differentiating Possible and Impossible Wave Functions: In the context of quantum mechanics, certain wave functions may be considered "allowed" because they satisfy certain physical conditions and boundary conditions. For example, wave functions must be single-valued, continuous, and square-integrable, among other requirements. Wave functions that violate these conditions may be considered "forbidden" or "impossible."

3. Connection with Schrödinger Equation: The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wave function of a quantum system evolves over time. It's a partial differential equation that relates the energy of the system to its wave function.

The Schrödinger equation provides a mathematical framework to determine the allowed wave functions for a given quantum system. Solutions to the Schrödinger equation that satisfy the physical conditions and boundary conditions are considered possible wave functions, while those that don't are considered impossible or forbidden.

4. Completeness and Information Content: The question of which approach contains more information about a quantum system—whether it's the statistical matrix approach or the Schrödinger equation approach—is a nuanced one. Both approaches provide different perspectives and tools for understanding quantum systems.

The statistical matrix (or density matrix) approach is particularly useful when dealing with mixed states or systems that are not in pure states. It allows you to describe the probabilities of different outcomes without needing to explicitly solve the Schrödinger equation.

On the other hand, the Schrödinger equation is a fundamental equation that provides a direct link between the energy of a system and its wave function. Solving the Schrödinger equation yields the allowed wave functions of a system, and these wave functions contain information about the energy levels and behavior of the system.

Both the statistical matrix approach and the Schrödinger equation approach are important and complementary tools for understanding quantum systems. The Schrödinger equation gives insight into the allowed wave functions and their energy properties, while the statistical matrix approach provides information about probabilities and states of mixed systems. The choice of approach depends on the specific problem and the information you are seeking to extract from the quantum system.

Answer II-continued.

This answer is meant to be a quick response to the important answer from our Indian friend.

I completely agree with his first three comments but not with the fourth.

First, our proposed statistical matrix approach,

Q=Sqrt(B)

is a new quantum theory different from the Schrödinger PDE (SE) and its derivatives.

You completely ignore SE and his interpretation of Bohr-Copenhagen as if they never existed.

However, there is a lot of confusion here, as explained in the answer:

"Completeness and informative content: The question of which approach contains the most information about a quantum system, whether it is the statistical matrix approach or the Schrödinger equation approach, is a nuanced question. The two approaches offer different perspectives and tools for understanding quantum systems.”

Our main difficulty here is how to explain a new quantum theory in less than a page?

The classical matrix representation of SE in Hilbert space which is mathematical and cannot convey more information than its parent SE is completely different from the approach

W(N)=Q^0+Q+Q^2+Q^3+. . . +Q^N. . . . . (1)

Where the proposed transition matrix Q is defined as follows:

Q=Sqrt(B) . . . . (2)

derives statistics from mother nature and therefore may contain more information than SE itself.

To be continued.

Answer III-continued.

The best way to test the proposed new statistical theory and validate that it conveys more information than SE is to compare its numerical results with those of the SE solution itself:

1- Although SE is not able to produce new mathematical formulas, the proposed statistical theory can validate several [1] such as,

the formula

1/[1-√(X)] can be expressed by the infinite integer power series , namely,

1/[1-√(X)] =X^0+√X+X + X^3/2 +x^2 +x^5/2. . . . +X^(N)/2 . . . . (1)

Where X is an element of the interval ]0,1].

We applied Equation 1 as a numerical validation of the Q transition matrix theory and the results were surprisingly accurate:

X  . . .  Formula 1/[1-SQRT(X)] . . .  Power series Eq 1

0.15  . .1.63211572  . . . . . . . . . . . . . 1.63211584

0.35 . . 2.44862771 . . . . . . . . . . . . .  2.445.0911

0.65 . . 5.16064501 . . . . . . . . . . .  .  .5.09116888

-The slight difference in the last value is due to a truncation error.

To be continued.

[1] - Q/A section of Researchgate, Is the quantum wave function Ψ a scalar, a vector or none of these?

Answer III-continued.

This answer aims to clear up any confusion and further explain the proposed statistical approach:

*1-In the Schrödinger equation SE, the wave function Ψ is complex and the time is imaginary where t (for example, in the heat equation or the emw equation) is practically replaced by i . t.

The solution unfolds when you transform time in the heat equation t to i .t, solve the complex heat equation, and then invert or transform back.

In the statistical approach, the wave function Ψ is complex and time is a dimensionless real integer.

In other words, the time t (in the heat equation or emw equation) is practically replaced by the number of iterations N, which means

that time t is not an external control separable from the geometric space x but that it is woven into a geometric space to form a 4D unit space.

**2- The proposed new technique that replaces the SE PDE is a way to predict reality without going through the SE PDE itself. This same vision was predicted by A Einstein, among other scientists, a century ago,

i-God/nature does not care about our mathematical difficulties. It integrates experimentally.

ii- Partial differential equations entered theoretical physics as servants, but gradually became masters.

***3-The so-called technical statistical method of Cairo, 2020, based on the chains of transition matrices B (B element of R) was able to numerically solve the partial differential equations of thermal conduction/diffusion and those of Poisson, and Laplace . . . etc.

The current Q chains (Q element of C) are a simple extension of the Cairo technique to deal with quantum mechanics and solve the SE-like PDE with prescribed boundary conditions and a number of free nodes.

Numerical analysis shows that an adequate complex transition matrix Q, corresponding to the transition matrix B exists and is equal to the square root of B.

Q=B^1/2

****4-The wave function Ψ and the state vector in the SE solution in Banach space are not the same thing.

The statistical wave function Ψ and its state vector in the complex space of transition chains are the same thing.

*****5-Physical and mathematical properties of the transition matrix Q

i-The transition matrix Q is symmetric Qij=Qji, therefore Q=Q*.

ii-all entries of the matrix Q have the form,

a+/-ia

which means that when multiplied by a complex conjugate number, the minus sign appears automatically.

In order not to worry too much about the details of the theory, let us go directly to the solution of Ψ in the form,

Ψ = W.BC

where W nxn is the transfer matrix and BC nxIT (IT=number of iterations) is the boundary conditions matrix.

note that in the case of a constant of potential applied over time, the BC matrix reduces to the vector nx1.

In the following answer, we present illustrative applications in some quantum mechanical situations

such as

1- Quantum particle in a one-dimensional infinite potential well with n free nodes

2- Quantum particle in a two-dimensional infinite potential well with n^2 equally spaced free nodes.

To be continued.

Answer III-continued

Iteratively solving the time-dependent PDE requires knowledge of the energy density eigenvalues ​​to use as input in the calculation of the wave function Ψ(x,t).

In the description of quantum mechanics via the Shrödinger equation, we have the only a priori data which is the applied potential V(x, t) as a function of space and time and there is a long path to arrive to eigenvalues ​​and eigenvectors.

However, in 1953, Numerov and Wolgang introduced the matrix description to numerically solve the time-independent Schrödinger equation (dΨ /dt]partial=E Ψ ]

Their matrix description is expressed as follows:

-h^2/2m = {Ψ(n-1)-2 Ψ(n)+Ψ(n+1)}/d^2 . . . . . (1)

which leads to the following Matrix-Form expression for the numerical solution of SE,

-h^2/2m d^2= M1.Ψ + M2.V . . . . (2)

where V is the intensity of the applied potential field, M1 is the tridiagonal transfer matrix M1 given by,

M1= -2 1 0 0 . . . 0 0

0 1-2 1 0 . 0 0

0 0 1-2 1 . 0 0

. . . . . .... . . . .

0 0 0 . . 0 -2 1

and M2 is the diagonal transfer matrix,

M2 is given by,

V1 0 0 . . . 0 0 0

0 V2 0 0 . . . 0 0

0 0 V3 0 . . . 0 0

. . . . . . . . . . . .

ooooo. . . . 0 0 VN

We assume that equations 1,2 are inadequate for the following reasons:

1-They fail or are incomplete because they only calculate the wave function in steady state and do not allow its transient temporal evolution to be studied.

2-When the number of nodes N is large, as it should be (e.g. N = 20-50), the tridiagonal matrix M1 is almost impossible to invert and solve the resulting linear algebraic equation N in order to obtain the values numerical values ​​of Ψ (X).

It should be noted that by using the real chains of the matrix B and the complex chains of the matrix Q, (via a temporal discretization which replaces Planck's energy quantization and should differentiate possible wave functions from impossible wave functions ) the inversion of the matrix M1 (M1^-1) emerges spontaneously.

Thanks for reading.

To be continued.