Dear Dr. Ismail Abbas you are right, there is no single one dimension mathematic that it does not recognize temperature and pressure can describe three dimension of nature of any kind.

On the other hand, "If"," assumption", "prediction"..etc.. is not science.

Read some of my papers for clarification

Thank you for sharing.

It wouldn't be a paradox, it would be an error. However this doesn't occur, so it would be useful to actually study quantum mechanics first (and, in fact, classical mechanics) before asking questions that don't make sense. Learning a subject also involves learning which questions make sense. The question asked just doesn't make sense.

The kinetic energy of a non-relativistic particle in classical mechanics is given by the expression E = (m/2)v.v = (m/2)(vx2+vy2+vz2) =

(m/2)vx2+(m/2)vy2+(m/2)vz2 that is, of course, free of any errors or paradoxes.

This expression can, also, be written as E = p2/(2m), upon introducing the conjugate momentum p=mv and can be, similarly, expressed as the sum of three terms, E = px2/(2m) + py2/(2m) + pz2/(2m). It is this expression that can be most easily identified with the corresponding expression in quantum mechanics, where the momentum operator p is defined as -ihbar nabla=ihbar(d/dx,d/dy,d/dz). No paradox or error there, either.

The potential energy, V(x) is a scalar function (which means it doesn't change under rotations) of the coordinates-that become the position operators in quantum mechanics. In neither classical nor in quantum mechanics is it true, in general, that V(x,y,z)=v(x)+u(y)+w(z). But that's not a problem.

Agreed with Stam Nicolis . The Ex, Ey and Ez notation is generally just an intermediate notation in some derivations to make them simply readable.

It has to be noted, that if operators commutate, you can separate the Schrödinger's equation, though. Then the solution wavefunction is a simple product of multiplication and the energies become additive. So, if the potential energy has a mathematical shape in which the spatial variables are truly unconnected, you can solve three separate Schrödinger equations and add the energy later.

This is actually done in normal mode analysis for molecular vibrations: you set up mass-weighed coordinates and diagonalize the associated mass-weighed Hessian matrix so you end up with mathematically fully separable normal coordinates, see e.g. Harris/Bertolucci: Symmetry and Spectroscopy.

This is a brief introductory response aimed initially at shedding light on the question and its answer.

Thank you to our fellow contributors for their useful response but not this negative response coming from our colleague from France.

His random response brings a lot of enthusiasm and little to no understanding of math and physics.

There are two different languages ​​or approaches to dealing with quantum mechanics problems:

i-Schrödinger equation with Bohr/Copenhagen interpretation and its derivatives.

ii- The chains of matrix B which are the statistical equivalence of SE.

One of the important reasons for replacing the Schrödinger equation with chains of statistical transition matrices is that you are moving from a field of SE where many questions remain unanswered to the field of modern statistical physics where almost all the questions have adequate answers.

The Schrödinger equation is a thing of the past while its equivalent B-matrix chains, even in their nascent stage, represent the future.

Concerning the question itself, the B-matrix strings offer a natural solution to the paradox based on the physical description of the source/sink (S) term of SE, namely:

S=Sx+Sy+Sz

And,

The term of the boundary conditions (BC) is,

BC=BCx+BCy+BCz

It is clear that the terms S and BC are the classic terms of classical physics and quantum mechanics but integrated over the imposed time step dt.

To be continued.

If by "B-matrix" you mean the representation in Heisenberg's matrix mechanics, there is literally a publication of Schrödinger titled "About the relationship between the Heisenberg-Born-Jordan quantum mechanics and mine" in which he concluded that the two were rather equivalent. In standard physics courses, that's also taught.

Article Über das Verhältnis der Heisenberg‐Born‐Jordanschen Quantenm...

When it comes to "terms of classical physics", that's called the correspondence proinciple.

Edit: I just googled the explicit term "B-matrix strings". The only results were this question and one thing written by you (and published in a Beall's list predatory journal https://www.researchgate.net/post/IJISRT_International_Journal_of_Innovative_Science_and_Research_Technology_is_good_to_publish_an_article_Journal/1) in which the term "quantization" does not occur. This is not a valid argumentation basis.

E=Ex+Ey+Ez is not a paradox, only a badly written equation.

If it was written Etotal=Ex+Ey+Ez it would be more meaningful.

Ex=E in vectorial direction x

Ey=E in vectorial direction y

Ez=E in vectorial direction z

"In quantum mechanics, we introduce the expression for total energy E=Ex+Ey+Ez, is this a paradox?"

Yes! It is a paradox and is in conflict with objective reality and the quantum phenomena. No matter how our “scientist serfs” (a term borrowed from the Bengali poet Tagore) try to rationalize with their causality-based and deterministic mathematical cobweb-spinning and eclectic flea-cracking! This paradox was realized (even if opportunistically denied) by one great man of post-quantum physics, namely, Albert Einstein; when he said, “Many physicists maintain - and there are weighty arguments in their favour – that in the face of these facts (quantum mechanical), not merely the differential law, but the law of causation itself - hitherto the ultimate basic postulate of all natural science – has collapsed”. A. Einstein, Essays in Science”, p. 38-39 (1934)

The reason for this "collapse" is that causality, formal logic and mathematics can deal only with an ideal and perfect geometrically closed system symbolized by the Pythagorean theorem and the quantitative relation (square of the sides) of a right-angled triangle. All the pre-quantum theories including, Newton's theory of universal gravitation, Maxwell’s electromagnetism, LTs, SR, GR and even post-quantum (official) energy-momentum relation or anything else of official physics; must conform to the essence of the Pythagorean theorem. This essentially follow the dictum “ex nihilo nihil fit” and means the theological and modern official sciences’ notion of only a onetime, closed and finite universe created by God in a “Big Bang” – and so this is end of the story and "science" must conform to this dictum of God!

But unfortunately, Reality, Nature and the Quantum Phenomena do not care much about dismay of one Einstein or the wishful fantasy of our modern Priests of theoretical physics and cosmology! This fact is demonstrated scientifically, mathematically and philosophically in the following RG question and the references cited therein:

"Ex nihilo nihil fit“? Are You Certain Mr. Einstein and Mr. Heisenberg?

Discussion Started April 24, 2019

Is the Uncertainty Principle just a measurement problem or a fundamental aspect of the sub-structure of objective reality? These are both scientific and philosophical questions brought to focus through some recent publications:

Scientific: "Are you certain Mr. Heisenberg"? https://phys.org/news/2012-01-heisenberg-deepen-quantum-uncertainty.html

Article Experimental demonstration of a universally valid error-dist...

https://www.nature.com/articles/nphys2194

Article Real/Virtual Exchange of Quantum Particles as a Basis for th...

Philosophical: The quantum vacuum, the "virtual particles" and their transformation to "real particles" may be understood in terms of the "Being-Nothing-Becoming" of Hegel's ontological triad :

Article The Philosophy of Space-Time: Whence Cometh "Matter" and "Motion"?

https://www.amazon.ca/Philosophy-Space-Time-Whence-Cometh-Matter/dp/984041884X

The conservation laws (of energy/mass) in physics and thermodynamics, including "Ex nihilo nihil fit" of philosophy are assumed a priori truths, but these are only valid at macroscopic scale. Nature at quantum level is fundamentally uncertain – it is not a statistical aberration – but a fact and a phenomena ascertained by Quantum Electro Dynamics (QED). The law of the conservation of mass/energy of thermodynamics and physics breaks down at quantum level and creation “ex nihilo” is practically feasible!

https://www.scientificamerican.com/article/something-from-nothing-vacuum-can-yield-flashes-of-light/

https://www.accessscience.com/content/electron-positron-pair-production-and-annihilation/224710

The quantum vacuum is full of "virtual particle/antiparticle pairs" that continuously pop in and out of existence and their effect on the spectral lines of atoms known as the "Lamb Shift" and the Lande factor of magnetic moment of electrons that relies on the concept of virtual particles can be very accurately measured. This effect can also be measured as the Casimir force with much less efficiency. The permittivity and the permeability of the classical vacuum can be attributed to collective effect of the momentary existence of infinite number of virtual particles of the quantum vacuum

The "virtual particles" can become "real particles" if sufficient energy equivalent to compensate for their mass is available. There is also a finite probability that a "virtual particle" can become "real" (even without sufficient energy for mass equivalence) through a phenomena known as "quantum tunnelling". In fact it is possible that this spontaneous creation (ex nihilo) of fundamental particles from the quantum vacuum (and not through the Big Bang) is how the galaxies (along with all other things) evolve, “come into being and pass out of existence” and maintained throughout this infinite and eternal universe as asserted by dialectics:

Article Ambartsumian, Arp and the Breeding Galaxies*

Article The Cosmic Gamma-Ray Halo - New Imperative for a Dialectical...

It has been made abundantly clear in these publications that the uncertainty principle is not just a simple measurement or statistical problem as is commonly assumed. Uncertainty and the breakdown of the conservation laws at the substructure of quantum level is a fundamental aspect of objective reality. At macroscopic (human) scale the objective reality is just a gross, averaged-out, summed-up and apparently stable superstructure of micro-level quantum uncertainty. Physics needs a revolutionary change of its epistemology - from traditional causality to dialectical materialism; to deal with this new revolutionary aspect of objective reality!

https://www.researchgate.net/post/Ex_nihilo_nihil_fit_Are_You_Certain_Mr_Einstein_and_Mr_Heisenberg2

Ismail Abbas : This question on uncertainty is not only valid in quantum scale, it is also valid in macroscopic scale as reflected by Leibniz's vis viva, against Newton's "First Impulse from God" in astrophysics and also the case of dialectical "chance and necessity" in biological and other macroscopic systems on Earth. Please see;

"KEPLER -NEWTON -LEIBNIZ -HEGEL Portentous and Conflicting Legacies in Theoretical Physics, Cosmology and in Ruling Ideas"

Article KEPLER -NEWTON -LEIBNIZ -HEGEL Portentous and Conflicting Le...

“The Mystery of the Lorentz Transform: A Reconstruction and Its Implications for Einstein's Theories of Relativity and cosmology” : INSPIRE>HEP: https://inspirehep.net/literature/2158754

"DIALECTICS NOT METAPHYSICS OF NATURE: FROM THE QUANTUM TO THE COSMIC": https://www.amazon.ca/DIALECTICS-NOT-METAPHYSICS-NATURE-QUANTUM-ebook/dp/B0BF5W9Q1N

Dear All, following this forum: Please see similar paradoxes in post quantum theories of causality based official theoretical physics and cosmology; being discussed in other RG forums at the following links:

"Were some scientists right in showing that the Lorentz Force brings to a paradox??" https://www.researchgate.net/post/Were_some_scientists_right_in_showing_that_the_Lorentz_Force_brings_to_a_paradox

"What is the actual physical meaning of the Lorentz Transformations?" : https://www.researchgate.net/post/What_is_the_actual_physical_meaning_of_the_Lorentz_Transformations

"Twin effect from relativity of simultaneity alone, provides a totally wrong prediction . Is somebody aware of it"? : https://www.researchgate.net/post/Twin_effect_from_relativity_of_simultaneity_alone_provides_a_totally_wrong_prediction_Is_somebody_aware_of_it

And many more!

Answer II – continued

It's not really a paradox if we assume that we currently treat the boundary conditions BC, the source term S and the thermal conductivity/diffusion k and D as scalar quantities when in fact they are tensors or second rank matrices having spatial components x, y, z.

for example, S=Sx+Sy+Sz and BC=BCx+BCy+BCz.

We recall what is called the paradox of the principal quantum number n,

Ex=Cn(x)

Ey=Cn(y)

Ez=Cn(z)

Where C is a constant,

And,

n^2=n(x)^2+n(y)+n(z)^2 . . . . ?

It should be noted that:

i-B matrix chains are a new theory born in 2020 that has no connection to past theories such as Heisenberg-Born-Jordan matrix quantum mechanics.

ii-B matrix chains and Markov matrix chains are the only statistical chains known today.

iii-In Markov matrix chains, we do not care about the energy density, the boundary conditions, the source term, the average properties of the medium, . . etc. whereas in the case of matrix B strings we do this.

To be continued.

Any causality-based category, philosophy, science etc., can only deal with the finite and a hermetically closed system, because it has to have a beginning in a "First Cause", i.e., the 'effect' of a 'cause' that is unknown or unknowable; hence a mystery. This can only lead to paradoxes, Kantian antinomies etc. With this limited epistemology, theoretical physics is forced to abstract from everyday-life sensuous objects to the realm of ideal thought, such that (Pythagorean) geometry forms the adequate basis for such abstractions. Malinowski-Einstein "spacetime" is such an abstract Pythagorean-geometry based construct. Please see: “The Mystery of the Lorentz Transform: A Reconstruction and Its Implications for Einstein's Theories of Relativity and cosmology” : INSPIRE>HEP: https://inspirehep.net/literature/2158754)

But once the abstraction is made "physicists" turn their abstracted phantom into a sensuous object - the "bending of spacetime"! Frederick Engels put in in a nice way: "It is the old story. First of all, one makes sensuous things into abstractions and then one wants to know them through, the senses, to see time and smell space. The empiricist becomes so steeped in the habit of empirical experience, that he believes that he is still in the field of sensuous experience when he is operating with abstractions".

The infinite is a category, where modern official theoretical physics finds itself miserably out of its depth. The situation is so desperate that the Harvard Physicists Max Tegmark wants to "Retire" the word "Infinite" from the vocabulary of science; thus glorifying self-impotence! The recognition of the quantum phenomena made Albert Einstein to completely deny this aspect of reality; but said the following in despair, “Many physicists maintain - and there are weighty arguments in their favour – that in the face of these facts (quantum mechanical), not merely the differential law, but the law of causation itself - hitherto the ultimate basic postulate of all natural science – has collapsed”. A. Einstein, “Essays in Science”, p. 38-39 (1934)

Hence it is increasingly becoming clear that causality, formal (Aristotelian) logic and ideal mathematics-based theoretical physics (which Hegel called "Metaphysics") is in an impasse and is incapable of dealing with the "Infinite", the quantum phenomena and other new and emerging developments in modern physics. Only a dialectical perspective can help physics to deal with the infinite and other new and emerging problems like the quantum phenomena:

"The Infinite - As a Hegelian Philosophical Category and Its Implication for Modern Theoretical Natural Science": Article The Infinite - As a Hegelian Philosophical Category and Its ...

The total energy paradox E=Ex+Ey+Ez can be resolved physically via the solution of the Schrödinger equation in 2D or 3D.

Another way to describe quantum particle dynamics is to use statistical transition matrices that completely ignore the Schrödinger equation as if it never existed in the same way that one solves the heat diffusion equation without going through thermal PDE itself.

today we only know one physical transition matrix which is the transition matrix B resulting from the so-called Cairo technique.

Step 1

Construct the 2D statistical matrix B corresponding to Figure 1 which represents a quantum particle in a 2D infinite potential well.

📷

Fig 1. A quantum particle in a 2D infinite potential well

The basis for generating an eigen or proper matrix is the 2D matrix B with 9 equidistant free nodes as shown in Figure 2, nodes 1-9.

Note that RO=0 because PE is zero.

It is expressed by the same matrix M1 explained previously in the example of 2D thermal conduction.

step 2

Compose the proper or eigen matrix M2 as given by,

M2=M1+S(x,y)

Where S(x,y ) is a diagonal matrix and S=C1*V(x,y)

The resulting eigenmatrix M2 will be given by,

M2=

1/14 1/4 0 1/4 0 0 0 0 0

1/4 4/14 1/4 0 1/4 0 0 0 0

0 1/4 1/14 0 0 1/4 0 0 0

1/4 0 0 4/14 1/4 0 1/4 0 0

0 1/4 0 1/4 9/14 1/4 0 1/4 0

0 0 1/4 0 1/4 4/14 0 0 1/4

0 0 0 1/4 0 0 1/14 1/4 0

0 0 0 0 1/4 0 1/4 4/14 1/4

0 0 0 0 0 1/4 0 1/4 1/14

Where C1 is substituted for by the factor 1/14.

step 3

The energy eigenvector E(x,y) is equal to the principal diagonal of the matrix A which gives the following eigenvector equation,

2/14 1/4 0 1/4 0 0 0 0 0

1/4 4/14 1/4 0 1/4 0 0 0 0

0 1/4 2/14 0 0 1/4 0 0 0

1/4 0 0 4/14 1/4 0 1/4 0 0

0 1/4 0 1/4 9/14 1/4 0 1/4 0

0 0 1/4 0 1/4 4/14 0 0 1/4

0 0 0 1/4 0 0 2/14 1/4 0

0 0 0 0 1/4 0 1/4 4/14 1/4

0 0 0 0 0 1/4 0 1/4 2/14

*

[2/14  4/14  2/14  4/14  9/14  4/14  2/14  4/14  2/14]T

is equal to,

[8/49 123/392 8/49 123/392 137/196 123/392 8/49 123/392 8/49] T

Showing that the energy eigenvector is=

[2/14 4/14 2/14 4/14 9/14 4/14 2/14 4/14 2/14 ] T

with a dominant eigenvalue almost equal to 1.

The reason why we multiply the nodes 1,3,6 and 9 by the factor 2 is that these nodes are located at the four intersections of the two axes x and y where the rule E=Ex+Ey applies.

This is called the total energy paradox.

The x-oriented eigenvectors and the y-oriented eigenvectors are shown in Figure 2 in black and red lines.