You write the energy operator for H and E remains the eigenvalue. The time dependent part of the wavefunction then leads to time independent probability density.

This is just a brief answer to show the physical and mathematical difference between the two eigenvalue matrix equations [1]:

H^ Ψ = E Ψ

And

( [B] + Constant. V[I] )= λ ( [B] + Constant. V[I] )

1-The first matrix equation describing Ψ is obtained by the variable separation method with time t considered as an external controller.

In the second equation, the method of separating variables is not necessary and time is considered as a dimensionless integer (N) woven into geometric space.

2-The eigenvalue in the first, which is the separation constant, is the total energy E while in the second an eigenvector appears (of main eigenvalue λ =1) for the total energy vector.

The physical meaning of λ = 1 is that the evolution of the solution over time leads directly to the final time-independent stationary solution on the shortest path.

This is the principle of least action.

3-The solution in the first matrix equation must be completed by the assumption of time dependence in the form Exp (-Alpha.t) while in the second the time dependence (N) follows automatically in the form Exp (-C t .2 Pi A/V).

where A and V are the surface area and volume of the system, respectively.

4-The normalization condition in the first matrix must be assumed while in the second it is inherent to the solution of Ψ^2 where the area under the curve is equal to 1.

[1] FALL and RISE by Matrix Mechanics, IJISRT journal, Researchgate, January 2024.

Using techniques from Cairo numerical statistical theory in the solution of Schrödinger's PDE (both time-dependent and time-independent) is not complicated but rather time-consuming and requires rigorous commitment to the rules of statistical mechanical matrices and lots of precise effort. follow them: Meaning is following the right path of the principle of least action.

Here we present an example of a SE statistical solution for a quantum particle in a 3D potential box.

The considered geometry of the 3D potential box is a cube discretized into 27 equidistant free nodes isolated from the external environment as shown in Figure 1.

📷

Fig.1A cube of 27 equally spaced free nodes containing a quantum particle.

Following the rules explained in reference 1, the 27x27 quantum matrix Q is expressed as follows:

????

????

MATRIX Entries.DD.. 0.187500000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.250000000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.187500000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.250000000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.375000000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.250000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.187500000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.250000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.187500000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.250000000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.375000000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.250000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.375000000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.500000000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.375000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.250000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.375000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.250000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.187500000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.250000000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.187500000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.250000000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.375000000 0.166666672 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.250000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.187500000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.250000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.166666672 0.00000000 0.166666672 0.187500000

Note that i-the determinant of the matrix Q is equal to zero and that ii-the condition,

([B] + Constant. V[I])= λ ([B] + Constant. V[I]). . . . . (1)

with the principal eigenvalue λ = 1 which is equivalent to the principle of least action is satisfied.

The above conditions i,ii are essential in the Schrödinger statistical solution PDE 1D, 2D, 3D via transition matrix chains B.

It is clear that the vector V replaces Ψ^2.

Furthermore, it can be shown that equation 1 predicts a time dependence (N) of the solution as follows:

Ψ^2(x,y,z,t)=Ψ^2(x,y,z,0) . [1.-Exp N2Pi A/V)]

which is the subject of the following answer.

To be continued.

1-A 3D numerical statistical solution for the time-independent Schrödinger equation

Researchgate, IJISRT Review, December 2023.

For interested contributors:

Please take a look at,

Cairo Technics Solution of Schrödinger's Partial Differential Equation - Time Dependence, Researchgate, IJISRT Journal, March 24.

We assume there is a detailed answer.