Rongqing Dai
Abstract
While it is a mundane mode for observers to be at rest with many macroscopic objects, the famous de Broglie formula for the wavelength of matter waves would go to infinity when the speed of the object equals zero. This paper provides an in depth analysis of how this defect was introduced by de Broglie in his original work and how we could fix this problem with the post special relativity physics.
Keywords: de Broglie, Matter Wave, Wavelength, Phase Speed, Schrödinger, Deterministic
1. Introduction
In his 1924 PhD thesis [[1]], de Broglie presented to the world his famous formula for the wavelength of what he called as “phase wave” of a moving body, which was later named as matter wave [[2]], as follows:
λ = h/p (1)
where λ is the wavelength of the matter wave, h is Planck’s constant, p = mv is the momentum of the body with its mass being m and velocity being v. By substituting the momentum expression into equation (1), we get:
λ = h/(mv) (1a)
Since then a popular enlightenment from (1a) has been taught around the world that the reason why we cannot observe the matter wavelength in the macroscopic world is because the mass m would be too large so that the wavelength λ would be too small in the macroscopic world as reflected in (1a).
However, it seems to have been ignored by all when delivering the above enlightenment that a basic characteristic of macroscopic objects is that they do not always move rapidly as microscopic particles are supposed to do, and accordingly it is a common thing for us to observe the world by sitting in a position that is at rest with many macroscopic objects so that their speed would be zero. With that in mind, we can see an obvious problem when applying (1a) to macroscopic objects with the speed of zero: their matter wavelengths would be infinity according to (1a), no matter how big the mass would be.
In the following sections we will see how the above defect of (1a) was introduced by de Broglie in his original work by applying the Lorentz transformation, and we will also see how to correct this problem by removing the Lorentz transformation in the derivation. Further we will see how the matter wavelength would be impacted by the revised mass-energy relationship.
2. The Impact of Special Relativity upon Matter Wavelength
Despite the claim made by de Broglie that he began his work on the phase wave by associating a stationary wave to a small body, his actual mathematical handling of the subject was to treat the body as a simple harmonic oscillator that oscillates in a sinusoidal pattern [[3]]. 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 the wave, and thus turned the phase speed into:
V = c2/v (2)
In the meantime, we have the following relationship between phase speed V, wavelength λ and frequency ν:
V = λν (3)
De Broglie also hypothesized that the total energy E of the body satisfies the formulation that Einstein proposed for photons:
E = hν (4)
By applying the Einstein mass-energy relation E = mc2, we have:
p = mv = mc2v/c2 = Ev/c2 = E/V = hν/ (λν) = h/λ (5)
From (5) we can get (1) and (1a).
In his presentations, m was expressed in the relativistic form γm0, with γ being the Lorentz factor and m0 being the so-called rest mass. That was nevertheless unnecessary for getting the wavelength formula as we could see from the above workout of (5).
2. Correction to de Broglie Wavelength Formula
After seeing how de Broglie reached his famous formula of matter wavelength, with the awareness that Lorentz time dilation is not physically real in nature [[4]] and accordingly a revised mass-energy relationship E = mc2/2 is needed [[5]], it would be quite straightforward for us to find the correction to fix the problem of infinitely large wavelength for macroscopic motionless objects as mentioned earlier.
Now without the Lorentz time dilation, the de Broglie wave would no longer be the complex wave as he originally dealt with, but only be a simple harmonic wave with the phase speed V equal to the speed v of the moving body instead, and we will have:
p = mv = 2mc2v/(2c2) = 2Ev/c2 = 2Ev2/ (Vc2) = 2(v2/c2)E/V=2(v2/c2)hν/ λν = 2(v2/c2) h/λ (6)
From (6) we have:
λ = 2(v2/c2) h/p = 2(v2/c2) h/(mv) (6a)
The appearance of the coefficient 2(v2/c2) in the corrected formula (6a) for the matter wavelength makes λ no longer be infinity for objects at rest.
2.1. Discussion
The correctional coefficient 2(v2/c2) in (6a) contains two factors: 2 and v2/c2, which come from two corrective steps in the new derivation: 2 comes from the adoption of the revised mass-energy relation and v2/c2 comes from the cessation of using the Lorentz transformation of special relativity. Obviously, those factors act in opposite directions for subluminal speed v, and thus we might see the corrected wavelength to be close to the value calculated with the original de Broglie formula for a range of speed v. For example, when v = (√2/2)c, we will have the same wavelength from (6a) as from (1a).
3. Further Remarks
In his landmark paper of 1926 for his celebrated equation [[6]], Schrödinger began with the claim that his study was based on the researches of de Broglie on what he called “phase-waves”, but ended with the claim that he did not employed the relativity modifications of classical mechanics. Consequently, Schrödinger’s undulatory mechanics would not be affected by the giving up of special relativity as de Broglie’s work on matter waves would. Nevertheless, the works of both de Broglie and Schrödinger on matter waves have been considered at least qualitatively confirmed by experiments. However, the deterministic views of both de Broglie and Schrödinger concerning quantum waves were soon challenged by Born’s statistical view [[7]] and then overthrown by the whole academia of physics that has been dominated by the so-called Copenhagen interpretation [[8]], according to which the quantum waves as initially investigated by de Broglie and Schrödinger are actually waves of probability instead of physical waves or motions of physical oscillators. In protest against the Copenhagen interpretation, Schrödinger pulled off his famous cat theory, which was then once again interpreted by the academia as a support to the Copenhagen interpretation.
However, the contention between Schrödinger and the Copenhagen school might be far from an end as the academia has supposed. While the 2022 Nobel Prize in Physics apparently sounds like the affirmation of the non-local statistical view of the Copenhagen interpretation, the 2023 Nobel Prize in Physics substantially sides with those proponents of the deterministic local view like Schrödinger by rewarding the experiments that help to deterministically capture the motions of electrons [[9]].
Now we are facing a very interesting situation: two philosophically opposite quantum interpretations seem to be experimentally proved both right and both wrong. The only rational interpretation to this situation should be that the existing human understanding of the quantum world is not complete, which would help to end another decades long argument in the quantum field. The nonlocal-local or statistical-deterministic dilemma obviously indicate the existence of some unknown variables for the quantum realm. Although these variables might not be literally the hidden variables referred to by Einstein, Podolsky, and Rosen in 1934 [[10]], at least the existence of these variables would concur with their claim that the existing quantum mechanics framed by the Copenhagen interpretation is not complete.
By the way, Schrodinger’s cat theory has a logical defect [[11]] due to his implicit assumption without proof that the macroscopic cause-effect relationships could be directly applied to the quantum superposition states. That defect would not invalidate his challenge against Born’s statistical interpretation of his undulatory mechanics, but could become a serious flaw if one takes the cat theory as fundamentally reflecting the nature of the quantum world.
References
[[1]] De Brogile, L. (1925). “On the Theory of Quanta”. Ann. de Phys., 10e s´erie, t. III (Janvier-F ´evrier 1925. Translated by A. F. Kracklauer 2004. Retrieved from: https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf
[[2]] Wikipedia. “Matter wave”. Retrieved from: https://en.wikipedia.org/wiki/Matter_wave. Last edited on 6 December 2023, at 10:29 (UTC).
[[3]] 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
[[4]] Dai, R. (2022). The Fall of Special Relativity and The Absoluteness of Space and Time. Retrieved from https://www.researchgate.net/publication/363582341_The_Fall_of_Special_Relativity_and_The_Absoluteness_of_Space_and_Time
[[5]] Dai, R. (2023). “Modifying Mass-Energy Relationship”. Retrieved from: https://www.researchgate.net/publication/370288880_Modifying_Mass-Energy_Relationship
[[6]] Schrödinger, 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
[[7]] Born, M (1926). “Zur Quantenmechanik der Stoßvorgänge” [On the quantum mechanics of collisions]. Zeitschrift für Physik. 37 (12): 863–867. Bibcode:1926ZPhy…37..863B. doi:10.1007/BF01397477. Reprinted as Born, Max (1983). “On the quantum mechanics of collisions”. In Wheeler, J. A.; Zurek, W. H. (eds.). Quantum Theory and Measurement. Princeton University Press. pp. 52–55. ISBN 978-0-691-08316-2. Retrieved from: http://www.ffn.ub.es/luisnavarro/nuevo_maletin/Born_1926_statistical_interpretation.pdf
[[8]] Wikipedia. “Copenhagen interpretation”. Retrieved from: https://en.wikipedia.org/wiki/Copenhagen_interpretation”. Last edited on 28 November 2023, at 03:37 (UTC).
[[9]] Dai, R. (2023). “Attosecond or Copenhagen?”. Retrieved from: https://www.researchgate.net/publication/374740086_Attosecond_or_Copenhagen
[[10]] Wikipedia. “Einstein–Podolsky–Rosen paradox”. Retrieved from: https://en.wikipedia.org/wiki/Einstein%E2%80%93Podolsky%E2%80%93Rosen_paradox. Last edited on 15 September 2023, at 16:01 (UTC).
[[11]] Dai, R. (2023). “The Genetic Defect of Schrödinger’s Cat”. Retrieved from: https://www.academia.edu/104037882/The_Genetic_Defect_of_Schr%C3%B6dingers_Cat
The World of Science and Science fiction of Ron Dai 
Leave a comment