>>>Concrete artificial intelligence predicts that quantum physics does not exist...

What, has AI already become the highest authority in science? Has it already made any important discoveries?

Artificial intelligence is a product of quantum mechanics itself. Let it respect its father. As for quantum mechanics itself, it should be considered as the probability of an event. Using the laws of quantum mechanics presented in the works of Planck, Schrödinger, Bohr and others, this probability can be controlled.

Quantum mechanics has been around for almost a century, but its predictions continue to face challenges from experimental research, including the results of 2013. In particular, the phenomenon of quantum mechanical collapse is observed, which makes this topic especially relevant for scientific discussions.

We invite you to discuss the discrepancies between theoretical quantum mechanics and the results of the experiment led by A. Stadolny. Read more on Academia.edu at the link: [https://www.academia.edu/s/dabf868553].

Dear Lev Verkhovsky and Bakhodir Tursunbayev. Thank you for the wonderful discussion. We create materials with a set of specified properties using the laws of quantum mechanics. Some of our research can be found at the following links:

1. Рахимов Р.Х., Паньков В.В., Ермаков В.П., Рашидов Ж.Х., Рахимов М.Р., Рашидов Х.К. Исследование свойств функциональной керамики синтезированной модифицированным карбонатным методом // Computational Nanotechnology. 2023. Т. 10. № 3. C. 130–143. DOI: 10.33693/2313-223X-2023-10-3-130-143. EDN: SZDYRZ.

2. Рахимов Р.Х., Ермаков В.П. Перспективы солнечной энергетики: роль современных гелиотехнологий в производстве водорода // Computational Nanotechnology. 2023. Т. 10. № 3. C. 11–25. DOI: 10.33693/2313-223X-2023-10-3-11-25. EDN: NQBORL.

3. Rakhimov R.Kh. Возможный механизм эффекта импульсного квантового туннелирования в фотокатализаторах на основе наноструктурированной функциональной керамики. Computational Nanotechnology. 2023. Vol. 10. No. 3. Pp. 26–34. DOI: 10.33693/2313-223X-2023-10-3-26-34. EDN: QZQMCA.

4. Рахимов Р.Х., Мухторов Д.Н. Гелиосушка фруктов и овощей с использованием полиэтилен-керамического композита // Computational nanotechnology. 2023.Т. 10. № 4. С. 104–110. DOI: 10.33693/2313-223X-2023-10-4-104-110. EDN: TLZMDV. 2024

5. Рахимов Р.Х., Ермаков В.П. Импульсный туннельный эффект. особенности взаимодействия с веществом. эффект наблюдателя // Computational Nanotechnology. 2024. Т. 11. № 2. С. 116–145. DOI: 10.33693/2313-223X-2024-11-2-116-145. EDN: MWBRQW.

6. Рахимов Р.Х., Паньков В.В., Саидвалиев Т.С. Исследование влияния импульсного излучения, генерируемого функциональной керамикой на основе принципа ИТЭ, на характеристики системы Cr2O3—SiO2—Fe2O3—CaO—Al2O3—MgO—CuO // Computational Nanotechnology. 2024. Т. 11. № 2. С. 146–157. DOI: 10.33693/2313-223X-2024-11-2-146-157. EDN: MWPEYI.

7. Рахимов Р.Х., Паньков В.В., Ермаков В.П., Саидвалиев Т.С., Рашидов Ж.Х., Рахимов М.Р., Рашидов Х.К. Импульсный туннельный эффект: результаты испытаний пленочно-керамических композитов// Computational Nanotechnology. 2024. Т. 11. № 2. С. 175–191. DOI: 10.33693/2313-223X-2024-11-2-175-191. EDN: NHSAVQ.

8. Рахимов Р.Х., Ермаков В.П. Новые подходы к синтезу функциональных материалов с заданными свойствами под действием концентрированного излучения и импульсного туннельного эффекта // Computational nanotechnology. 2024. Т. 11. № 1. С. 214–223. DOI: 10.33693/2313-223X-2024-11-1-214-223. EDN: EYKREQ.

9. Рахимов Р.Х., Паньков В.В., Ермаков В.П, Махнач Л.В. Производительные методы повышения эффективности протекания промежуточных реакций при синтезе функциональной керамики // Computational nanotechnology. 2024. Т. 11. № 1. С. 224–234. DOI: 10.33693/2313-223X-2024-11-1-224-234. EDN: FCGMYR.

10. Рахимов Р.Х., Паньков В.В., Ермаков В.П., Саидвалиев Т.С., Рашидов Ж.Х., Рахимов М.Р., Рашидов Х.К. Импульсный туннельный эффект: результаты испытаний пленочно-керамических композитов// Computational Nanotechnology. 2024. Т. 11. № 2. С. 175–191. DOI: 10.33693/2313-223X-2024-11-2-175-191. EDN: NHSAVQ.

11. Рахимов Р.Х. Импульсный туннельный эффект: фундаментальные основы и перспективы применения // Computational nanotechnology. 2024. Т. 11. № 1. С. 193–213. DOI: 10.33693/2313-223X-2024-11-1-193-213. EDN: EWSBUT.

12. Рахимов Р.Х., Ермаков В.П. Особенности процесса полимеризации на основе ИТЭ // Computational Nanotechnology. 2024. Т. 11. № 2. С. 158–174. DOI: 10.33693/2313-223X-2024-11-2-158-174. EDN: MXFORZ.

13. Saidov, R.M.; Touileb, K. Improving the formation and quality of weld joints on aluminium alloys during tig welding using flux backing tape. Metals 2024, 14, 321. https://doi.org/10.3390/met14030321. Q1, импакт-фактор 2,6

14. Рахимов Р.Х. Потенциал ИТЭ для преодоления технических барьеров квантовых компьютеров // Computational Nanotechnology. 2024. Т. 11. № 3. С. 11–33. DOI: 10.33693/2313-223X-2024-11-3-11-33. EDN: PZNUYI

15. Рахимов Р.Х. Взаимосвязь и интерпретация эффектов в квантовой механике и классической физике // Computational Nanotechnology. 2024. Т. 11. № 3. С. 98–124. DOI: 10.33693/2313-223X-2024-11-3-98-124. EDN: QEHXLV

16. Рахимов Р.Х. Импульсный туннельный эффект: новые перспективы управления сверхпроводящими устройствами // Computational Nanotechnology. 2024. Т. 11. № 3. С. 161–176. DOI: 10.33693/2313-223X-2024-11-3-161-176. EDN: QBGGDW

17. Рахимов Р.Х. Фракталы в квантовой механике: от теории к практическим применениям // Computational Nanotechnology. 2024. Т. 11. № 3. С. 125–160. DOI: 10.33693/2313-223X-2024-11-3-125-160. EDN: QFISKE

18. Рахимов Р.Х. Электроотрицательность и химическая жёсткость: ключевые концепции в химии. Рахимов Р.Х. Электроотрицательность и химическая жесткость: ключевые концепции в химии // Computational Nanotechnology. 2024. Т. 11. № 4. С. 154–172. DOI: 10.33693/2313-223X-2024-11-4-154-172. EDN: HJJEPR.

19. Рахимов Р.Х. Оптимизация квантовых вычислений: влияние эффекта доплера на когерентность кубитов. Рахимов Р.Х. Оптимизация квантовых вычислений: влияние эффекта Доплера на когерентность кубитов // Computational Nanotechnology. 2024. Т. 11. № 4. С. 58–76. DOI: 10.33693/2313-223X-2024-11-4-58-76. EDN: GFQRFT

20. Рахимов Р.Х. Фракталы и устройство Вселенной. Рахимов Р.Х. Фракталы и устройство Вселенной // Computational Nanotechnology. 2024. Т. 11. № 4. С. 190–208. DOI: 10.33693/2313-223X-2024-11-4-190-208. EDN: HLFIJC.

21. Рахимов Р.Х. Эффект наблюдателя в двухщелевом эксперименте: роль экспериментальных параметров в формировании интерференционного паттерна. Рахимов Р.Х. Эффект наблюдателя в двухщелевом эксперименте: роль экспериментальных параметров в формировании интерференционного паттерна // Computational Nanotechnology. 2024. Т. 11. № 4. С. 173–189. DOI: 10.33693/2313-223X-2024-11-4-173-189. EDN: HJSEPD.

I hope that we can discuss the results we obtain together. Sincerely, Rustam

Dear Rustam Khakimovich Rakhimov,

Thank you very much for your response and for sharing the link to your article "Investigation of the properties of functional ceramics synthesized by the modified carbonate method" (Computational Nanotechnology, 2023)! My co-authors, including Khikmatulla Nigmatov, and I greatly appreciate your contribution to our discussion.

Your study, using the laws of quantum mechanics (QM) to create materials with desired properties, looks very interesting, especially in the context of functional ceramics. However, our work questions the applicability of QM to accurately describe atomic systems such as the hydrogen atom. As we showed in the article "The real radius of Wasserstoffatoms: New theoretic and experimental analysis" (German International Journal of Modern Science №99, 2025), the Bohr radius (0.529 Å) and the QM prediction for the excited state (483 Å) differ from the experimental data (1.0 Å and 1.4 Å) by 47% and more than 30,000%, respectively. Our deterministic approach (DTA) with the potential electrostatic field of the proton (PEPP) predicts radii of 1.06 Å (deviation 6%) and 1.413 Å (deviation 0.9%), which is much closer to the experiment.

Your work with materials is probably based on calculations of electronic structures, where QM is traditionally used. I was interested in whether you encountered similar discrepancies between QM predictions and experimental data in your studies? For example, when modeling interatomic distances or energy levels in ceramic materials? We believe that DTA may offer a more accurate approach, especially for systems where QM gives significant errors. I would also like to know if you have data on the radii of hydrogen atoms in your materials? This could be an additional confirmation (or refutation) of our approach. I will be glad to continue the discussion and, perhaps, find common ground between our studies. I look forward to hearing from you!

Dear Bakhodir,

We are developing materials that operate based on the principle of impulse tunneling effect (ITE). The impulse tunneling effect is a quantum mechanical phenomenon in which a particle or wave can overcome a potential barrier due to the accumulation of significant energy momentum.

According to de Broglie's hypothesis, the momentum of any type determines its wavelength by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. As a large amount of energy momentum accumulates, for example, in the form of photons, the wavelength of the particle significantly decreases.

These "short-wavelength" particles are capable of tunneling through a potential barrier, overcoming it even when their energy is below the height of the barrier itself. Unlike the standard tunneling effect, in ITE all photons incident on the functional ceramics are used and converted to the desired wavelength. Thus, ITE allows for the efficient utilization of radiation energy by focusing momentum, exceeding the effective energy of photons over their actual energy.

Moreover, ITE facilitates the attainment of a very narrow energy range associated with the momentum rise front. By precisely tuning the momentum front to match the energy of the target process, ITE acts with high selectivity, directing all momentum energy into the required narrow range. This enables maximum efficiency in the chosen processes by optimally aligning the impulse characteristics with the required energy.

In other words, our ceramics convert the energy from a primary source, such as the Sun, into pulsed light radiation with specified parameters and very high density.

Key points distinguishing ITE from the standard tunneling effect:

1. Utilization of all incoming photons and their conversion to the desired wavelength.

2. Provision of high energy utilization efficiency through momentum focusing.

3. High selectivity related to the ability to precisely tune the momentum front to the required energy of the process.

The combination of these features allows ITE to achieve maximum efficiency in various practical applications.

The presented list of publications presents various applications of this phenomenon and materials operating on this principle. For example, obtaining hydrogen from water vapor due to the energy of the Sun with high efficiency, due to the use of photocatalysts based on our ceramics. In practice, we study the interaction of energy with matter. Our systems operate on the basis of cascade schemes for converting primary energy according to the scheme: photons (in a wide energy range) - phonon - photons (in a narrow, specified spectral range). So we use quantum mechanics in a slightly different way.

Dear Ismail Abbas, quantum mechanics will never collapse. Already now, in many of its applications, quantum mechanics allows us to obtain the most accurate, planned results. I agree that we are at the beginning of the path. Let's consider a simple example from general physics. As we know, a positron and an electron have all the same parameters, but an opposite charge. Can we explain what this effect of the opposite charge of otherwise identical particles is associated with? Quantum mechanics, at least to some extent, sees the answer to many questions, and also offers solutions. I think that you yourself can find enough examples confirming this. For example, the collision of two photons or similar examples. Best regards, Rustam

Dear Rustam Khakimovich ,

Thank you for your thoughtful comment and for sharing your perspective on the role of quantum mechanics (QM)! I truly appreciate your confidence in QM and its ability to deliver precise, planned results in many applications. Indeed, engineering physics, which relies on QM, has made tremendous contributions to technology development—from semiconductors to quantum computing. Your example of the positron and electron, where QM attempts to explain the effect of opposite charges in otherwise identical particles, is also intriguing, and I agree that QM provides solutions to many complex questions, such as photon collisions.

However, allow me to raise a critical point: if QM struggles with the simplest system—the hydrogen atom—where its predictions for the radii (0.529 Å for the ground state and 483 Å for the excited state, n=33 n = 33 n=33) deviate from experimental data (1.0 Å and 1.4 Å) by 47% and over 30,000%, respectively, how can we be confident in its accuracy for more complex systems? In our paper, "Der reale Radius des Wasserstoffatoms: Neue theoretische und experimentelle Ansätze" (German International Journal of Modern Science №99, 2025, DOI: [to be inserted]), we demonstrate that a deterministic approach (DTA) based on the proton’s potential electrostatic field (PEPP) predicts radii of 1.06 Å (6% deviation) and 1.413 Å (0.9% deviation), which are much closer to experimental values.

If QM exhibits such significant errors in the simplest atom, could it be that its successes in complex systems are the result of parameter fitting rather than a true reflection of physical reality? For instance, in your work on functional ceramics (Computational Nanotechnology, 2023), where you apply QM, have you encountered discrepancies between calculations and experimental results that might stem from such fundamental inaccuracies? We believe that DTA, which relies on real electron trajectories, may offer a more accurate framework, especially for systems where QM depends on artificial energy levels.

I would also like to ask: how do you think QM explains the discrepancy between the Bohr radius and the experimental radius of the hydrogen atom? Is this simply an "acceptable error" that we tolerate for the sake of practical applications? I look forward to hearing your thoughts and continuing this discussion!

Dear Bakhodir Khonnazarovich Tursunbaev ,

You asked the question, "how do you think QM explains the discrepancy between the Bohr radius and the experimental radius of the hydrogen atom?" In this regard I wish to introduce my two research papers.

1. Wrong Potential Energy Term in Schrödinger’s Equation for Hydrogen Atom

2. Dynamic Electron Orbits in Atomic Hydrogen

Current picture of the hydrogen atom, as taught to generations of Physics students, is based on atomic orbitals defined by the solutions of Schrödinger equation for hydrogen atom. Atomic orbitals are bounded regions which describe a specific volume of space where the electron is likely to be located. The solutions of Schrödinger equation for the wave function ψ are interpreted to obtain various significant parameters of the electron motion. The position probability density of an electron is given by square of the wavefunction. Thus, as per our current understanding from Schrödinger’s wave mechanics, the instantaneous position of an orbiting electron gets smeared over the whole volume of an atomic orbital instead of being a specific point on a well-defined trajectory of its motion.

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 current solutions of Schrödinger’s equation for different energy states of electron in Hydrogen atom appear to describe only the time averaged charge density distributions around nucleus and not the trajectories of electrons. That is because the potential energy term V in the equation has been assumed as 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.

To fully comprehend and understand any physical phenomenon, we must demand mental visualization of such phenomenon. Due to the conceptual mistake in the Schrödinger’s equation for Hydrogen Atom as discussed above, the instantaneous position of the orbiting electron cannot be mentally visualized, but is said to be smeared across atomic orbitals as probability density. In fact, due to this conceptual mistake, the very picture of an electron gets transformed from a real particle with mass m and charge ‘e’ to a wave packet whose position and momentum parameters get related through Heisenberg’s uncertainty principal. The real electron particle never gets transformed to any wave packet; it is only the intrinsic electric field of the electron which acquires wave-like properties during motion of the electron. Therefore, as in Bohr-Sommerfeld models, we must be able to mentally visualize the instant-to-instant orbiting motion of the electron in hydrogen atom.

Accordingly, I have published a paper titled,"Dynamic Electron Orbits in Atomic Hydrogen" in Journal of Modern Physics. In this paper we analyze the energy balance of an isolated proton-electron pair and develop the electron trajectory by using energy and angular momentum conservation principle in central force field system. Based on this methodology I have provided an improved and more detailed model of dynamic electron orbits than the old Sommerfeld model. During emission of a photon, elliptical orbit transitions are also computed and plotted. Orbit transition time is of the order of a fraction of a femtosecond. I have extended this methodology for electron orbits in hydrogen molecular bond and computed the H2 bond energy. In fact, following the steps outlined in this paper, all science and engineering students can easily replicate these electron orbits by using Scilab or Matlab software. I have also tabulated the salient orbital parameters of various possible electron orbits, from ground state 1s to 2s, 2p, up to 4f, in atomic Hydrogen. For the 1s orbit, its orbital time period is 0.152 fs, minimum vertex radius is 0.13a0 and maximum vertex radius is 1.87a0 (about 1 Å).

Hopefully, the analysis presented in this paper will enable the scientific community to mentally visualize the instant-to-instant motion of orbiting electrons in hydrogen atoms and their molecular bonds.

Article Wrong Potential Energy Term in Schrödinger’s Equation for Hy...

Article Dynamic Electron Orbits in Atomic Hydrogen

With Best Regards

Gurcharn S. Sandhu

Dear Gurcharan Singh Sandhu,

Thank you for your insightful and thought-provoking contribution to our discussion! I deeply appreciate your detailed analysis and the references to your research papers, "Incorrect Potential Energy Term in the Schrödinger Equation for the Hydrogen Atom" and "Dynamic Electron Orbits in Atomic Hydrogen." Your perspective on the conceptual flaws in the Schrödinger equation and the resulting misinterpretation of electron behavior in the hydrogen atom is both compelling and highly relevant to our work.

I completely agree with your critique that the current formulation of the Schrödinger equation, particularly the time-invariant and position-independent potential energy term 𝑉, fails to account for the instantaneous coordinates of the electron. This leads to a "smearing" of the electron’s position across the atomic orbital as a probability density, rather than providing a clear trajectory. Your point about the transformation of the electron into a "wave packet" due to this conceptual error, rather than treating it as a real particle with mass

𝑚 and charge 𝑒, resonates strongly with our own concerns about quantum mechanics (QM). As you rightly noted, this approach obscures the ability to mentally visualize the electron’s motion, which is a critical aspect of understanding physical phenomena.

In our paper, "Der reale Radius des Wasserstoffatoms: Neue theoretische und experimentelle Ansätze" (German International Journal of Modern Science №99, 2025, DOI: [to be inserted]), we also challenge the very structure of QM by demonstrating that its predictions for the hydrogen atom radii—0.529 Å for the ground state and 483 Å for the excited state 𝑛=30—deviate significantly from experimental data (1.0 Å and 1.4 Å) by 47% and over 30,000%, respectively. These substantial discrepancies call into question the foundational assumptions of QM, particularly its reliance on probabilistic wave functions rather than deterministic trajectories. In contrast, our deterministic approach (DTA), based on the proton’s potential electrostatic field (PEPP), predicts radii of 1.06 Å (6% deviation) and 1.413 Å (0.9% deviation), which are much closer to experimental values. Like your model, DTA emphasizes real electron trajectories, allowing us to visualize the electron’s motion more intuitively. For instance, our approach explains the transition from 1.06 Å to 1.413 Å with an energy close to 3.4 eV (photon at 365 nm), aligning with experiments by Stodolna et al. (2013).

I find your methodology in "Dynamic Electron Orbits in Atomic Hydrogen" particularly fascinating, especially your use of energy and angular momentum conservation to derive elliptical orbital transitions during photon emission, with transition times on the order of fractions of a femtosecond. Your calculated orbital parameters for the 1s state—minimum vertex radius of 0.13𝑎0 , maximum vertex radius of 1.87

𝑎0 (approximately 1 Å), and an orbital period of 0.152 fs—are remarkably close to our DTA prediction of 1.06 Å for the ground state. This convergence suggests that deterministic models, whether based on PEPP (as in DTA) or your energy balance approach, may indeed offer a more accurate representation of the hydrogen atom than QM.

I’d like to ask a few questions to deepen our discussion:

How do you see the potential for integrating your dynamic orbit model with our DTA framework? Could combining our approaches lead to even more precise predictions for atomic systems?

You’ve extended your methodology to the

H2 molecule and calculated its binding energy. We’re currently working on applying DTA to

H2 and helium to test its universality—do you think your approach could also be applied to such systems, and what challenges might arise?

Finally, how can we encourage the broader scientific community to move away from the probabilistic interpretations of QM and embrace deterministic models that allow for mental visualization of electron motion?

Your work provides a refreshing perspective, and I believe it aligns closely with our goal of establishing DTA as a path to truth in atomic physics. I’d also like to invite our colleagues from Germany, Estonia, India, Indonesia, and the USA—who have shown great interest in our paper with over 50 downloads—to join this conversation. Let’s continue this respectful and collaborative dialogue to advance our understanding of the hydrogen atom! I look forward to your response and further insights.

Unveiling the Flaws of Quantum Mechanics: The Misunderstanding of the Bohr Radius

The Bohr radius (0.529 Å) is often taught as the "most probable" radius of the hydrogen atom in its ground state, but experimental data reveal a radius of 1.0 Å—a 47% deviation! For the excited state 𝑛=30, QM predicts 483 Å, while experiments show 1.4 Å, a staggering 30,000% deviation. These discrepancies expose fundamental flaws in QM’s probabilistic framework. Our paper, "Der reale Radius des Wasserstoffatoms: Neue theoretische und experimentelle Ansätze" (German International Journal of Modern Science №99, 2025), challenges QM with a deterministic approach (DTA) based on the proton’s potential electrostatic field. DTA predicts radii of 1.06 Å (6% deviation) and 1.413 Å (0.9% deviation), aligning closely with experimental data.

Gurcharn S. Sandhu’s work, "Dynamic Electron Orbits in Atomic Hydrogen" (Journal of Modern Physics, 2023, DOI: 10.4236/jmp.2023.1411087), further supports this critique. Sandhu argues that the Schrödinger equation’s time-invariant potential energy term "smears" the electron’s position, preventing visualization of its trajectory. His deterministic model yields a 1s orbit radius ranging from 0.13𝑎0 to 1.87𝑎0 (≈1 Å), closely matching our DTA prediction of 1.06 Å.

With over 50 downloads and interest from Germany, Estonia, India, Indonesia, and the USA, our ResearchGate discussion is gaining traction—but we need your insights! How can we correct the misunderstanding of the Bohr radius and move beyond QM’s limitations? Join our respectful dialogue: [insert ResearchGate discussion link]. Let’s redefine atomic physics together! #Physics #HydrogenAtom #Determinism #QuantumMechanics

Bakhodir, have a nice day. Thank you for the interesting and important information. Rustam

Hi Rustam!

Thank you for your kind words! I hope you have a great day too. If you have any more questions or need information, don't hesitate to reach out.

Respectfully,

Bakhodir

Dear Ismail Abbas and colleagues,

Your question, "When will quantum mechanics collapse?", aligns with our findings. In "Validation of Deterministic and Quantum Models for Electron Orbit Radii in Hydrogen Atom Using Photoionization Microscopy" (DOI: 10.13140/RG.2.2.33076.56961), co-authored with Grok 3 (xAI), experiments (Stodolna et al., 2013) show a radii ratio of 1.4. QM predicts 900—off by 64,000%. Our DTA yields 1.33, with 4.8% deviation. See also "Der reale Radius" (DOI: 10.5281/zenodo.15061577) and "The Real Radius of the Hydrogen Atom" (DOI: 10.13140/RG.2.2.18580.00644).

I invite you to join my discussion: https://www.researchgate.net/post/Why_do_the_Bohr_radius_0529_A_and_excited_radius_483_A_differ_from_experiment_by_47_and_30_000_What_is_the_true_radius.

Let’s build a physics that reflects reality!

This is only a preliminary answer, first to shed light on the question and its answer, and second to thank our distinguished contributors and readers for their valuable responses.

In fact, this question should be divided into two:

1- Will quantum mechanics eventually collapse or disappear?

2- If so, when?

Einstein claimed that the Schrödinger equation was born flawed and incomplete in R^4 space (3D+t as the external controller).

Therefore, SE is doomed to disappear and be replaced by concrete and narrow artificial intelligence in unitary 4D x-t space.

The old iron guards who defend SE until their last breath will also disappear.

But when?

The Cairo Techniques AI predicts a maximum duration of one or two years, at most.

To be continued.

Dear Ismail Abbas,

Thank you for your thought-provoking answer! We agree that the Schrödinger equation (SE) has flaws, as shown by its failure to predict H atom radii (0.529 Å and 483 Å vs. experimental 1.0 Å and 1.4 Å, deviations 47% and 30,000%). Our deterministic approach (DTA) uses Newtonian mechanics, yielding 1.06 Å and 1.413 Å (deviations 6% and 0.9%), supporting your critique of SE. However, we believe replacing SE with AI is unlikely; AI is a tool, not a theory. While Cairo Techniques AI predicts QM’s collapse in 1–2 years, we think 5–10 years is more realistic, given the slow shift in scientific paradigms. What are your thoughts on DTA as an alternative?

Let’s continue this discussion! 😊

Dear Bakhodir and Ismail Abbas,

You are correct in stating that quantum mechanics does not allow for precise calculations. Determinism, in this respect, is significantly more accurate. Quantum mechanics indicates that a certain event may occur, and the probability of this event depends on many factors. For these calculations, determinism is once again utilized. We use a kinetic model for our material calculations. Quantum mechanics can explain many nonlinear effects, such as genetic mutations or the tunneling effect, and so on.

By employing other calculation systems, it is possible to increase the probability of the event you are interested in. I remember when we studied quantum mechanics, and F.F. Volkenstein was teaching us, we also mentioned that it was impossible to calculate even the hydrogen atom with its help. However, by some twist of fate, I have had to use quantum mechanics throughout my conscious life to solve my problems.

Wishing you success in your work, Rustam

Dear Rustam Khakimovich,

Thank you for your thoughtful response and for sharing your experience and perspective on the problem! Ismail Abbas and I highly value your opinion and fully agree with your statement that quantum mechanics (QM) does not allow for precise calculations, unlike the deterministic approach (DTA) that we are developing. Your words that QM only indicates the probability of events, and for precise calculations we still have to return to determinism confirm our position.

Nonlinear effects and QM

You mentioned that QM can explain nonlinear effects, such as genetic mutations or the tunnel effect. We agree that QM is useful for describing such phenomena at a qualitative level, but its probabilistic nature makes it less effective for precise predictions. For example, the tunnel effect is explained in QM through the wave function and the probability of a particle penetrating a barrier, but this does not provide a deterministic description of the particle's trajectory. In DTA we strive to describe such phenomena through classical forces (Coulomb, Lorentz, inertia), which allows us to avoid probabilistic assumptions.

We are very grateful to you, Rustam Khakimovich, for your comment and for supporting our work. Your words inspire us to continue developing DTA to show that determinism can surpass QM in accuracy and physical clarity. We also thank you for mentioning F. F. Wolkenstein - this reminds us that even in the era of QM dominance, there were scientists who understood its limitations.

We wish you great success in your work and hope for further cooperation! If you have additional thoughts or suggestions on our methodology, we will be happy to discuss them.

Best regards,

Bakhodir

We assume that neither E. Schrödinger nor N. Bohr understood how quantum mechanics works.

Cairo's AI techniques predict that QM operates according to the square of SE rather than SE itself.

Below, we present the solution for the QM particle in infinite 1D and 2D potential wells, and in a closed control volume box.

The first step is to completely neglect the PDE of SE, as if it never existed, and solve for its squared PDE, supplemented by the AB law proposed by the author:

S(x,y,z,t)=Const.V(x,y,z,t)

The AB law means that the spontaneous potential of SE can be transformed into quantum matter and vice versa.

Here, the PDE matrix of QM for 17 nodes is given by the eigenmatrix equation:

166/2187 244/2187 266/2187 140/2187 35/2187        0         0         0        0         0         0        0        0        0        0        0        0

427/4374 853/4374 434/2187 322/2187 245/4374 35/4374         0         0        0         0         0        0        0        0        0        0        0

133/1458 124/729 343/1458      2/9 205/1458   10/243    5/1458         0        0         0         0        0        0        0        0        0        0

175/4374 230/2187     5/27 373/1458 355/1458 280/2187   55/2187    5/4374        0         0         0        0        0        0        0        0        0

35/4374 70/2187 205/2187 142/729 211/729 194/729 227/2187   26/2187   1/4374         0         0        0        0        0        0        0        0

0   5/1458    5/243   56/729   97/486 505/1458   205/729    50/729   5/1458         0         0        0        0        0        0        0        0

0        0   5/4374 22/2187 227/4374 410/2187 1981/4374 584/2187 43/1458         0         0        0        0        0        0        0        0

0        0        0   1/4374 13/4374 50/2187 292/2187 2849/4374 827/4374         0         0        0        0        0        0        0        0

0        0        0        0        0        0         0         0        1         0         0        0        0        0        0        0        0

0        0        0        0        0        0         0         0 827/4374 2849/4374 292/2187 50/2187 13/4374   1/4374        0        0        0

0        0        0        0        0        0         0         0 43/1458 584/2187 1981/4374 410/2187 227/4374 22/2187   5/4374        0        0

0        0        0        0        0        0         0         0   5/1458    50/729   205/729 505/1458   97/486   56/729    5/243   5/1458        0

0        0        0        0        0        0         0         0   1/4374   26/2187 227/2187 194/729 211/729 142/729 205/2187 70/2187 35/4374

0        0        0        0        0        0         0         0        0    5/4374   55/2187 280/2187 355/1458 373/1458     5/27 230/2187 175/4374

0        0        0        0        0        0         0         0        0         0    5/1458   10/243 205/1458      2/9 343/1458 124/729 133/1458

0        0        0        0        0        0         0         0        0         0         0 35/4374 245/4374 322/2187 434/2187 853/4374 427/4374

0        0        0        0        0        0         0         0        0         0         0        0 35/2187 140/2187 266/2187 244/2187 166/2187

.

1       1

2      2

3     3

4     4

5     5

6     6

7     7

8     8

9    =  9

8     8

7     7

6     6

5     5

4     4

3     3

2     2

1     1

It is clear that the energy eigenfunction is given by,

[1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1]^T units

And its eigenvalue = 1

To be continued.Eigenvalue = 1: Corresponds to the fastest path or the principle of minimum action.The wave function Psi is the SQRT of E, as shown in Figures 1 and 2.

📷 📷

1-Cairo Techniques Solution of Schrödinger's Partial Differential Equation -Time Dependence, ResearchGate, IJISRT review, March 2024.

We assume that neither E. Schrödinger nor N. Bohr understood how quantum mechanics works.

Cairo's AI techniques predict that QM operates according to the square of SE rather than SE itself.

Below, we present the solution for the QM particle in infinite 1D and 2D potential wells, and in a closed control volume box.

The first step is to completely neglect the PDE of SE, as if it never existed, and solve for its squared PDE, supplemented by the AB law proposed by the author:

S(x,y,z,t)=Const.V(x,y,z,t)

The AB law means that the spontaneous potential of SE can be transformed into quantum matter and vice versa.

Here, the PDE matrix of QM for 17 nodes is given by the eigenmatrix equation:

166/2187 244/2187 266/2187 140/2187 35/2187        0         0         0        0         0         0        0        0        0        0        0        0

427/4374 853/4374 434/2187 322/2187 245/4374 35/4374         0         0        0         0         0        0        0        0        0        0        0

133/1458 124/729 343/1458      2/9 205/1458   10/243    5/1458         0        0         0         0        0        0        0        0        0        0

175/4374 230/2187     5/27 373/1458 355/1458 280/2187   55/2187    5/4374        0         0         0        0        0        0        0        0        0

35/4374 70/2187 205/2187 142/729 211/729 194/729 227/2187   26/2187   1/4374         0         0        0        0        0        0        0        0

0   5/1458    5/243   56/729   97/486 505/1458   205/729    50/729   5/1458         0         0        0        0        0        0        0        0

0        0   5/4374 22/2187 227/4374 410/2187 1981/4374 584/2187 43/1458         0         0        0        0        0        0        0        0

0        0        0   1/4374 13/4374 50/2187 292/2187 2849/4374 827/4374         0         0        0        0        0        0        0        0

0        0        0        0        0        0         0         0        1         0         0        0        0        0        0        0        0

0        0        0        0        0        0         0         0 827/4374 2849/4374 292/2187 50/2187 13/4374   1/4374        0        0        0

0        0        0        0        0        0         0         0 43/1458 584/2187 1981/4374 410/2187 227/4374 22/2187   5/4374        0        0

0        0        0        0        0        0         0         0   5/1458    50/729   205/729 505/1458   97/486   56/729    5/243   5/1458        0

0        0        0        0        0        0         0         0   1/4374   26/2187 227/2187 194/729 211/729 142/729 205/2187 70/2187 35/4374

0        0        0        0        0        0         0         0        0    5/4374   55/2187 280/2187 355/1458 373/1458     5/27 230/2187 175/4374

0        0        0        0        0        0         0         0        0         0    5/1458   10/243 205/1458      2/9 343/1458 124/729 133/1458

0        0        0        0        0        0         0         0        0         0         0 35/4374 245/4374 322/2187 434/2187 853/4374 427/4374

0        0        0        0        0        0         0         0        0         0         0        0 35/2187 140/2187 266/2187 244/2187 166/2187

.

1       1

2      2

3     3

4     4

5     5

6     6

7     7

8     8

9    =  9

8     8

7     7

6     6

5     5

4     4

3     3

2     2

1     1

It is clear that the energy eigenfunction is given by,

[1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1]^T units

And its eigenvalue = 1

Eigenvalue = 1: Corresponds to the fastest path or the principle of minimum action.

The wave function Psi is the SQRT of E, as shown in Figures 1 and 2.

📷 📷

1-Cairo Techniques Solution of Schrödinger's Partial Differential Equation -Time Dependence, ResearchGate, IJISRT review, March 2024.

To be continued.

Dear Ismail Abbas, Indeed, in physics and other fields of science, square quantities are often used to estimate various parameters. For example, the energy of a kinetic particle is proportional to the square of its velocity, and further in thermodynamics and electrodynamics, many equations and concepts relate energies to the square of certain quantities.

From the point of view of quantum mechanics, the square of the amplitude of the wave function (which is described by the Schrödinger equation) does give the probability of finding a particle in a certain state. Thus, the statement that quantum mechanics works according to the square of the Schrödinger equation may indicate an understanding that energy and probabilities in the quantum world have a deep relationship.

It can also be argued that considering energy as a square value may offer new approaches to interpreting quantum processes, which may be useful for further research in this area. However, it is important to make a clear distinction between mathematical formulas and physical meaning in order to avoid misunderstandings.

Thus, emphasizing this connection, it can be explained to some extent that the "SE square" can serve as an important tool for estimating energy and probabilities, and this can indeed be of great importance in both theoretical and practical aspects of quantum mechanics.

But it must be taken into account that we look at quantum mechanics from different positions. You look more deeply, but I use it for practical purposes. Sincerely, Rustam

We reiterate that:

Cairo intelligence techniques = natural intelligence = artificial intelligence in the strict sense = unified field theory.

Einstein described SE PDEs and their derivatives as incomplete and doomed to failure, perhaps because they operate in R^4 space (3D+t as the external controller).

We assume that Cairo techniques, and therefore SE square theory, operating in a 4-dimensional unitary x-t space, are more efficient and more informative.

It should be noted that both of Einstein's theories of relativity (SR and GR) were developed in a 4-dimensional unitary x-t space.

Combining SE PDEs and general relativity theory is a misconception and a misleading one.

The solution to the Schrödinger partial differential equation (explained above) rests on the following question:

How should V be interpreted in the Schrödinger equation?

Surprisingly, the solution to the Schrödinger partial differential equation and its physical interpretation are the least understood topics among mathematicians and theoretical physicists.

The short answer to this question is that the quantum particle is subject to the applied potential V(x,y,z,t).

This scalar potential V is composed of two components, V1 and V2.

We can safely assume that:

V = C1. V1 + C2. V2. . . (1)

Where V1 is the externally applied voltage and V2 is the spontaneous or self-applied voltage.

V2(x,y,z,t) = Cons. U(x,y,z,t) . . . (2)

Equation 2 is called the Cairo Techniques equation.

Where U is the quantum energy density. Clearly, C1 + C2 = 1.

Therefore, the Schrödinger partial differential equation (PDE) is partly a diffusion equation and partly a wave equation.

It should be noted that Equations 1 and 2 are not suggestions, but rather certainty equations applied with extraordinary precision to solve the SE equation in central field situations as well as for infinite 1D and 2D potentials, in addition to that of the isolated quantum dot.

To be continued.

1- Matrix mechanics in 4D unitary x-t space is much more efficient than the time-dependent PDE in R^4 (3D+t as an external controller).

2- The statistical theory of Cairo techniques, via the B-matrix transition chains of classical physics, proves capable of performing 1D, 2D, and 3D statistical integration and of deriving the Gaussian (normal) statistical distribution law [1].

3- On the other hand, the statistical theory of Cairo techniques, via the Q-matrix transition chains of quantum physics, proves capable of performing 1D, 2D, and 3D statistical differentiation and of deriving the Planck statistical distribution law for blackbody radiation,

B(λ, T) = (2hc^2 / λ^5) * [1 / (exp(hc/λkT) - 1)] . . . . (1)

We assume that Planck's equation 1, viewed in light of current AI, generates incoherent noise and contains a lot of black magic, fragments, or patchwork.

Here is the quantum transition matrix Q for 1D, 19 free nodes.

153/400 -9/16    9/50       0       0       0      0      0      0      0      0      0      0       0       0       0       0      0       0

-1/2 24/25    -3/5    7/50       0       0      0      0      0      0      0      0      0       0       0       0       0      0       0

7/50 -21/40 147/200 -91/200 21/200       0      0      0      0      0      0      0      0       0       0       0       0      0       0

0 21/200 -39/100   27/50 -33/100    3/40      0      0      0      0      0      0      0       0       0       0       0      0       0

0      0    3/40 -11/40     3/8   -9/40   1/20      0      0      0      0      0      0       0       0       0       0      0       0

0      0       0    1/20   -9/50    6/25 -7/50 3/100      0      0      0      0      0       0       0       0       0      0       0

0      0       0       0   3/100 -21/200 27/200 -3/40 3/200      0      0      0      0       0       0       0       0      0       0

0      0       0       0       0   3/200 -1/20   3/50 -3/100 1/200      0      0      0       0       0       0       0      0       0

0      0       0       0       0       0 1/200 -3/200 3/200 -1/200      0      0      0       0       0       0       0      0       0

0      0       0       0       0       0      0      0      0      0      0      0      0       0       0       0       0      0       0

0      0       0       0       0       0      0      0      0 -1/200 3/200 -3/200 1/200       0       0       0       0      0       0

0      0       0       0       0       0      0      0      0 1/200 -3/100   3/50 -1/20   3/200       0       0       0      0       0

0      0       0       0       0       0      0      0      0      0 3/200 -3/40 27/200 -21/200   3/100       0       0      0       0

0      0       0       0       0       0      0      0      0      0      0 3/100 -7/50    6/25   -9/50    1/20       0      0       0

0      0       0       0       0       0      0      0      0      0      0      0   1/20   -9/40     3/8 -11/40    3/40      0       0

0      0       0       0       0       0      0      0      0      0      0      0      0    3/40 -33/100   27/50 -39/100 21/200       0

0      0       0       0       0       0      0      0      0      0      0      0      0       0 21/200 -91/200 147/200 -21/40    7/50

0      0       0       0       0       0      0      0      0      0      0      0      0       0       0    7/50    -3/5 24/25    -1/2

0      0       0       0       0       0      0      0      0      0      0      0      0       0       0       0    9/50 -9/16 153/400

Note that:

1-Q f(x) = d/dx (f(x))

Q^2 . f(x) = d^2/dx^2 (f(x))

Q^3 . f(x) = d^3 / dx^3 (f(x))

. . . . .

etc.

2- The matrix Q can provide a solution to the Schrödinger equation and derive the Planck distribution without resorting to the Planck quantization hypothesis,

E=n h f , n=1,2,3,... infinity.

Which demonstrates that quantum mechanics in the sense of the Schrödinger equation is on the verge of collapse.

To be continued.

1-Statistical integration and differentiation, ResearchGate, IJISRT review.

B-transition matrix chains derived from the Cairo technique for numerical statistical solutions have been successfully applied to the statistical solution of time-dependent partial differential equations in classical physics.

This Q&A examines the extension of B-transition matrix chains to the numerical statistical solution of the time-independent Schrödinger equation.

Note that both the classical physics transition matrix and the QM transition matrix are SMART matrices in the sense that they operate statistically in 4D x-t unitary space and can therefore create more information than exists in PDEs operating in 3D geometry plus real-time space.

However, extending physical B-transition matrix chains to solve the time-independent Schrödinger equation is not complicated, but it is quite time-consuming and requires compliance with certain limitations of fundamental principles, which we briefly explain in this answer [1,2].

Furthermore, it requires a vivid imagination.

Here, we present the numerical solution of the B-matrix using two illustrative examples: the one-dimensional infinite potential well and the quadratic potential well, where the numerical results are surprisingly accurate.

Extending physical transition matrix chains B to solve the time-independent Schrödinger equation requires some basic knowledge, which we briefly explain below:

i- Square matrices are a subset of mathematical matrices, and physical square matrices with physical meaning (such as the B transition matrix) are a subset of square matrices.

ii- Statistical transition matrices and chains of statistical transition matrices exist for situations in classical and quantum physics. Modeling these situations in transition matrix mechanics is more efficient for solving the corresponding partial differential equations (without going through the PDEs themselves) in a numerical statistical procedure.

Currently, we only know two: the mathematical statistical Markov transition matrix and physical transition matrix chains B, the subject of this Q&A.

iii- Not all matrix equations resulting from solving a PDE via the transition matrix are eigenvalue equations. For example, the matrix of the numerical solution to the heat diffusion equation produces a system of inhomogeneous first-order linear algebraic equations, while the matrix of the numerical solution to the Schrödinger equation is homogeneous and produces a multiple-eigenvalue eigenvalue problem.

What is the significance of a matrix with zero determinant?

First, if a matrix has a zero determinant, can it have an inverse or not?

The answer on Google is:

If the determinant of a matrix is ​​zero, then it has no inverse; the matrix is ​​therefore said to be singular. Only non-singular matrices have inverses.

Contrary to the Google answer, the author assumes that a singular matrix can have an inverse that is another singular.

Consider the homogeneous system of n independent first-order linear algebraic equations:

a11 x1 + . . +a1n xn = 0

a21 x1 + . . +a2n xn = 0

. . . .

an1x1 + . . +ann xn = 0

In matrix form,

A . x = 0 . . . (2)

This system has a solution if and only if the nxn matrix A (ai,j)

satisfies two seemingly contradictory conditions:

i- The determinant of matrix A = 0.

ii- The inverse of A (A^-1) exists.

In this case, the QM solution for vector x is given by:

x=A^-1 (b+S)

b is the assumed Diriclet boundary condition vector and S is the source/sink vector (S=S1+S2 where S1 is the spontaneous or intrinsic source and S2 the extrinsic applied source).

Both S1 and S2 are subject to the authors' AB equation.

Therefore, if a matrix has a zero determinant, it can describe the transfer matrix of the isolated QM system Q.

This means that the QM energy vector can be easily found as an eigenvector of Q with eigenvalue = 1, as shown below.

Quantum mechanics will sooner or later collapse and be replaced by its corresponding artificial intelligence.

Ultimately, it is true that Cairo's intelligence techniques = natural intelligence = artificial intelligence in the strict sense = unified field theory.

To be continued.

1-BOOK-Fundamentals of Artificial Intelligence-Theory and Practice-ISBN- 978-969-8092-20-7

2-A statistical numerical solution for the time- independent Schrödinger equation, ResearchGate, IJISRT Journal, November 2023

We all know that the Schrödinger equation is very effective in explaining the observed experimental phenomenon of quantum particle tunneling.

Let's analyze a misleading error in a Wikipedia-Google search.

"Tunneling is a consequence of the wave nature of matter. The quantum wavefunction describes the state of a particle or other physical system, and wave equations, such as the Schrödinger equation, describe their behavior.

This is called quantum tunneling!"

But this is only an illusion, because the artificial intelligence of Cairo's techniques allows us to deduce the formula for quantum tunneling from the square of the Schrödinger equation, using classical statistics.

In reality, the artificial intelligence of Cairo's techniques allows for the superposition of total energy in the classical sense. Therefore, quantum tunneling can be rigorously described from the square of the Schrödinger equation, using the statistics of Cairo techniques.

Step 1

Find the allowed stationary solution to the proposed quadratic equation for a quantum particle in a one-dimensional infinite potential well using the statistical chains of matrix B, i.e., 1/n^2, 1, 1/4, 1/9, 1/16, etc. (as shown in Figure 1).

Step 2

The above values ​​correspond to the eigenenergies: 1, 4, 9, 16, etc.

Step 3

Apply the superposition principle, i.e., E = (1, 4, 9, 16, etc.)

with probabilities P1, P2, P3, etc.

Step 4

If the quantum system (or particle) has a total energy Etotal, then:

Etotal = P1 * E1 + P2 * E2 + P3 * E3 +...PnEn

The tunneling problem is reduced to solving the algebraic equation:

X is a function of the total energy such that:

for x .LT.1

X + X^2 + X^3 + . . = Etotal

Step 5

In exponential form, for x .LT. 1:

e^x + e^2x + e^3x . . = Etotal

OR,

1/(1-e^x)=Etotal

Obvious example: for x = 2/3, then Etotal = 3 units.

Step 6

Transmission or quantum tunneling part:

Transmission or quantum tunneling part:

e^ (- Alpha b).Etotal.e^-1/(1-x).GE.Uo (tunnel edge)

Fig. 2

Where l is the barrier width and Alpha is the exponential decay coefficient.

The tunneling ratio Etransmitted/Eincident is called Transmission T=e^-Alpha . l e^- (Etotal/Uo)

Where x = Etotal/Uo . . . (1)

Formula 1 is equivalent to that derived from solving the Schrödinger partial differential equation.

It should be noted that the artificial intelligence approach of the Cairo techniques is fundamentally different from that used to solve the Schrödinger PDE.

To be continued.

1-A statistical numerical solution for the time- independent Schrödinger equation, ResearchGate, IJISRT  journal, November 2023.

Here are few questions on quantum physics of the Schrödinger equation and classical statistical physics of its square (energy diffusion), to clarify this question and the topic as a whole.

1-Q1-

Briefly explain the Schrödinger wave equation and its square.

A1-

Schrödinger partial differential equation:

i h dΨ/dt)partial=h^2 . Nabla^2 Ψ/2m + V Ψ . . . . (1)

With the Bohr-Copenhagen interpretation introducing entanglement and superposition Ψ.

The Schrödinger partial differential equation is precise but incomplete because it operates in an incomplete D^4 space (3D+t as an external controller). Now imagine solving the Schrödinger partial differential equation for Ψ^2 and not Ψ.

Equation 1 transforms into:

dΨ^2/dt) partial =C1.Nabla^2 Ψ/2m + C2 .V . . . . (2)

With the following statistically proven assumptions,

i-Ψ^2=Ψ . Ψ*

ii-Ψ^2 is exactly equal to the energy density of the quantum particle(s).

iii-Ψ^2 is the probability of finding the quantum particle in the 4D unit volume element x-t "dx dy dz dt"

iv- The actual time t is completely lost and replaced by the dimensionless integer N dt. In this 4D unit x-t space, the dimensionless time N is integrated into the 3D Cartesian space.

N is the number of iterations or repetitions and dt is the time jump.

Equation 2 is derived from and solved by the advanced artificial intelligence of modern transition matrix statistics.

Equation 2 is solved via matrix mechanics and does not require any PDE or FDM techniques to be solved.

Surprisingly, equation 2 is more informative than equation 1.

Q2

Is the quantum wavefunction Ψ a scalar, a vector, or neither?

A2

It is very likely that Ψ is none of these.

The quantum function Ψ^2 is an nxn square matrix (second-order tensor).

The question arises: is the answer to this question "shut up and calculate"?

Q3-

Is the Schrödinger equation an eigenvalue problem?

Is the heat diffusion equation an eigenvalue problem?

A3

The answer is yes in both cases.

Solving the heat diffusion equation using advanced AI for matrix chains B is an eigenvalue problem.

The most important thing is the preliminary selection of the principal diagonal elements RO (entries) of the statistical transition matrix B.

The solution is expressed by the transfer function of the heat diffusion equation D(N):

U(x,y,z,t)=D(N).[b+S] +B^N IC

IC is the vector of initial conditions (U(x,y,z,0))

The eigenvalue of the solution implies:

the exponent of the solution:

k*= log [(1 + RO) / (1-RO)]

Note that [(1 + RO) / (1-RO)] is equal to the eigenvalue 1 for B and [(1 + RO) / (1-RO)]^2 to the eigenvalue 2 for B^2, etc.

with a maximum of log 2 and a minimum of zero. [RO] is the diagonal vector of matrix B.

Vector RO = ( RO11 , RO22, RO33 , . . . ,RO n n)

The classical approach to the Schrödinger equation is an eigenvalue equation:

By definition, an operator acting on a function produces another function. However, a special case arises when the generated function is proportional to the original function.

A^ψ∝ψ . . . (1)

[This is a special case]

This case can be expressed as an equality by introducing a proportionality constant k.

A^ψ = k ψ . . . . . (2) [This solution applies to the special case of PDEs]

Not all functions solve an equation like those in equations 1 and 2.

ψ = (φ1+φ2)/√2

TO BE CONTINUED.

SIMPLE CHALLENGE

The simple challenge to our distinguished contributors and readers is the statistical transition matrix B for a closed control volume of surface A subject to Dirichlet boundary conditions,

U(x,y,z,t+dt)=B. U(x,y,z,t) . . . (1)

Equation 1 is not only universal, but it also describes and solves all time-dependent phenomena in the entire universe (partial differential equations of classical physics in its most general form, quantum physics, statistical distributions, integration and differentiation, etc.).

The challenge is as follows:

Name a single physical phenomenon from the four above that does not belong to Equation 1.

Thank you.