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

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

o o o . . . . 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 temoral 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 quantification) the inversion of the matrix M1 (M1^-1) emerges spontaneously.

Thanks for reading.

To be continued.

Ismail Abbas

If it helps - the discovery of h I like to view in the sense that it is found that the action is essentially countable, i.e. regardless of its value, and in that capacity dimensionless.

In that case, simply put, ∫E∙dt (as well as ∫p∙ds) is equally dimensionless.

In such context, if time is thought be discretized, it would not quite indicate a quantized, but otherwise normal time flow, but rather represent a comprehensive, simultaneous state of the system.

Specifically, if h=S=∫L∙dt , the potential energyV implies a spatial dimension, which in this case is geometrized.

In summary I'm suggesting a geometrized approach (e.g. of the SE), whereby the action is countable, and time indicates a simultaneous state.

Article The Hydrogen Atom as an Integrative Eigenstate of the Bifurc...

Is it true that real-time t-quantization exists and can somehow replace Planck energy quantization?

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 SE exists, we call it SESE for distinction.

Surprisingly, SESE is more revealing and comprehensive than SE itself.

However, solving SE via quantized time intervals is not complicated but rather requires caution.

*I-DERIVATION OF THE STATISTICAL EQUIVALENCE OF THE 1D SCHRODINGER EQUATION

--------------------

The statistical equivalence of the time-dependent 1D Schrödinger equation

\hbar i \frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2m} \nabla ^2 \psi + V(x,y,z) \psi

..............(1)

is a simple extension of the 1D B matrix chains of the Cairo technique with the applied potential V(x) as source/sink term.

it is also simple to show that the ratio V(X)/E where E is the total energy is expressed by the diagonal element RO (RO is the element of [0,1] ref 1,2,3) of the matrix chains B.

The SESE is therefore given by:

U(x,t)=B^0+B+B^2 . . . +B^N. . . . .(2)

Where the 1-D statistical transition matrix B is expressed as follows:

B= RO 1/2-RO 0 0 0 0 0... . . . . ...0 . . . . . . . . .(3)

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

Note that,

Equation 2 for U(x,t) Defines the solution of the transient transfer function D(x,N) where the statistical solution is given by,

.U(x,t)=D(x,N). V(x)

It is worth mentioning that equation 2.3 replaces and completely ignores SE (Eq.1).

II. FEATURES OF THE STATISTICAL SOLUTION

-------------------------------------------------- ---------

Equations 1.2 define five different solution criteria for the statistical equivalence of the Schrödinger equation (equation 2.3) and therefore for SE itself (equation 1):

A: 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 emw energy in a limited medium [1].

B: RO element of the interval ]0.1/2[

Here, the statistical solution is reduced to solving a time-dependent PDE for heat diffusion in a limited medium [2].

C: 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.

D: 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 section 3.

E: RO=1

The elements of matrices B and D diverge infinitely and therefore the solution does not exist. and consequently the transfer matrix D reduces to the unitary matrix I.

III. ILLUSTRATIVE EXAMPLE

-----------------------------------------

Here are the numerical results for the stationary oscillatory matrix (N is large enough) B for 11 nodes and RO = 0.6

2.6767 -0.7136 0.1881 -4.8344E-2 1.189E-2 -2.7363E-3 0 0 0 0 0

-0.7136 2.8647 -0.7619 0.19996 -5.108E-2 1.2466E-2 -2.846E-3 m 0 0 0 0

0.1881 -0.7619 2.8766 -0.7647 0.2005 -5.1189E-2 1.2484E-2 -2.8484E-3 0 0 0

-4.8344E-2 0.2 -0.765 2.8772 -0.7648 0.2006 -5.1192E-2 1.2485E-2 -2.848E-3 0 0

1.189E-2 -5.108E-2 0.2005 -0.7648 2.8772 -0.7648 0.2006 -5.1192E-2 1.2483E-2 -2.8457E-3 0

-2.7363E-3 1.2466E-2 -5.1189E-2 0.2006 -0.7648 2.8772 -0.7648 0.2006 -5.1189E-2 1.2466E-2 -2.7363E-3

0 2.8457E-3 1.2484E-2 -5.1192E-2 0.2006 -0.7648 2.8772 -0.7648 0.2005 -5.108E-2 1.1889E-2

0 0 2.848E-3 1.2485E-2 -5.1192E-2 0.2006 -0.7648 2.8772 -0.7647 0.2 -4.8344E-2

0 0 0 -2.8484E-3 1.2484E-2 -5.1189E-2 0.2005 -0.7647 2.8766 -0.7619 0.1881

0 0 0 0 -2.8457E-3 1.2466E-2 -5.108E-2 0.2 -0.7619 2.8647 -0.7136

0 0 0 0 0 -2.7363E-3 1.1889E-2 -4.8344E-2 0.1881 -0.7136 2.6766

It is obvious that when the above matrix is ​​multiplied by a non-zero potential V(x)  , oscillations appear.

To be continued.

1-A numerical statistical solution to the partial differential equations of Laplace and Poisson.

2-a statistical numerical solution for the time-dependent 3D heat diffusion problem without the need for the PD heat equation or its FDM techniques.

3-Theory and design of audio rooms-A statistical view.

Please take a look at,

Fall and rise of matrix mechanics - IJISRT Review, Researchgate, January 2024.

We hope there will be a more detailed answer there.