Contribution of Linguistic Inspiration over Logic
Rongqing Dai
Abstract
By reviewing how the quantum mechanics was made through de Broglie matter wave, Schr?dinger wave function and equation, Born interpretation of the wave function, Heisenberg uncertainty principle, and Dirac’s work, this writing will unveil a linguistic link, instead of logic connection, between each step of the early development of quantum mechanics, which has been largely ignored by the academia but of profound significance to the subject of modern physics, as well as to modern sciences by and large.
Keywords: Quantum Mechanics, Linguistic Inspiration, Matter Wave, Wave Function, Quantization
1. Introduction
While quantum mechanics has been portrayed as the most rigorously constructed theoretical system, if we carefully look back to how its foundation was laid in 1920s step by step, we might draw a very different conclusion.
1.1. The “wave” of de Broglie
Despite the claim made by de Broglie in his Nobel Lecture (de Broglie 1929) [[1]] that he began his work on the phase wave by associating a stationary wave to a small body, as noticed by Dai (2023a)[[2]], his actual mathematical handling of the subject was to treat the body as a simple harmonic oscillator that oscillates in a sinusoidal pattern. The “wave” was generated by making that body to move with a speed v relative to the observer. In a normal Galilean system, this “wave” would be a simple harmonic wave with the phase speed equal to v; however, de Broglie transformed this wave into a complex wave by applying the Lorentz time dilation to it, and thus turned the phase speed into V = c2/v, which then leads to the famous de Broglie wavelength formula:
λ = h/p= h/(mv) (1)
As pointed out by Dai in 2023 that an obvious defect of (1) is that when applying (1) to macroscopic objects with the speed of zero, their so-called matter wavelengths would be infinity according to (1), no matter how big the mass would be. By eliminating the relativity factor, we can turn (1) into (Dai 2023a)[2]:
λ = 2(v2/c2) h/p = 2(v2/c2) h/(mv) (2)
Formula (2) no longer goes to infinity problem even if v goes to zero.
1.2. The work of Schr?dinger
Inspired by de Broglie, in 1926 Schr?dinger published his work (Schr?dinger 1926) [[3]] on the state equation of subatomic particles. In order to work out that equation, he needed to have a wave function. While he was supposed to model the behavior of the wave of de Broglie based on the vague notion of “matter wave”, in his 1926 report, Schr?dinger did not seem to be very sure of what aspect of the wave he was working on at the beginning. He had to figure out an interpretation of the wave function in his report after he worked out his equation, and then chose it to be particle charge density.
Nevertheless, Schr?dinger demonstrated that from his derivation he could give out the exact same de Broglie formula for the matter wavelength. However, he was not careful enough about the difference in the conditions of his derivation and de Broglie’s. That difference determined that he should not have reached the same result as de Broglie since he excluded the special theory of relativity as he explicitly announced in his report (Dai 2024a) [[4]]. This reckless defect alone was enough to tell that Schr?dinger did not work out his equations rigorously following the Hamiltonian mechanics as he claimed, but rather through some kind of heuristic means as Feynman commented later (Gottlieb and Pfeiffer1965) [[5]].
1.3 The probability wave of the Copenhagen school
Unhappy with Schr?dinger’s interpretation of the meaning of his own wave, Born of the Copenhagen school modified the meaning of the wave function ψ in Schr?dinger equation (Wikipedia [[6],[7]]; Born 1954a,b [[8],[9]]). Born reinterpreted the modulus of Schr?dinger’s wave function ψ as the probability density of a particle’s appearance in a certain condition, and the wave spreads over the whole space before its necessary collapse to make real physical sense.
1.4. Heisenberg’s uncertainty
Subsequent to Born, as a direct application of the probabilistic wave dynamics, Heisenberg proposed his famous uncertainty principle (Wikipedia [[10]]) by struggling to work on the probability together with the kinematic position and velocity (Heisenberg 1927[[11]]).
Many years later, Frank Wilczek assigned a new meaning to Heisenberg’s uncertainty in his work of quarks and gluons (Wilczek 2005[[12]]; Wikipedia[13]]) by saying “Very roughly speaking, the required uncertainty in position is accommodated by allowing for the possibility that the act of measurement can involve the creation of several particles, each indistinguishable from the original, with different positions”, which obvious goes way beyond the original proposal of Heisenberg.
1.5. Quantization and Dirac’s work
Despite the unsatisfactory interpretation for the meaning of his wave function, Schr?dinger’s work is of critical importance to the quantum mechanics because it paved the way to a fundamental means of later quantum mechanics, which is quantization (Dai 2024b)[[14]]. Unlike the common perception of dividing a macroscopic object into tiny microscopic particles, the so-called quantization in quantum mechanics does not do any dividing at all…..it is just a scheme to turn a macroscopic equation that looks ordinary into a quantized equation through symbolic operations that could be expressed as following:
where E is total energy, pr is momentum component, and all the differential operations are applied to the Schr?dinger’s wave function ? based on Born interpretation, and then multiplied to the reduced Planck constant and the imaginary number i.
Obviously, the logical foundation of the above scheme is the assumption of the validity of Schr?dinger’s equation despite the original meaning of Schr?dinger’s wave function ? was abandoned and a new meaning was given by Born of the Copenhagen school.
By applying the above quantization operation and starting from relativistic Hamiltonian (or relativistic energy conservation), Dirac worked out his Nobel prize winning equation, the famous Dirac equation (Dirac 1928) [[15],[16] ]:
Despite Dirac’s effort of trying to make his derivation look like rigorously based on the Hamiltonian mechanics, it is far from it. First of all, as its quantization foundation, the Schr?dinger’s equation was not a product of rigorous Hamiltonian mechanics; secondly, its relativistic foundation is not Hamiltonian at all (Dai 2023b)[[17]]; thirdly and more obviously, he made several heuristic constructive maneuvers during his derivation (Dai 2024b)[14].
One famous point that has made the physics community very proud of Dirac’s work was that during his derivation, he predicted the existence of positron. However, by looking into his fundamentally heuristic derivation we might find that his so-called prediction was just his constructed equation contains both positive and negative energy solutions. Considering that there was no any experimental evidence at the time to support that result, Dirac’s such prediction obviously should not become an example for college students not to discard solutions that do not get along with common knowledge unless they have some particular reasons, if they want to pass class tests or examinations of qualification.
2. The Role of Linguistic Inspiration instead of Logical Connection in Making Quantum Mechanics
From the above section 1 we can observe the following dramatic evolutionary building process of the foundation of quantum mechanics:
de Broglie called the pattern traced by a simple harmonic oscillator moving through space a “wave” è the physics community dubbed this as “matter waves” è aiming to establish a dynamical mathematical framework for matter waves, Schr?dinger cobbled together his famous equation, laying the theoretical foundation for quantum mechanics, and interpreted the wave function as “particle charge density” è Born reinterpreted Schr?dinger's wave function as probability density, thereby transforming de Broglie waves into probability waves è based on Born's interpretation, the renowned Heisenberg uncertainty principle can be derived from the Schr?dinger equationè by reassigning meaning to Heisenberg uncertainty principle, Frank Wilczek worked his theory concerning quarks and gluons, for which he was awarded the 2004 Nobel Prize in Physics.
The Schr?dinger-equation-based quantization è Dirac incorporated Pauli's spin matrices and special relativity into the Hamiltonian, yielding the famous Dirac equation; the presence of both positive and negative energy solutions within it led to the later claim that it predicted the existence of the positron.
2.1 Discussion
The most striking feature of the above dramatic process is that the connections between each step of its development were more of linguistic inspiration, instead of commonly supposed strict logical causality.
1) Inspired by the research of Planck, Einstein, and other physicists on the wave-particle duality of light, de Broglie introduced the notion of wave into the realm of particles of mass. Lacking a formal theoretical foundation for this move, he treated the waveform generated by a simple harmonic oscillator translating through space as the wave of a fundamental particle, a notion that the physics community dubbed as matter wave.
The term matter wave not only opened a linguistic door to viewing all matter as wave-like, but more importantly ignited the imagination. This act of imagination itself established a kind of self-evident logic, allowing the wave-particle duality of matter to formally enter the mainstream of physics.
2) With the notion of de Broglie waves as a starting point, Schr?dinger was able to “logically” proceed to “derive” a dynamical equation for the wave associated with particles of mass. The crucial element here was simply the word wave from de Broglie's notion; the specific nature of the wave no longer mattered, as Schr?dinger’s wave was distinct from de Broglie’s—it was mainly a function of an undefined nature, yet a wave function nonetheless.
3) With Schr?dinger’s equation as a foundation, the physics community was free to further exercise its imagination. For them, the specific physical significance of either de Broglie waves or Schr?dinger waves was irrelevant; they were well aware that de Broglie had conjured something out of nothing and that Schr?dinger had essentially borrowed a name to launch a new concept, neither approach rested on a concrete physical basis. Consequently, they (led by Max Born) interpreted Schr?dinger’s wave as a probability wave.
4) An important consequence of accepting the notion that a matter wave spreads throughout the entire (phase) space is this: if a particle is actually a wave distributed throughout the entire (phase) space, then Heisenberg’s inequality, known as the Uncertainty Principle, can be derived through rigorous mathematical deduction[10, 11].
5) Compared to de Broglie’s so-called matter waves or the interpretation of the wave function as charge density that Schr?dinger proposed, a major advantage of the probabilistic interpretation is its suitability for integral calculus. The direct consequence of combining Schr?dinger’s equation with Born’s probabilistic interpretation was the emergence of the symbolic operations associated with “quantization”.... Here the “quantization” referred to is not the notion of breaking macroscopic objects down into microscopic particles (as often conveyed in popular science); rather, it is a specific formal treatment arising from the synthesis of Schr?dinger’s equation and Born’s concept of probability wave.
6) The Uncertainty Principle holds another fascinating semantic significance: the term “uncertainty” served as the “logical” basis for Frank Wilczek to conclude that measuring the motion of a single quark within an elementary particle could result in the creation of multiple quarks [12,13]—a discovery for which he won the 2004 Nobel Prize in Physics. The crucial point here is that this application bears almost no relation to the mathematical formulation or the original physical meaning of Heisenberg’s Uncertainty Principle at all; the only common link is the word “uncertainty” itself.
2.2. Transforming quantum mechanics through symbolic operations
Despite the quandary of lacking of solid empirical and theoretical foundation faced by its pioneers, nowadays quantum mechanics has been portrayed as the best theory with a solid experimental foundation and a rigorous theoretical framework. This fundamental transformation in the image of quantum mechanics is largely attributed to the quantization scheme.
2.2.1 The veil cast over quantization by the mainstream academic community
The mainstream academic community, including the realm of popular science, has shrouded quantization in a veil that obscures its true nature from the layperson in two ways:
a) Dirac was the key figure who perfected the symbolic formalism of quantization. He developed the “bra-ket” notation system, effectively imbuing quantum mechanics with an air of mystique that makes it difficult for the average person to grasp its true essence.
b) Popular science often conveys the impression that quantization is merely the subdivision of the macroscopic world into the microscopic scale. It typically fails to mention that the actual process of quantization in quantum mechanics involves no direct division between the macroscopic and microscopic realms… rather, it is simply a matter of symbolic manipulation based on the Schr?dinger equation and Born's probabilistic interpretation. While this approach in popular science is often a necessary concession to avoid technical complexity, the practical effect is that the true nature of quantization remains elusive to the general public.
2.2.2. Quantum mechanics transformed: shedding the shadow of intrinsic defects
With the above protective measures, quantum mechanics has emerged as a system boasting a “solid experimental foundation and a rigorous theoretical framework.” It is now difficult to perceive, within Dirac equation, the predicament faced by pioneers like de Broglie, Schr?dinger, and Born—who produced quantum wave theory in the absence of a firm theoretical basis.
2.3. Heuristic Illusion
Heuristic theories that are “fitted” to some known data often appear the most robustly verified; for instance, if one fits a curve to n data points and then tests that curve against those same n points, it will inevitably match the experimental results 100%. This could create an illusion of the perfectness of some heuristic theories, including the heuristic theories of quantum mechanics.
Of course, one cannot rule out the possibility that experimental results from laboratories unknown to the outside world at the time of de Broglie, Schr?dinger, Born, and Dirac provided indirect inspirations for their theories; however, the developmental trajectory clearly presented in their papers is not a rigorous logical chain, but rather a chain of linguistic inspiration.
3. Theoretical Defects versus Technological Advances
As mentioned my books “When Philosophy Is Disparaged”[ [18]] and “The Cracking Scientific Foundation”[[19]], whenever I discuss the defects in the theoretical foundation of modern sciences, an inevitable question will arise: if the theoretical foundation is really so messy as you are saying, how could we have so many dazzling technological advancement?
A brief answer to this question will be provided in the rest of this section.
3.1. Direct causes
In reality, the relationship between technological progress and theoretical advancement is not the simple, linear cause-and-effect connection that is commonly assumed. To better understand the drivers behind technological progress, let us consider the following factors:
1) Even if the existing framework of physical theory contains numerous flaws, it still could encompass a wealth of valuable knowledge. Combining existing knowledge with new experimental results often gives rise to new theories; even if both established and new theories are only partially correct, they can still help improve existing technologies or facilitate the conception and realization of new ones.
2) Regarding technological development, people often overlook a crucial point: it is a process of gestation and evolution within a global family of technologies spanning millennia. This is because the influence of established, older technologies on new developments often far outweighs that of new, speculative theories. For instance, prior to the advent of automation, engineers, technicians, and artisans across various industries—regardless of their specific products—required a set of fundamental craft skills; these skills were themselves the product of millennia of accumulated practical experience. These long-standing manual and semi-manual techniques subsequently provided essential tools and standards for the emergence of automation technology.
3) Technological development is heavily influenced by cross-disciplinary advancements—advancements that often proceed independently of the theoretical evolution within the target field itself. For instance, the most significant advancements in human science over the past century did not stem from physical theories and their associated technologies; rather, they arose from computer technology, automation and artificial intelligence, related telecommunications and information technologies, precision instrumentation and machining, optics, chemistry, biochemistry, medicine, geology, and so forth. The evolution of computing—particularly in algorithms and programming—proceeded largely independently of developments in frontier physics; while frontier physics did exert some influence on computer hardware, that impact was, on the whole, quite limited. Regarding hardware, classical physics has played a far greater role to date than cutting-edge quantum physics. Conversely, however, the development of computer technology, automation, and artificial intelligence has clearly had a profound impact on the progress of physics itself (whether experimental or theoretical).
4) Even when guided by theories touted as state-of-the-art, “trial and error” remains the norm in technical design and application, often yielding results that deviate significantly from established theory. In fact, throughout human history, instances where technological development diverged from existing theory have played a crucial role in the birth of new theories. Scientific discoveries are frequently the result of sheer chance—or even unexpected gifts gained after the failure of seemingly unrelated experiments—often bearing no direct connection to the theoretical progress of the time.
5) Although certain general theories may harbor serious logical flaws, they can still yield correct results under specific conditions where the factors causing their errors do not come into play.
6) Some theories are established precisely to simulate specific experimental phenomena. This is akin to fitting an experimental curve to a set of data points; while the curve may not capture every aspect of the relevant data, it represents the specific points used for the fit quite well. For instance, now we know that the Schr?dinger equation was not derived from fundamental physical laws but was instead constructed to fit the facts; at the time of its formulation, the existence of “half-integer” quantum numbers in certain experiments was already known, so the choice of the equation’s form and parameters was guided by the aim of yielding such values. Indeed, the wave equation itself was conceived specifically to align with the de Broglie wavelength formula. Ironically, theories constructed in this manner often prove to be the most robust under verification. If, as mentioned earlier, you fit a curve to n experimental data points and then test that curve against those same points, it will inevitably match the experimental results perfectly.
3.2. The role of empirical formulas
Furthermore, empirical formulas derived from experimental observations have played a crucial role in the development of modern science. Although the proponents of these formulas did not (unlike de Broglie, Schr?dinger, Dirac, the Copenhagen School, and relativity theorists) claim to be unveiling the fundamental mysteries of the universe, the formulas they produced often accurately reflected natural laws in specific contexts. To a large extent, these formulas provided the theoretical foundation required for engineering and technology—a foundation that the so-called "fundamental theories" (which were often riddled with errors and detached from reality) failed to supply.
The banner of “unveiling the fundamental mysteries of the universe” often leads the general public to believe that the so-called fundamental theories of physics are the most precise, and that they provide the justification for the existence of empirical formulas which are then viewed merely as “approximate solutions” to those fundamental theories. The reality, however, is quite the opposite: it was the proven validity of these empirical formulas that garnered public trust for theories that were error-ridden and yet frequently awarded Nobel Prizes for supposedly revealing the universe's fundamental secrets. For the layperson, the correctness of empirical formulas creates the impression that the underlying fundamental theories must be even more profound. Regrettably, this misconception is not limited to the general public; it has also been embraced by the professional physics community. Not only does the physics community accept this view, but it also actively promotes this illusion worldwide—including through popular science outreach—convincing ordinary people (who might otherwise be indifferent to the subject) to believe in this narrative.
3.3. Luck
Beyond the factors mentioned above—which involve relatively clear logical cause-and-effect relationships—we cannot rule out the role of sheer luck. It is entirely possible for a fundamentally flawed line of reasoning to (by a stroke of coincidence) yield a result that is correct in certain respects.
3.4. The semiotic scaffolding role of scientific theories
The traditional models likening the development of knowledge to a relay race or the construction of a building bear little resemblance to the actual trajectory of the progress of civilization witnessed since the dawn of the 20th century. In fact, even the seemingly more plausible adage that “failure is the mother of success” fails to accurately capture the nature of the development of civilization in the modern era.
Here, we must consider a model distinct from the three mentioned above: the semiotic scaffolding model. Here semiotic scaffolding refers to the phenomenon where the concepts employed within a theory (or even the literal meanings of the terms used) serve to organize future theories or catalyze technological advancement. In other words, while scientific progress is often likened to erecting a building, scientific activity does not always contribute directly to the structure's masonry; sometimes, its role is to linguistically construct scaffolding, enabling future generations to climb it and build the actual edifice.
4. Misdirection and Delays Caused by Flawed Theory
While the development of all technologies—including those in experimental science and engineering—might not necessarily depend on the advancement of theoretical foundations but rather benefit from an inherent technical coherence, a flawed theoretical framework will inevitably mislead humanity's progress in understanding the natural world and severely hinder its advancement.
Although the existing framework of quantum mechanics could serve as a form of symbolic scaffolding for the advancement of experimentation and technology, I believe that anyone who truly understands the history of quantum theory’s development would hardly be naive enough to think a solid edifice of quantum physics could be constructed upon such a flawed foundation.
Nevertheless, not all quantum theories rely on those so-called fundamental equations or other mathematical models built upon them... However, the issue is that the scattered theories existing outside the theoretical frameworks built upon fundamental equations do not constitute the core of quantum theory; the heart of quantum mechanics is precisely what is constructed upon those foundational theories. In particular, the Standard Model of elementary particles—repeatedly hailed as one of the most elegant achievements in the history of human civilization—can be said to rest entirely within the framework of the aforementioned foundational theories, even though we cannot entirely rule out the positive, heuristic influence of certain experimental results.
5. Final Remarks
While it might be the case that, absent the various errors in theoretical physics, human science and technology could have advanced hundreds of times beyond current levels, humanity has still achieved leaps in technological progress since the late 19th century, even with those errors present. Yet, simultaneously, human social and cultural development has lagged far behind... We might well ask: if human science and technology were indeed hundreds of times more advanced than they are now, would that be a blessing or a curse, given our current level of social and cultural maturity?
Can humanity’s current social and cultural state peacefully accept and assimilate a level of technology that is a hundred times—or more—advanced than what we have today?
Perhaps having evaded from the tragedy of the combination of poor social and cultural state with very advanced science and technology is the most positive, albeit unintended, consequence of the disarray found in the current theoretical physics?
Accordingly, we need a new discipline, the “Relationship between Theory and Technology”, to study the conditions under which social and cultural systems can constructively embrace advanced levels of technological development.
Endnotes
[[1]]de Brogile, L. (1929). “The wave nature of the electron”. Nobel Lecture, December 12, 1929. Retrieved from: https://www.nobelprize.org/uploads/2016/04/broglie-lecture.pdf
[[2]]Dai, R. (2023a). Correction to de Broglie Wavelength. Retrieved from: https://www.researchgate.net/publication/376679293_Correction_to_De_Broglie_Wavelength
[[3]]Schr?dinger, E. (1926). “An Undulatory Theory of the Mechanics of Atoms and Molecules”, The Physical Review, Vol. 28, No. 6, December, 1926. Retrieved from: https://web.archive.org/web/20081217040121/http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf
[[4]]Dai, R. (2024a). A Short Survey of the Defects in Schr?dinger’s Derivation. Retrieved from: https://www.researchgate.net/publication/383751232_A_Short_Survey_of_the_Defects_in_Schrodinger's_Derivation
[[5]]Gottlieb, M.A. and Pfeiffer, R. (1965). The Dependence of Amplitudes on Position. Feynman Lectures Ch 16. Retrieved from: https://www.feynmanlectures.caltech.edu/III_16.html#Ch16-S1
[[6]]Wikipedia. Wave function. Retrieved from: https://en.wikipedia.org/wiki/Wave_function. Last edited on 18 August 2024, at 12:29 (UTC).
[[7]]Wikipedia. Max Born. Retrieved from: https://en.wikipedia.org/wiki/Max_Born. Last edited on 3 October 2023, at 09:25 (UTC).
[[8]]Born, M. (1954a). Quantum Mechanics of Collision Processes. Retrieved from: https://web.archive.org/web/20201201173255/http://www.ymambrini.com/My_World/History_files/Born_1.pdf
[[9]]Born, M. (1954b). The statistical interpretation of quantum mechanics. Nobel Lecture, December 11, 1954. Retrieved from: https://web.archive.org/web/20121019194414/http://www.nobelprize.org/nobel_prizes/physics/laureates/1954/born-lecture.pdf
[[10]]Wikipedia. Uncertainty principle. Retrieved from: https://en.wikipedia.org/wiki/Uncertainty_principle. Last edited on 11 October 2023, at 17:29 (UTC).
[[11]]Heisenberg, W. (1927). The Actual Content of Quantum Theoretical Kinematics and Mechanics, Z. Phys. 43 (3–4): 172–198. Retrieved from: https://ntrs.nasa.gov/api/citations/19840008978/downloads/19840008978.pdf
[[12]]Wilczek, F. (2005) Asymptotic Freedom: From Paradox to Paradigm. Retrieved from https://arxiv.org/pdf/hep-ph/0502113
[[13]]Wikipedia. Frank Wilczek. Retrieved from https://en.wikipedia.org/wiki/Frank_Wilczek. Last edited on 14 April 2026, at 12:26 (UTC).
[[14]] Dai, R. (2024b). Schr?dinger Equation and Quantization. Retrieved from: https://www.researchgate.net/publication/385379588_Schrodinger_Equation_and_Quantization
[[15]]Dirac, P. A. M. (1928). "The Quantum Theory of the Electron". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 117 (778): 610–624. Retrieved from: https://royalsocietypublishing.org/doi/epdf/10.1098/rspa.1928.0023
[[16]]Wikipedia. Dirac equation. Retrieved from https://en.wikipedia.org/wiki/Dirac_equation. Last edited on 14 August 2026, at 01:19 (UTC).
[[17]]Dai, R.(2023b). The Faulty System of Relativistic Momentum and Energy. Retrieved from: https://www.researchgate.net/publication/368357262_The_Faulty_System_of_Relativistic_Momentum_and_Energy
[[18]]Dai, R. (2024) When Philosophy is Disparaged. Scholars’ Press. 2024. Final Remarks, pp. 101-106. ISBN: 978-620-6-77202-6.
[[19]]Dai, R. (2026) Metaphysical Symphony Series:Book One, The Cracking Scientific Foundation. Amazon. 2026. Paperback ISBN: 979-8249046958, ASIN: B0GPPD2ZHM. eBook ASIN: B0GQX9G6TV.