Mother nature has only one face to show, still sought after as a unified physical theory.
Classical diffusion theory describes the diffusion of energy density while quantum mechanical theory describes the diffusion of the square root of energy density.
The question arises: can one of the two direct the other?
These are just a few lines to explain the question and its proper answer.
Thermal diffusion is a physical domain where quantum mechanics QM and classical physics CPh meet.
This is a good example to explain how the laws of physics are similar for the QM approach of wavelengths down to a few nanometers and the CPh approach of diffusing mechanical sound waves in audio rooms of length characteristic up to several tens of meters.
In all cases, the diffusion laws are the same and also have the same dependence on the boundary conditions over time.
This shows that Mother Nature has only one face to show, still sought after as a unified physical theory.
To be continued.
Ref;
I. Abbas, theory and design of audio rooms, a statistical view.
Yes to the extent that neither one of them work. They provide fuzzy probabilities and that is a nice way of saying the wrong answer. It accepted y main stream as good enough. Its not. Our reality doesn't run on fuzzy probabilities. Go tell an electrician to use Schrodinger equation to calculate his next job. He will never do it because it could kill him. He needs exact information and the equations and functions he uses are exact and calculate the same thing as Schrodinger trying to do. But physicist don't die if they get wrong. its another fabled attempt at the truth that failed but yo all think its ok. Its not O.K. everything in mainstream is not O.K. There is no place i can check that contains any truth any solutions how come no one seams to notice you have solved a damn thing in physics nothing.
Here is a brief answer intended to quickly respond to our friend from Argentina,
"No, it's quite different.
The Schrödinger equation SE is of the eigenvalue type.
HΨ = EΨ"
Our explanation is that SE was invented to describe the behavior of a quantum subatomic particle in a (negative) binding potential such as a hydrogen atom, but not to describe diffusion phenomena.
However, recently appeared preliminary attempts to describe the diffusion phenomena via the Shrödinger equation,
1-Mita, Katsunori,
Schrödinger's equation can be thought of as a diffusion equation with a diffusion coefficient β2 = ℏ/2 m. In this article, we explore the implications of this view. By rewriting the wave function in polar form and transforming Schrödinger's equation into two real equations, we show that one reduces to the continuity equation, and the other, to a nonlinear dynamical equation for the probability density. Considering these two equations as if they were the basic equations of quantum mechanics, we apply them to several one-dimensional quantum systems. We show the dispersive properties in the probability densities of the stationary states of a particle in a rigid box and in the harmonic potential; quasi-classical Gaussian probability densities of a free particle; and the coherent states and compressed states of the harmonic oscillator. We also present the soliton as a quantum mechanical representation of a free particle. We discuss the significance of the diffusion coefficient β 2 for each quantum system using a density plot.
*American Journal of Physics, Volume 89, Number 5, p.500-510
May 2021
2-Schroedinger's equation is not quite the diffusion equation because of the complex term. It is, however, a diffusion equation in a complex space.
Physics Stack Exchange, November 4, 2014.
3-Our own statistical approach which we consider to be the best and simplest.
It bypasses SE itself and offers the solution in matrix form like,
Ψ = W. BC . . . . (1)
where Wnxn is the transfer matrix and BC nxIT is the boundary conditions matrix (IT=number of iterations N)
It is clear that equation 1 represents an eigenvalue problem but not necessarily in the case of diffusion phenomena.
Yes there are some similarities if one cares to check carefully.
The intended use is quite different.
Diffusion is to drop ink in a pale of water and watch it spread, or fill a container
only half with a gas and watch the gas fill up the whole space. But these
situations are not quantum.
The diffusion equation is a classical one.
Interesting connection to the problem of random walks.
How do you get a scalar out of a matrix multiplication? Except row column inner product?
Mark
At least credit us with the eigenvalue predictions, if you dont like the fuzzy.
Or do better yourself!
De Broglie's hypothesis of matter waves implied that the dynamic characteristics of a micro particle in motion, can be ascribed to the wave characteristics of the wavelet accompanying the particle. The Schrödinger equation models the matter-wave interactions through wavefunction ψ and effectively serves as the foundation of QM.
However, the Schrödinger equation is founded on a conceptual mistake in the representation of Potential Energy. The Coulomb potential energy of the proton electron pair in Hydrogen atom, which is inversely proportional to their instantaneous separation distance, has not been correctly modeled in the Schrödinger equation. The potential energy term V in the equation has been assumed to be time invariant and not dependent on the instantaneous position coordinates of the electron. Since the position coordinates of the electron have been wrongly omitted in the input to the equation, naturally the exact position of the electron is lost in the final solution. This has created all the weirdness in subsequent interpretations of QM. This was one of the wrong turns of fundamental physics which diverted our focus from physical reality to the mathematical formulations that are supposed to describe physical reality.
I have explained it in detail in my paper titled, "Wrong Potential Energy Term in Schrödinger’s Equation for Hydrogen Atom".
Article Wrong Potential Energy Term in Schrödinger’s Equation for Hy...
"...quantum mechanical theory describes the diffusion of the square root of energy density."
A system subject to a time-dependent external influence does not have an energy density.
Answer III continued
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].
It is true that the Schrödinger equation and the diffusion equation of classical physics are somehow the same thing?
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.
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, Researchgate, IJISRT review.
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, Researchgate, IJISRT review.
3-Theory and design of audio rooms-A statistical view, Researchgate, IJISRT review.
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