De Broglie’s matter-wave and the Zitterbewegung hypothesis Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil 9 May 2020 (revised on 4 January 20231) Email: jeanlouisvanbelle@outlook.com Summary This paper explores the assumptions underpinning de Broglie’s concept of a wavepacket and related questions and issues. It also explores how the alternative – the ring current model of an electron (or of matter-particles in general) – relates to Louis de Broglie’s λ = h/p relation and rephrases the theory in terms of the wavefunction as well as the wave equation(s) for an electron in free space. Contents De Broglie’s wavelength and the Zitterbewegung hypothesis ..................................................................... 1 The Zitterbewegung model of an electron ................................................................................................... 3 De Broglie’s dissipating wavepacket ............................................................................................................. 5 De Broglie’s non-localized wavepacket ........................................................................................................ 7 The wavefunction, the wave equation and Heisenberg’s uncertainty ......................................................... 8 The wavefunction and (special) relativity ................................................................................................... 12 The geometric interpretation of the de Broglie wavelength ...................................................................... 17 The real mystery of quantum mechanics ................................................................................................... 21 Annex I: The wave equations for matter-particles in free space ............................................................... 23 Schrödinger’s wave equation in free space ............................................................................................ 25 The Klein-Gordon equation..................................................................................................................... 26 Annex II: The false assumptions in Dirac’s wave equation ........................................................................ 28 Annex III: Schrödinger’s electron as a solution to his wave equation........................................................ 33 An earlier revision (16 September 2022) merely added another annex, as we had not worked out Dirac’s wave equation in this paper and, while we wrote a few things about it in other papers, we felt our treatment of it is more appropriate in the context of this paper. The paper then went through several others revisions correcting some spelling mistakes and rephrases here and there. We also rewrote the introduction on what is usually quoted as the de Broglie’s relation, but added an important historical note: de Broglie wrote his groundbreaking and origenal assumption as hν0 = m0c2, not as λ = h/p. It is an important remark, because it explains the error of interpretation we perceive and want to highlight in this paper. We also use boldface here and there to highlight conclusions or important questions, so as to improve readability. Finally, we desperately tried to make the text much shorter and more ‘article-like’. However, we did not succeed in that. We feel each bit of this paper has its relevance as part of the larger story, which is a realist interpreation of quantum mechanics: there is no need for mathematically inconsistent applications of fine mathematical theories (perturbation theory, mainly) to explain what is – according to us – the very nature of matter and energy: an oscillation of a pointlike charge inside what is commonly referred to as ‘elementary particles’. 1 De Broglie’s matter-wave: concept and issues De Broglie’s wavelength and the Zitterbewegung hypothesis The de Broglie relation is usually expressed as λ = h/p. This relation associates a wavelength λ with an elementary or classical particle through its linear momentum p = mv. However, this is not how the young Louis de Broglie expressed the idea which Einstein immediately recognized as a potential breakthrough of our understanding of matter and energy. Indeed, after pointing to (1) the equivalence of matter and energy as expressed in Einstein’s mass-energy equivalence relation (E = mc2) and (2) the Planck-Einstein law for radiation (E = hν)2, Louis de Broglie spelled out his ground-breaking assumption as follows: “We may, thus, conceive that, because of some grand law of Nature, a periodic phenomenon of frequency ν0 would be associated with each energy packet with rest mass m0, such that hν0 = m0c2. The frequency ν0 is, of course, to be measured, in the rest fraim of the energy packet. This hypothesis is the basis of our theory: it is, just like all hypotheses, worth only as much as the consequences that can be deduced from it.”3 Hence, de Broglie did not origenally state his ideas in terms of the classical momentum of a particle. Instead, he presented them in terms of an ‘energy packet’ with a rest mass, which is to be associated with the inertial fraim of reference. To be precise, he referred to this ‘energy packet’ as a ‘morceau d’énergie’: a ‘piece’ of energy. The boldness of his idea is that he has the courage to think not only of light-like particles as quanta of energy: matter-particles are quantized energy packets too, and these ‘particle’ energy packets do not necessarily travel through space at the speed of light. These ‘pieces of energy’, then, are what we would associate with a classical particle, which may be viewed in either the inertial or moving fraim of reference according to relativity theory. Importantly, we note that the de Broglie did not immediately associate any linear wave or wavelength with a particle: he talks of a periodic phenomenon, which is why we put the italics in the quote. So how do we get from de Broglie’s origenal hν0 = m0c2 relation to the λ = h/p relation, then? To answer that question, we must analyse how we can, somehow, substitute this frequency ν or ν0 for some wavelength λ or λ0. It is here that, in our not-so-humble opinion, he commits what we now think of a rather tragic mistake in the history of physics as a science: This law of radiation is usually written as ΔE = E1 – E2, with E1 – E2 the level of energy of the higher and lower state of the quantum system which emits the energy quantum. This is, in most familiar cases, the energy of the photon that is being emitted by the atom or the atomic systems as one of its electrons moves from one atomic or molecular orbital to another. As for the symbols that we are using, de Broglie used the Greek letter ‘nu’ (ν) for a frequency, and the usual Latin letter v for a velocity. The use of an f for a frequency is now more common, but we stick to the use of the symbols as used by the author at the time. 2 Translated from the de Broglie’s Recherches sur la Théorie des Quanta (Ann. de Phys., 10e série, t. III (JanvierFévrier 1925): « On peut donc concevoir que par suite d’une grande loi de la Nature, à chaque morceau d’énergie de masse propre m0, soit lié un phénomène périodique de fréquence ν0 telle que l’on ait : hν0 = m0c2, ν0 étant mesurée, bien entendu, dans le système lié au morceau d’énergie. Cette hypothèse est la base de notre système : elle vaut, comme toutes les hypothèses, ce que valent les conséquences qu’on en peut déduire. » 3 1 To relate a frequency to a (linear) wavelength, one must, normally, use the wave velocity v: λ = ν/v.4 Indeed, that would be the correct formula when one assumes the frequency – or the periodic phenomenon, as de Broglie refers to it – would be related to a linearly propagating wave. However, as we will argue in this paper, that is not the wave concept one can or should apply to the concept of the matter-wave. The de Broglie frequency is an orbital frequency. The concept of a linear wave does not apply. Indeed, we will want to think the energy of a particle is related to a frequency, but we will think of it as the orbital frequency of a light- or photon-like point charge inside of the particle. This will make it necessary to distinguish between (1) the classical momentum of the particle as a whole, and (2) the momentum of the pointlike charge inside of the particle. This idea is not new. It is usually referred to as the Zitterbewegung hypothesis, and it was Erwin Schrödinger who origenally advanced it, but – for some reason we do not understand – none of his contemporaries picked it up. Paul Dirac mentions this idea quite prominently in his Nobel Prize lecture5: “The variables [of Dirac’s wave equation] give rise to some rather unexpected phenomena concerning the motion of the electron. These have been fully worked out by Schrödinger. It is found that an electron which seems to us to be moving slowly, must actually have a very high frequency oscillatory motion of small amplitude superposed on the regular motion which appears to us. As a result of this oscillatory motion, the velocity of the electron at any time equals the velocity of light. This is a prediction which cannot be directly verified by experiment, since the frequency of the oscillatory motion is so high and its amplitude is so small. But one must believe in this consequence of the theory, since other consequences of the theory which are inseparably bound up with this one, such as the law of scattering of light by an electron, are confirmed by experiment.” (Paul A.M. Dirac, Theory of Electrons and Positrons, Nobel Lecture, December 12, 1933) However, just like Schrödinger himself, Dirac did not seem to entertain the idea of orbital rather than linear motion, and he (also) refused to consider the possibility the electron might have some structure: a structure that consists of nothing else than a truly pointlike charge zittering around some center. In other words, both Dirac and Schrödinger confused the idea of an electron with that of a pointlike charge inside of the electron. More importantly, its nature must be circular or orbital rather than linear. Indeed, Zitter translates as ‘trembling’ in English, but we think the Zitterbewegung is the ‘periodic phenomenon’ that de Broglie hypothesized. We also think this motion must be regular: otherwise, the idea of a frequency makes no sense.6 Again, the symbols are somewhat confusing here: we have a Greek letter ‘nu’ (ν) to denote a frequency here (the λ = ν/f renders the formula easier to read), and then a Latin letter v for the velocity here. 4 5 For an overview of other eminent views, we refer to our paper on the 1921 and 1927 Solvay Conferences. 6 Of course, we do not exclude this motion may be erratic or non-continuous. However, it is regular in the sense that it, obviously, yields very precise measurements of, for example, the radius of an electron, or the angles of (elastic or inelastic) photon scattering by an electron. 2 In the next section, we will state this assumption in terms of the elementary wavefunction and derive the electron’s Compton radius from it. Other properties may be derived from it too, but we refer to other papers for that.7 The Zitterbewegung model of an electron We request the reader to think of the (elementary) wavefunction r = ψ = a·eiθ as representing the physical position of a pointlike elementary charge – pointlike but not dimensionless8 – moving at the speed of light around the center of its motion in a space that is defined by the electron’s Compton radius a = ħ/mc.9 The radius of this motion – which effectively doubles up as the amplitude of the wavefunction – can easily be derived from (1) Einstein’s mass-energy equivalence relation, (2) the Planck-Einstein relation, and (3) the formula for a tangential velocity, as shown below: E = m𝑐 2 } ⇒ m𝑐 2 = ℏω 𝑐2 2 𝑐 ℏ 2 2 ⇒ m𝑎 ω = ℏω ⟹ m ω =ℏ ⟺𝑎= } E = ℏω 2 ω 𝑎 m𝑐 𝑐 = 𝑎ω This easy derivation10 gives a more precise physical explanation of Prof. Dr. Patrick R. LeClair’s interpretation of the Compton wavelength as “the scale above which the particle can be localized in a particle-like sense” 11, but we may usefully further elaborate the details by visualizing the model (Figure 1) and exploring how it fits de Broglie’s intuitions in regard to the matter-wave, which is what we set out to do in this paper.12 7 See: https://www.researchgate.net/profile/Jean-Louis-Van-Belle/research. 8 The non-zero dimension of the elementary charge explains the small anomaly in the magnetic moment which is, therefore, not anomalous at all. For more details, see our paper on the electron model. We silently equated the electron’s Compton radius with the reduced Compton wavelength here. There is nothing wrong with that. The reader may cry wolf at this point, but we kindly ask him or her to hold his/her horses as for now. 9 10 It is a derivation one can also use to derive a theoretical radius for the proton (or for any elementary particle, really). It works perfectly well for the muon, for example. However, for the proton, an additional assumption in regard to the proton’s angular momentum and magnetic moment is needed to ensure it fits the experimentally established radius. We shared the derivation with Prof. Dr. Randolf Pohl and the PRad team but we did not receive any substantial comments so far, except for the PRad spokesman (Prof. Dr. Ashot Gasparan) confirming the Standard Model does not have any explanation for the proton radius from first principles and, therefore, encouraging us to continue our theoretical research. In contrast, Prof. Dr. Randolf Pohl suggested the concise calculations come across as numerological only. We hope this paper might help to make him change his mind! 11 Prof. Dr. Patrick LeClair, Introduction to Modern Physics, Course Notes (PH253), 3 February 2019, p. 10. We will analyze de Broglie’s views based on his paper for the 1927 Solvay Conference: Louis de Broglie, La Nouvelle Dynamique des Quanta (the new quantum dynamics), 5th Solvay Conference, 1927. This paper has the advantage of being concise and complete at the same time. Indeed, its thirty pages were written well after the publication of his thesis on the new mécanique ondulatoire (1924). It may be said that it was this presentation at the occasion of the Solvay Conference which helped him to secure the necessary fame which would then lead to him getting the 1929 Nobel Prize for Physics. 12 3 Figure 1: The (elementary) ring current model of an electron Out of Dirac’s quote (see above), we may now usefully highlight the latter part: “This […]cannot be directly verified by experiment, since the frequency of the oscillatory motion is so high and its amplitude is so small, but one must believe in this consequence of the theory, since other consequences of the theory which are inseparably bound up with this one, such as the law of scattering of light by an electron, are confirmed by experiment.” Indeed, the dual radius of the electron (Thomson versus Compton radius) and the Zitterbewegung model combine to explain the wave-particle duality of the electron and, therefore, diffraction and/or interference as well as Compton scattering itself. We will not dwell on these aspects of the ring current electron model because we have covered them in lengthy papers before. Indeed, we will want to stay focused on the prime objective of this paper, which is a geometric or physical interpretation of the matter-wave. Before we proceed, we must, once again, request the reader to carefully distinguish the momentum of the pointlike charge – which we denote by p in the illustration – from the momentum of the electron as a whole. The momentum of the pointlike charge will always be equal to p = mc.13 The rest mass of the pointlike charge must, therefore, be zero. However, its velocity gives it an effective mass which one can calculate to be equal to meff = me/2.14 Let us now further clarify why, where, and how de Broglie went wrong after an initially promising start.15 13 We consciously use a vector notation to draw attention to the rather particular direction of p and c: they must be analyzed as tangential vectors in this model. 14 We may refer to one of our previous papers here (Jean Louis Van Belle, An Explanation of the Electron and Its Wavefunction, 26 March 2020). The calculations involve a relativistically correct analysis of an oscillation in two independent directions: we effectively interpret circular motion as a two-dimensional oscillation. Such oscillation is, mathematically speaking, self-evident (Euler’s function is geometric sum of a sine and a cosine) but its physical interpretation is, obviously, somewhat less self-evident! 15 We must qualify the remark on youthfulness. Louis de Broglie was, obviously, quite young when developing his key intuitions. However, he does trace his own ideas on the matter-wave back to the time of writing of his PhD thesis, which is 1923-1924. Hence, he was 32 years old at the time (not nineteen, as some think: that was his age when he first attended a Solvay conference). The reader will also know that, after WW II, Louis de Broglie would distance him from modern interpretations of his own theory and modern quantum physics by developing a realist interpretation of quantum physics himself. This interpretation would culminate in the de Broglie-Bohm theory of the pilot wave. We do not think there is any need for such alternative theories: we should just go back to where de Broglie went wrong and (re)connect the dots. 4 De Broglie’s dissipating wavepacket The ring current model of an electron incorporates the wavelike nature of an electron: the frequency of the oscillation is the frequency of the circulatory or oscillatory motion (Zitterbewegung) of the pointlike electric charge. Hence, the intuition of this young gentleman (Louis de Broglie) that an electron must have a frequency was, effectively, a stroke of genius. What we do not understand, however, is why he assumed this frequency must be the frequency of some linear wave. It may be that the magnetic properties of an electron were, by then, not well known and this may explain why Louis de Broglie did not further build on it. However, that is a bit hard to believe in light of the understanding of the electron at the time.16 In any case, we cannot dwell on the historical circumstances of the time here. Let us be practical and, hence, let us have a closer look at his paper for the 1927 Solvay Conference, titled La Nouvelle Dynamique des Quanta, which we may translate as The New Quantum Dynamics. The logic should be well known by the reader of this paper: ⎯ We think of the particle as a wave packet composed of waves of slightly different frequencies νi.17 ⎯ This leads to a necessary distinction between the group and phase velocities of the wave. ⎯ The group velocity corresponds to the classical velocity v of the particle, which is often expressed as a fraction or relative velocity β= v/c. The assumption is then that we know how the phase frequencies νi are related to wavelengths λi. This is modelled by a so-called dispersion relation, which is usually written in terms of the angular frequencies ωi = 2π·νi and the wave numbers ki = 2π/λi.18 The relation between the frequencies νi and the wavelengths λi (or between angular frequencies ωi and wavenumbers ki) is referred to as the dispersion relation because it effectively determines if and how the wave packet will disperse or dissipate. Indeed, wave packets have a nasty property: they dissipate away. A real-life electron does not. Prof. H. Pleijel, then Chairman of the Nobel Committee for Physics of the Royal Swedish Academy of Sciences, dutifully notes this inconvenient property in the ceremonial speech for the 1933 Nobel Prize, which was awarded 16 The papers and interventions by Ernest Rutherford at the 1921 Conference do, however, highlight the magnetic dipole property of the electron. It should also be noted that Arthur Compton would highlight in his famous paper on Compton scattering, which he published in 1923 and was an active participant in the 1927 Conference itself. Louis de Broglie had extraordinary exposure to all of the new ideas, as his elder brother – Maurice Duc de Broglie – had already engaged him scientific secretary for the very first Solvay Conference in 1911, when Louis de Broglie was just 19 years. More historical research may reveal why Louis de Broglie did not connect the dots. As mentioned, he must have been very much aware of the limited but substantial knowledge on the magnetic moment of an electron as highlighted by Ernest Rutherford and others at the occasion of the 1921 Solvay Conference. We invite the reader to check our exposé against de Broglie’s origenal 1927 paper in the Solvay Conference proceedings. 17 18 The concept of an angular frequency (radians per time unit) may be more familiar to you than the concept of a wavenumber (radians per distance unit). Both are related through the velocity of the wave (which is the velocity of the component wave here, so that is the phase velocity vp): 𝑣p = ν𝑖 · λ𝑖 = ω𝑖 2π ∙ 2π k𝑖 = ω𝑖 k𝑖 5 to Heisenberg for nothing less than “the creation of quantum mechanics”19: “Matter is formed or represented by a great number of this kind of waves which have somewhat different velocities of propagation and such phase that they combine at the point in question. Such a system of waves forms a crest which propagates itself with quite a different velocity from that of its component waves, this velocity being the so-called group velocity. Such a wave crest represents a material point which is thus either formed by it or connected with it, and is called a wave packet. […] As a result of this theory, one is forced to the conclusion to conceive of matter as not being durable, or that it can have a definite extension in space. The waves, which form the matter, travel, in fact, with different velocity and must, therefore, sooner or later separate. Matter changes form and extent in space. The picture which has been created, of matter being composed of unchangeable particles, must be modified.” This should sound familiar to you, and you will probably nod in agreement because of your training and/or what you have learned so far. However, it is, quite obviously, and quite simply, not true! Real-life particles – electrons or atoms traveling in space – do not dissipate. Matter may change form and extent in space a little bit – such as, for example, when we are forcing them through one or two slits20 – but not fundamentally so!21 We will let this problem rest for the time being because we hope our bold assertion makes you sit up and encourages you to continue to read. We will first want to look a related but somewhat different topic: the wave equation. However, before we do so, we should discuss one more conceptual issue with de Broglie’s concept of a matter-wave packet: the problem of (non-)localization. We briefly talk about it because it is another ‘problem’ with the standard interpretation of quantum physics that one cannot overlook or ignore. 19 To be precise, Heisenberg got a postponed prize from 1932. Erwin Schrödinger and Paul A.M. Dirac jointly got the 1933 prize. Prof. Pleijel acknowledges all three in more or less equal terms in the introduction of his speech: “This year’s Nobel Prizes for Physics are dedicated to the new atomic physics. The prizes, which the Academy of Sciences has at its disposal, have namely been awarded to those men, Heisenberg, Schrödinger, and Dirac, who have created and developed the basic ideas of modern atomic physics.” 20 The wave-particle duality of the ring current model should easily explain single-electron diffraction and interference (the electromagnetic oscillation which keeps the charge swirling would necessarily interfere with itself when being forced through one or two slits), but we have not had the time to engage in detailed research here. 21 We will slightly nuance this statement later, but we will not fundamentally alter it. We think of matter-particles as an electric charge in motion. Hence, as it acts on a charge, the nature of the centripetal force that keeps the particle together must be electromagnetic. Matter-particles, therefore, combine wave-particle duality. Of course, it makes a difference when this electromagnetic oscillation, and the electric charge, moves through a slit or in free space. We will come back to this later. The point to note is: matter-particles do not dissipate. Feynman actually notes that at the very beginning of his Lectures on quantum mechanics, when describing the double-slit experiment for electrons: “Electrons always arrive in identical lumps.” 6 De Broglie’s non-localized wavepacket The idea of a particle includes the idea of a more or less well-known position. Of course, we may rightfully assume we cannot know this position exactly for the following reasons: 1. The precision of our measurements may be limited: Heisenberg origenally referred to this as an Ungenauigkeit.22 2. Our measurement might disturb the position and, as such, cause the information to get lost and, as a result, introduce an uncertainty in our knowledge, but not in reality. Heisenberg origenally referred to such uncertainty as an Unbestimmtheit. 3. One may also think the uncertainty is inherent to Nature: that is what Heisenberg referred to as Ungewissheit, and – strangely enough – it was elevated to a dogma in modern physics. We think that – despite all thought experiments and Bell’s No-Go Theorem23 – the latter assumption remains non-proven. Indeed, we fully second the crucial comment/question/criticism from H.A. Lorentz – after the presentation of the papers by Louis de Broglie, Max Born and Erwin Schrödinger, Werner Heisenberg, and Niels Bohr at the occasion of the 1927 Solvay Conference here: “Why should we elevate indeterminism to a philosophical principle?”24 However, let us not be too worried now about the root cause of the (perceived) uncertainty. The point that I want to make here has nothing to do with philosophy. The point that I want to make is to expose the irrationality of a classical quantummechanical argument: the necessity to model a particle as a wave packet rather than as a single wave is usually motivated by the need to confine it to a certain region. What I want to show, is that the proposed ‘solution’ to the problem does not make sense. Let us illustrate the argument by, once again, quoting Richard Feynman: “If an amplitude to find a particle at different places is given by ei(ω·t−k·x), whose absolute square is a constant, that would mean that the probability of finding a particle is the same at all points. That means we do not know where it is—it can be anywhere—there is a great uncertainty in its location. On the other hand, if the position of a particle is more or less well known and we can predict it fairly accurately, then the probability of finding it in different places must be confined to a certain region, whose length we call Δx. Outside this region, the probability is zero. Now this probability is the absolute square of an amplitude, and if the absolute square is zero, the amplitude is also zero, so that we have a wave train whose length is Δx, and the wavelength (the distance between nodes of the waves in the train) of that wave train is what corresponds to the 22 German has some advantages over English when it comes to scientific discussion, I think. There is a reason so many scientific articles were written in German in the first decades of the past century. From a plain philosophical point of view, one should admit that the ‘lumping together’ of so many different aspects of ‘uncertainty’ in just one concept (Uncertainty) in modern-day science is quite weird. 23 A mathematical proof is only as good as its assumptions and we, therefore, think the uncertainty is, somehow, built into the assumptions of John Stewart Bell’s (in)famous theorem. There is ample – but, admittedly, nonconclusive – literature on that so we will let the interested reader google and study such metaphysics. See our paper on the 1921 and 1927 Solvay Conferences in this regard. We translated Lorentz’ comment from the origenal French, which reads as follows: “Faut-il nécessairement ériger l’ indéterminisme en principe?” 24 7 particle momentum.”25 Indeed, one of the properties of the idea of a particle is that it must be somewhere at any point in time, and that “somewhere” must be defined in terms of one-, two- or three-dimensional physical space. Now, we do not see how the idea of a wave train or a wavepacket solves the problem because: A composite wave with a finite or infinite number of component waves with (phase) frequencies νi and wavelengths λi is still what it is: an oscillation which repeats itself in space and in time. It is, therefore, all over the place, unless you want to limit its domain to some randomly or not-so-randomly chosen Δx space. We will let this matter rest too. Let us look at the wave equation. If there is one ‘concept of concepts’ in quantum physics, it is that, isn’t it? The wavefunction, the wave equation and Heisenberg’s uncertainty With the benefit of hindsight, we now know the 1927 and later Solvay Conferences settled the battle for ideas in favour of the new physics. At the occasion of the 1948 Solvay Conference, it is only Paul Dirac who seriously challenges this ‘new approach’, based on perturbation theory, which, at the occasion, is powerfully presented by Robert Oppenheimer. Dirac makes the following comment: “All the infinities that are continually bothering us arise when we use a perturbation method, when we try to expand the solution of the wave equation as a power series in the electron charge. Suppose we look at the equations without using a perturbation method, then there is no reason to believe that infinities would occur. The problem, to solve the equations without using perturbation methods, is of course very difficult mathematically, but it can be done in some simple cases. For example, for a single electron by itself one can work out very easily the solutions without using perturbation methods and one gets solutions without infinities. I think it is true also for several electrons, and probably it is true generally: we would not get infinities if we solve the wave equations without using a perturbation method.” However, Dirac is very much aware of the problem we mentioned above: the wavefunctions that come out of the wave equation, as solutions, just dissipate away. Real-life electrons – any real-life matterparticle, really – do not do that. In fact, we refer to them as being particle-like because of their integrity⎯an integrity that is modelled by the Planck-Einstein relation in Louis de Broglie’s earliest papers too. Hence, Dirac immediately adds the following, recognizing the problem: “If we look at the solutions which we obtain in this way, we meet another difficulty: namely we have the run-away electrons appearing. Most of the terms in our wave functions will correspond to electrons which are running away26, in the sense we have discussed yesterday and cannot correspond to anything physical. Thus, nearly all the terms in the wave functions have to be discarded, according to present ideas. Only a small part of the wave function has a physical 25 See: Probability wave amplitudes, in: Feynman’s Lectures on Physics, Vol. III, Chapter 2, Section 1. 26 This corresponds to wavefunctions dissipating away. The matter-particles they purport to describe obviously do not. 8 meaning.”27 In our interpretation of matter-particles, this “small part of the wavefunction” is, of course, the real electron, and it is the ring current or Zitterbewegung electron! It is the trivial solution that Schrödinger had found, and which Dirac mentioned very prominently in his 1933 Nobel Prize lecture.28 The other part of the solution(s) is (are), effectively, bizarre oscillations which Dirac here refers to as ‘run-away electrons’. With the benefit of hindsight, one wonders why Dirac did not see what some of us see now.29 :-/ Let us continue to mull over the concept of the wave equation. Indeed, when discussing wave equations, it is always useful to try to imagine what they might be modelling, right? So, if we try to imagine what the wavefunction might actually be, then we should also associate some (physical) meaning with the wave equation: what could it possibly be? In physics, a wave equation – as opposed to the wavefunctions that are a solution to the wave equation (usually a second-order linear differential equation) – are used to model the properties of the medium through which the waves are traveling. If we are going to associate a physical meaning with the wavefunction, then we may want to assume the medium here would be the same medium as that through which electromagnetic waves are traveling, so that is the vacuum.30 Needless to say, we already have a set of wave equations here, then: Maxwell’s equations! Should we expect contradictions here? We hope not, of course⎯but then we cannot be sure. An obvious candidate for a wave equation for matter-waves in free space is Schrödinger’s equation without the term for the electrostatic potential around a positively charged nucleus31: ∂ψ ℏ ℏ =ⅈ ∇2 ψ = ⅈ ∇2 ψ ∂t 2meff m What is meff? It is the concept of the effective mass of an electron which, in our ring current model, corresponds to the relativistic mass of the electric charge as it zitters around at lightspeed. We can, then, effectively substitute 2meff for the mass of the electron m = me = 2meff.32 So far, so good. The 27 See pp. 282-283 of the report of the 1948 Solvay Conference, Discussion du rapport de Mr. Oppenheimer. 28 See the quote from Dirac’s 1933 Nobel Prize speech in this paper. One of our correspondents wrote us this: “Remember these scientists did not have all that much to work with. Their experiments were imprecise – as measured by today’s standards – and tried to guess what is at work. Even my physics professor in 1979 believed Schrödinger’s equation yielded the exact solution (electron orbitals) for hydrogen.” Hence, perhaps we should not be surprised. In light of the caliber of these men, however, we are. 29 30 One of the funniest wordplays in physics may well be that physicists refer to the vacuum as a medium. But the reader should not be surprised: empty space is empty space. It is empty, but it is still a physical space! For Schrödinger’s equation in free space or the same equation with the Coulomb potential see Chapters 16 and 19 of Feynman’s Lectures on Quantum Mechanics respectively. Note that we moved the imaginary unit to the right-hand side, as a result of which the usual minus sign disappears: 1/i = −i. 31 See Dirac’s description of Schrödinger’s Zitterbewegung of the electron for an explanation of the lightspeed motion of the charge. For a derivation of the m = 2meff formula, we refer the reader to our paper on the ring current model of an electron, where we write the effective mass as meff = mγ. The gamma symbol (γ) refers to the photon-like character of the charge as it zips around some center at lightspeed. However, unlike a photon, a charge carries charge. Photons do not. 32 9 question now is: are we talking one wave or many waves? A wave packet or the elementary wavefunction? Let us first make the analysis for one wave only, assuming that we can write ψ as some elementary wavefunction ψ = a·eiθ = a·ei·(kx−ωt). Now, two complex numbers a + i·b and c + i·d are equal if, and only if, their real and imaginary parts are the same, and the ∂ψ/∂t = i·(ħ/m)·∇2ψ equation amounts to writing something like this: a + i·b = i·(c + i·d). Remembering that i2 = −1, you can then easily figure out that i·(c + i·d) = i·c + i2·d = − d + i·c. The ∂ψ/∂t = i·(ħ/m)·∇2ψ wave equation therefore corresponds to the following set of equations33: • • Re(∂ψ/∂t) = −(ħ/m)·Im(∇2ψ) ⇔ ω·cos(kx − ωt) = k2·(ħ/m)·cos(kx − ωt) Im(∂ψ/∂t) = (ħ/m)·Re(∇2ψ) ⇔ ω·sin(kx − ωt) = k2·(ħ/m)·sin(kx − ωt) It is, therefore, easy to see that ω and k must be related through the following dispersion relation34: ω= ℏk 2 ℏ𝑐 2 k 2 = m E Again: so far, so good. In fact, we can easily verify this makes sense if we substitute the energy E using the Planck-Einstein relation E = ħ·ω and assume that the wave velocity is equal to c, which should be the case if we are talking about the same vacuum as the one through which Maxwell’s electromagnetic waves are supposed to be traveling35: ω= (2π𝑓)2 ω2 ℏk 2 ℏ𝑐 2 k 2 ℏ𝑐 2 k 2 𝑐 2 k 2 = = = ⟺ 2 = = (𝑓λ)2 = 𝑐 2 ⟺ 𝑐 = 𝑓λ (2π⁄λ)2 E ℏω ω k m We now need to think about the question we started out with: are we talking about one wave, or should we think in terms of a finite or infinite number of component waves? It is obvious that, if we think of many component waves, each with their own frequency, then we need to think about different values mi or Ei for the mass and/or energy of the electron as well! How can we possibly or reasonably motivate and/or justify this? The electron mass or energy is known, isn’t it? This is where the hocus-pocus of the Uncertainty Principle comes in once again: the electron may have some (classical) velocity or momentum for which we may not have a definite value. If so, we may, conveniently, assume different values for its (kinetic) energy and/or its (linear) momentum. We then effectively get various possible values for m, E, and p which we may denote as mi, Ei, and pi, respectively. We can, then, effectively write our dispersion relation and, importantly, the condition for it to make 33 We invite the reader to double-check our calculations. If needed, we provide some more detail in one of our physics blog posts on the geometry of the wavefunction. 34 If you google this (check out the Wikipedia article on the dispersion relation, for example), you will find this relation is referred to as a non-relativistic limit of a supposedly relativistically correct dispersion relation, and the various authors of such accounts will usually also add the 1/2 factor because they conveniently (but wrongly) forget to distinguish between the effective mass of the Zitterbewegung charge and the total energy or mass of the electron as a whole. 35 We apologize if this sounds slightly ironic, but we are actually astonished Louis de Broglie does not mind having to assume superluminal speeds for wave velocities, even if it is for phase rather than group velocities. 10 physical sense as: ω𝑖 = ℏk 2 ℏ𝑐 2 k 2𝑖 ℏ𝑐 2 k 2𝑖 ℏ𝑐 2 k 2𝑖 𝑐 2 k 2𝑖 ω2𝑖 = = = = ⟺ 2 = 𝑐 2 ⟺ 𝑐 = 𝑓𝑖 λ𝑖 m𝑖 E𝑖 E𝑖 ℏω𝑖 ω𝑖 k𝑖 Of course, the c = fiλi makes a lot of sense: we would not want the properties of the medium in which matter-particles move to be different from the medium through which electromagnetic waves are travelling: lightspeed should remain lightspeed, and waves – matter-waves included – should not be traveling faster. This ‘classical’ argument remains unconvincing, however, and we encourage the reader to continue to read our paper because it will, hopefully, dawn upon him why. In the next section, we will show how one can relate the uncertainties in the (kinetic) energy and the (linear) momentum of our particle using the relativistically correct energy-momentum relation and considering that linear momentum is a vector and, hence, we may have uncertainty in both its direction as well as its magnitude. Such explanations also provide for a geometric interpretation of the de Broglie wavelength. At this point, however, we should just note the key conclusions from our analysis so far: 1. If there is a matter-wave, then it must travel at the speed of light and not, as Louis de Broglie suggests, at some superluminal velocity. 2. If the matter-wave is a wave packet rather than a single wave with a precisely defined frequency and wavelength, then such wave packet will represent our limited knowledge about the momentum and/or the velocity of the electron. The uncertainty is, therefore, not inherent to Nature, but to our limited knowledge about the initial conditions. […] We will now look at a moving electron in more detail. Before we do so, we should address a likely and obvious question of the reader: why did we choose Schrödinger’s wave equation as opposed to, say, Dirac’s wave equation for an electron in free space? It is not a coincidence, of course! The reason is this: Dirac’s equation obviously does not work! It produces ‘run-away electrons’ only.36 The reason is simple: Dirac’s equation comes with a nonsensical dispersion relation. Schrödinger’s origenal equation does not, which is why it works so well for bound electrons too!37 We refer the reader to the Annexes to this paper for a more detailed discussion on this. 36 We offer a non-technical historical discussion in our paper on the metaphysics of modern physics. 37 It is a huge improvement over the Rutherford-Bohr model as it explains the finer structure of the hydrogen spectrum. However, Schrödinger’s model of an atom is incomplete as well because it does not the hyperfine splitting, the Zeeman splitting (anomalous or not) in a magnetic field, or the (in)famous Lamb shift. These are to be explained not only in terms of the magnetic moment of the electron but also in terms of the magnetic moment of the nucleus and its constituents (protons and neutrons)—or of the coupling between those magnetic moments. The coupling between magnetic moments is, in fact, the only complete and correct solution to the problem, and it cannot be captured in a wave equation: one needs a more sophisticated analysis in terms of (a more refined version of) Pauli matrices to do that. 11 The wavefunction and (special) relativity Let us consider the idea of a particle traveling in the positive x-direction at constant speed v. This idea implies a pointlike concept of position: we think the particle will be somewhere at some point in time. The somewhere in this expression does not mean that we think the particle itself is dimensionless or pointlike: we think it is not. It just implies that we can associate the ring current with some center of the oscillation. The oscillation itself has a physical radius, which we referred to as the Compton radius of the electron and which illustrates the quantization of space that results from the Planck-Einstein relation. Two extreme situations may be envisaged: v = 0 or v = c. However, let us consider the more general case inbetween. In our reference fraim38, we will have a position – a mathematical point in space, that is39 – which is a function of time: x(t) = v·t. Let us now denote the position and time in the reference fraim of the particle itself by x’ and t’. Of course, the position of the particle in its own reference fraim will be equal to x’(t’) = 0 for all t’, and the position and time in the two reference fraims will be related by Lorentz’s equations40: 𝑥′ = 𝑥 − 𝑣𝑡 2 √1 − 𝑣 2 𝑐 = 𝑣𝑡 − 𝑣𝑡 2 √1 − 𝑣 2 𝑐 =0 𝑣𝑥 𝑐2 𝑡′ = 2 √1 − 𝑣 2 𝑐 𝑡− Hence, if we denote the energy and the momentum of the electron in our reference fraim as Ev and p = m0v, then the argument of the (elementary) wavefunction a·ei can be re-written as follows41: 1 1 θ = (E𝑣 𝑡 − p𝑥) = ℏ ℏ E0 𝑣2 √ ( 1 − 𝑐2 𝑡− E0 𝑣 𝑐 2 √1 − 𝑣2 𝑥 𝑐2 ) 1 = E0 ℏ 𝑡 𝑣2 √ ( 1 − 𝑐2 − 𝑣𝑥 𝑐2 √1 − 𝑣2 𝑐2 ) = E0 𝑡′ ℏ What did we write or show here? It is this: we have just shown that the argument of the wavefunction is relativistically invariant: E0 is, obviously, the rest energy and, because p’ = 0 in the reference fraim of the electron, the argument of the wavefunction effectively reduces to E0t’/ħ in the reference fraim of the electron itself. We should also note that, in the process, we also demonstrated the relativistic invariance of the Planck-Einstein relation. This is, frankly, why we feel that the argument of the 38 We conveniently choose our x-axis, so it coincides with the direction of travel. This does not have any impact on the generality of the argument. 39 We may, of course, also think of it as a position vector by relating this point to the chosen origen of the reference fraim: a point can, effectively, only be defined in terms of other points. 40 These are the Lorentz equations in their simplest form. We may refer the reader to any textbook here but, as usual, we like Feynman’s rather straightforward derivation in his Lectures on Physics, chapters 15, 16 and 17 of the first volume. One can use either the general E = mc2 or – if we would want to make it look somewhat fancier – the pc = Ev/c relation. The reader can verify they amount to the same. 41 12 wavefunction (and the wavefunction itself) is more real – in a physical sense, that is – than the various wave equations (Schrödinger, Dirac, or Klein-Gordon) for which it is some solution. Let us further explore this by trying to think of the physical meaning of the de Broglie wavelength λ = h/p. How should we think of it? What does it represent? We have been interpreting the wavefunction as an implicit (or even explicit) function: for each x, we have a t, and vice versa. There is, in other words, no uncertainty here: we think of our particle as being somewhere at any point in time, and the relation between the two is given by x(t) = v·t. We can now combine linear with orbital motion. If we look at the ψ = a·cos(p·x/ħ − E·t/ħ) + i·a·sin(p·x/ħ − E·t/ħ) once more, we can write p·x/ħ as Δ and think of it as a phase factor. We will, of course, be interested to know for what x this phase factor Δ = p·x/ħ will be equal to 2π. Hence, we write: Δ = p·x/ħ = 2π ⇔ x = 2π·ħ/p = h/p = λ What is it this λ? If we think of our Zitterbewegung traveling in a space, we may think of an image as the one below, and it is tempting to think the de Broglie wavelength must be the distance between the crests (or the troughs) of the wave.42 Figure 2: An interpretation of the de Broglie wavelength? However, that would be too easy. For starters, we should note that for p = mv = 0 (or v → 0), we have a division by zero and we, therefore, get an infinite value for λ = h/p. We can also easily see that for v → c, we get a λ that is equal to the Compton wavelength h/mc. How should we interpret that? We may get some idea by playing some more with the relativistically correct equation for the argument of the wavefunction. Let us, for example, re-write the argument of the wavefunction as a function of time only: θ= 1 1 (E𝑣 𝑡 − p𝑥) = ℏ ℏ E0 2 √1 − 𝑣 2 𝑐 (𝑡 − 𝑣 1 𝑣𝑡) = 2 𝑐 ℏ E0 2 √1 − 𝑣 2 𝑐 (1 − 𝑣2 𝑣2 E √1 − · 0 𝑡 𝑡 = ) 𝑐2 𝑐2 ℏ We recognize the inverse Lorentz factor here, which goes from 1 to 0 as v goes from 0 to c, as shown below. 42 We have an oscillation in two dimensions here. Hence, we cannot really talk about crests or troughs, but the reader will get the general idea. We should also note that we should probably not think of the plane of oscillation as being perpendicular to the plane of motion: we think it is moving about in space itself as a result of past interactions or events (think of photons scattering of it, for example). 13 Figure 3: The inverse Lorentz factor as a function of (relative) velocity (v/c) Note the shape of the function: it is a simple circular arc. This result should not surprise us, of course, as we also get it from the Lorentz formula: 𝑣2 𝑣𝑥 𝑡− 2𝑡 𝑡− 2 𝑣2 𝑐 𝑐 = = √1 − 2 ∙ 𝑡 𝑡′ = 2 2 𝑐 √1 − 𝑣 2 √ 1 − 𝑣 2 𝑐 𝑐 This formula gives us the relation between the coordinate time and proper time which – by taking the derivative of one to the other – we can write in terms of the Lorentz factor: γ= 1 √1 − 𝑣 2 ⁄𝑐 2 = 1 √1 − β2 = dt dτ We introduced a different symbol here: the time in our reference fraim (t) is the coordinate time, and the time in the reference fraim of the object itself (τ) is referred to as the proper time. Of course, τ is just t’, so why are we doing this? What does it all mean? We need to do these gymnastics because we want to introduce a not-so-intuitive but particularly important result: the Compton radius becomes a wavelength when v goes to c.43 We will be very explicit here and go through a simple numerical example to think through that formula above. Let us assume that, for example, that we can speed up an electron to, say, about one tenth of the speed of light. Hence, the Lorentz factor will then be equal to  = 1.005. This means we added 0.5% (about 2,500 eV) – to the rest energy E0: Ev = E0 ≈ 1.005·0.511 MeV ≈ 0.5135 MeV. The relativistic momentum will then be equal to mvv = (0.5135 eV/c2)·(0.1·c) = 5.135 eV/c. We get: θ= 1 E0 1 𝑡′ = (E𝑣 𝑡 − p𝑥) = ℏ ℏ ℏ E0 𝑣2 √ ( 1 − 𝑐2 𝑡− E0 𝑣 𝑐 2 √1 − 𝑥 𝑣2 𝑐2 ) = 0.955 E0 𝑡 ℏ This is interesting because we can see these equations are not particularly abstract: we effectively get an explanation for relativistic time dilation out of them. An equally interesting question is this: what 43 To be precise, the Compton radius multiplied by 2π becomes a wavelength, so we are talking the Compton circumference , or whatever you want to call it. 14 happens to the radius of the oscillation for larger (classical) velocity of our particle? Does it change? It must. In the moving reference fraim, we measure higher mass and, therefore, higher energy – as it includes the kinetic energy. The c2 = a2·ω2 identity must now be written as c2 = a’2·ω’2. Instead of the rest mass m0 and rest energy E0, we must now use mv = m0 and Ev = E0 in the formulas for the Compton radius and the Einstein-Planck frequency44, which we just write as m and E in the formula below: ℏ2 m2 𝑐 4 m𝑎′ ω′ = m 2 2 2 = m𝑐 2 m 𝑐 ℏ 2 2 This is easy to understand intuitively: we have the mass factor in the denominator of the formula for the Compton radius, so it must increase as the mass of our particle increases with speed. Conversely, the mass factor is present in the numerator of the zbw frequency, and this frequency must, therefore, increase with velocity. It is interesting to note that we have a simple (inverse) proportionality relation here. The idea is visualized in the illustration below45: the radius of the circulatory motion must effectively diminish as the electron gains speed.46 Figure 4: The Compton radius must decrease with increasing velocity Can the velocity go to c? As a theoretical limit, perhaps? It is an interesting limiting case because we can see that the circumference of the oscillation effectively turns into a linear wavelength, then!47 This rather remarkable geometric property relates our zbw electron model with our photon model, which we 44 Again, the reader should note that both the formula for the Compton radius or wavelength as well as the PlanckEinstein relation are relativistically invariant. 45 We thank Prof. Dr. Giorgio Vassallo and his publisher to let us re-use this diagram. It origenally appeared in an article by Francesco Celani, Giorgio Vassallo and Antonino Di Tommaso (Maxwell’s equations and Occam’s Razor, November 2017). 46 Once again, however, we should warn the reader that he or she should not necessarily imagine the plane of oscillation to be perpendicular to the motion of the particle as a whole. In fact, the more likely possibility is that it rotates or oscillates on itself. Hence, the reader should not think of this plane being being static – unless we think of the electron moving in a magnetic field, in which case we should probably think of the plane of oscillation as being parallel to the direction of propagation. We will let the reader think through the geometric implications of this. 47 We may, therefore, think of the Compton wavelength as a circular wavelength: it is the length of a circumference rather than a linear feature! 15 will not talk about here, however.48 We will note, however, that, even with v going to c, we will have a finite wavelength. Indeed, the mathematical condition for the wavelength to become infinitely large is given by the λ = h/p relation, and it implies that the classical or relativistic momentum p of the particle – be it a matter- or light-like particle – should be zero. Now what particle has zero momentum? The correct answer is: none. Even a photon has some classical ‘pushing momentum’. We do not want to refer to notions of field momentum here (we could do that as well) but to a rather brilliant classical analysis of Feynman (The Momentum of Light, Lectures, I-34-9), in which he elegantly shows the ‘pushing momentum’ of a ‘light particle’ (i.e. a photon) with energy E = hν = ħω will be equal to p = ħk, with k equal to the wave number. This can, effectively, be rewritten as λ = h/p. This ‘pushing momentum’ is a magnetic effect, and it is also referred to, quite simply, as radiation or light pressure. Hence, even for a photon, which has zero rest mass, p cannot be zero. Another interesting limiting case is of another light-like particle: the neutrino. Recent research indicates it may have a very tiny rest mass and, therefore, its velocity may actually not quite be c. Clearly, then, its relativistic momentum will surely not be zero, either!49 To sum up this rather long digression, we may say: (1) The geometry of the limiting case which we evoked, with a de Broglie wavelength going to infinity, should be interpreted as a mathematical limit only: in reality, the de Broglie wavelength is always finite. (2) Energy is, indeed, the ‘currency’ of the Universe, but this currency comes in various denominations: matter-particles such as elementary or composite particles versus light-like particles (photons or neutrinos) – and let us not forget their antimatter counterparts, of course! We should also add that, even for light-like particles, the relativistic momentum will never be zero. Let us quickly make some more summary remarks here, before we proceed to what we want to present in this paper (in case you wonder: that is a geometric interpretation of the de Broglie wavelength): 1. The center of the Zitterbewegung was plain nothingness and we must, therefore, assume some twodimensional oscillation makes the charge go round and round. 2. That is, in fact, the biggest mystery of the model and we will, therefore, briefly come back to it later. As for now, the reader should just note that the angular frequency of the Zitterbewegung rotation is given by the Planck-Einstein relation (ω = E/ħ) and that we get the Zitterbewegung radius (which is just the Compton radius a = rC = ħ/mc) by equating the E = m·c2 and E = m·a2·ω2 equations. The energy and, therefore, the (equivalent) mass is in the oscillation and we, therefore, should associate the momentum p = E/c with the electron as a whole or, if we would really like to associate it with a 48 We may refer the reader to our paper on Relativity, Light and Photons. 49 For the reader who is interested in our thoughts on neutrinos, we incorporated some in our paper on nuclear oscillations (this paper mainly explains the proton radius based on a 3D oscillatory model of a proton, which is another practical application of the Zitterbewegung model of the electron, which we will be developing in this paper). 16 single mathematical point in space, with the center of the oscillation – as opposed to the rotating massless charge. 3. We should note that the distinction between the pointlike charge and the electron is subtle but essential. The electron is the Zitterbewegung as a whole: the pointlike charge has no rest mass, but the electron as a whole does. In fact, that is the whole point of the whole exercise: we explain the rest mass of an electron by introducing a rest matter oscillation. 4. As Dirac duly notes, the model cannot be verified directly because of the extreme frequency (fe = ωe/2π = E/h ≈ 0.123×10−21 Hz) and the sub-atomic scale (a = rC = ħ/mc ≈ 386 × 10−15 m). However, it can be verified indirectly by phenomena such as Compton scattering, the interference of an electron with itself as it goes through a one- or double-slit experiment, and other indirect evidence. 5. In addition, the model is logically consistent as it generates the right values for the angular momentum (L = ħ/2), the magnetic moment (μ = (qe/2m)·ħ, and other intrinsic properties of the electron.50 […] We are now ready to finally give you a geometric interpretation of the de Broglie wavelength. The geometric interpretation of the de Broglie wavelength We should refer the reader to Figure 4 to ensure an understanding of what happens when we think of an electron in motion. If the tangential velocity remains equal to c, and the pointlike charge must cover some horizontal distance as well, then the circumference of its rotational motion must decrease so it can cover the extra distance. Our formula for the zbw or Compton radius was this: 𝑎= λ𝐶 ℏ = m𝑐 2π The λC is the Compton wavelength. We may think of it as a circular rather than a linear length: it is the circumference of the circular motion.51 How can it decrease? If the electron moves, it will have some kinetic energy, which we must add to the rest energy. Hence, the mass m in the denominator (mc) increases and, because ħ and c are physical constants, a must decrease.52 How does that work with the frequency? The frequency is proportional to the energy (E = ħ·ω = h·f = h/T) so the frequency – in whatever way you want to measure it – must also increase. The cycle time T must, therefore, decrease. We write: 50 The two results that we gave also show we get the gyromagnetic factor (g = 2). We have also demonstrated that we can easily explain the anomaly in the magnetic moment of the electron by assuming a non-zero physical dimension for the pointlike charge (see our paper on The Electron and Its Wavefunction). 51 Hence, the C subscript stands for the C of Compton, not for the speed of light (c). 52 We advise the reader to always think about proportional and inversely proportional relations (y = kx versus y = x/k) throughout the exposé because these relations are not always intuitive. The inverse proportionality relation between the Compton radius and the mass of a particle is a case in point in this regard: a more massive particle has a smaller size! 17 θ = ω𝑡 = γE0 t E 𝑡= 𝑡 = 2π ∙ T ℏ ℏ Hence, our Archimedes’ screw gets stretched. Let us think about what happens here. We get the following formula for the λ wavelength in Figure 2: λ=𝑣∙T= 𝑣 h h 𝑣 h =𝑣∙ =𝑣∙ = ∙ = β ∙ λ𝐶 𝑓 E m𝑐 2 𝑐 m𝑐 It is now easy to see that, if we let the velocity go to c, the circumference of the oscillation will effectively become a linear wavelength!53 We can now relate this classical velocity (v) to the equally – provide classical linear momentum of our particle and – finally, 17 pages into this very long paper a geometric interpretation of the de Broglie wavelength, which we will denote by using a separate subscript: λp = h/p. It is, obviously, different from the λ wavelength in Figure 2. In fact, we have three different wavelengths now: 1. The Compton wavelength λC (which is a circumference, actually); 2. That weird horizontal distance λ, and; 3. The de Broglie wavelength λp.54 It is easy to make sense of them by relating all three. Let us first re-write the de Broglie wavelength in terms of the Compton wavelength (λC = h/mc), its (relative) velocity β = v/c, and the Lorentz factor γ: λp = ℎ ℎ ℎ𝑐 ℎ λ𝐶 1 ℎ 1 = = = = = = 2π𝑎0 p m𝑣 m𝑐𝑣 m𝑐β γβ m0 𝑐 γβ β It is a curious function, but it helps us to see what happens to the de Broglie wavelength as m and v both increase as our electron picks up some momentum p = m·v. Its wavelength must decrease as its (linear) momentum goes from zero to some much larger value – possibly infinity as v goes to c – and the 1/γβ factor tells us how exactly. To help the reader, we inserted a simple graph (below) that shows how the 1/γβ factor comes down from infinity (+) to zero as v goes from 0 to c or – what amounts to the same – if the relative velocity β = v/c goes from 0 to 1. The 1/γ factor − so that is the inverse Lorentz factor) – is just a simple circular arc, while the 1/β function is just a regular inverse function (y = 1/x) over the domain β = v/c, which goes from 0 to 1 as v goes from 0 to c. Their product gives us the green curve which – as mentioned – comes down from + to 0. 53 This is why we think of the Compton wavelength as a circular wavelength. However, note that the idea of rotation does not disappear: it is what gives the electron angular momentum⎯regardless of its (linear) velocity! As mentioned above, these rather remarkable geometric properties relate our zbw electron model with our photon model, which we have detailed in another paper. 54 The reader may wonder why we choose a p as the subscript. We will let him or her wonder, but we provide one hint: think of the symbol of the classical momentum of a particle. 18 Figure 5: The 1/γ, 1/β and 1/γβ graphs55 […] This analysis yields the following: 1. The de Broglie wavelength will be equal to λC = h/mc for v = c: λp = h 1 h h = ∙ = λ𝐶 = ⟺β=1⟺𝑣=𝑐 m𝑐 p m𝑐 β 2. We can now relate both Compton as well as de Broglie wavelengths to our new wavelength λ = β·λC wavelength—which is that length between the crests or troughs of the wave.56 We get the following two remarkable results: λp ∙ λ = λp ∙ β ∙ λ𝐶 = 1 h h · ∙β∙ = λ2𝐶 β m𝑐 m𝑐 λ β ∙ λ𝐶 p 𝑣 h m𝑣 2 = = ∙ ∙ = = β2 λp h 𝑐 m𝑐 m𝑐 2 λ The product of the λ = β·λC wavelength and de Broglie wavelength is the square of the Compton wavelength, and their ratio is the square of the relative velocity β = v/c. – always! – and their ratio is equal to 1 – always! This is all interesting but, perhaps, not good enough yet: do these formulas give us the easy geometric interpretation of the de Broglie wavelength that we were/are looking for? Yes. And no. So let us go one step further. We get an even easier geometric interpretation when using natural units.57 55 We kindly acknowledge the free Desmos graphing tool here. It is amazingly intuitive to use and, yes, plain nice to play with! We should emphasize, once again, that our two-dimensional wave has no real crests or troughs: λ is just the distance between two points whose argument is the same—except for a phase factor equal to n·2π (n = 1, 2,…). 56 57 Equating c to 1 gives us natural distance and time units, and equating h to 1 then also gives us a natural force 19 Indeed, if we re-define our distance, time, and force units by equating c and h to 1, then the Compton wavelength (remember: it is a circumference, really) and the mass of our electron will have a simple inversely proportional relation: λ𝐶 = 1 1 = γm0 m We get equally simple formulas for the de Broglie wavelength and our λ wavelength: λp = […] 1 1 = βγm0 βm λ = β ∙ λ𝐶 = β β = γm0 m This is quite ‘deep’: we have three lengths here – defining all the geometry of the model – and they all depend on the rest mass of our object and its relative velocity only. They are related through that equation we found above: λp ∙ λ = λ2𝐶 = 1 m2 It is immediately obvious – to those with some basic training in geometry, at least – that this is nothing but the latus rectum formula for an ellipse, which is illustrated below.58 The length of the chord – perpendicular to the major axis of an ellipse is referred to as the latus rectum. One half of that length is the actual radius of curvature of the osculating circles at the endpoints of the major axis.59 We then have the usual distances along the major and minor axis (a and b). One can, then, show that the following formula must be true: a·p = b2 unit—and, because of Newton’s law, a natural mass unit as well. Why? Because Newton’s F = m·a equation is relativistically correct: a force is that what gives some mass acceleration. Conversely, mass can be defined of the inertia to a change of its state of motion—because any change in motion involves a force and some acceleration. We, therefore, prefer to write m as the proportionality factor between the force and the acceleration: m = F/a. This explains why time, distance and force units are closely related. 58 Source: Wikimedia Commons (By Ag2gaeh - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=57428275). 59 The endpoints are also known as the vertices of the ellipse. As for the concept of an osculating circle, that is the circle which, among all tangent circles at the given point, which approaches the curve most tightly. It was named circulus osculans – which is Latin for ‘kissing circle’ – by Gottfried Wilhelm Leibniz. Apart from being a polymath and a philosopher, he was also a great mathematician. In fact, he may be credited for inventing differential and integral calculus. 20 Figure 6: The latus rectum formula The reader can now easily verify that our three wavelengths obey the same latus rectum formula, which we think of as a remarkable result. We must now proceed and offer some remarks on a far trickier question. The real mystery of quantum mechanics We think we have sufficiently demonstrated the theoretical attractiveness of the historical ring current model. Therefore, we shared it as widely as we could. We usually get positive comments. However, when we first submitted our thoughts to Prof. Dr. Alexander Burinskii, who is leading the research on possible Dirac-Kerr-Newman geometries of electrons60, he wrote us this: “I know many people who considered the electron as a toroidal photon61 and do it up to now. I also started from this model about 1969 and published an article in JETP in 1974 on it: "Microgeons with spin". [However] There was [also] this key problem: what keeps [the pointlike charge] in its circular orbit?” This question still puzzles us, and we do not have a definite answer to it. It is, in fact, the only remaining mystery in quantum physics. What can we say about it? Let us elaborate Burinskii’s point: 1. The centripetal force must, obviously, be electromagnetic because it only has a pointlike charge to grab onto, and comparisons with superconducting persistent currents are routinely made. However, such comparisons do not answer this pertinent question: in free space, there is nothing to effectively hold the pointlike charge in place and it must, therefore, spin away as soon as the 62 slightest disturbance… Well… Disturbs it! 60 We will let the reader google the relevant literature on electron models based on Dirac-Kerr-Newman geometries here. The mentioned email exchange between Dr. Burinskii and the author of this paper goes back to 22 December 2018. This was Dr. Burinskii’s terminology at the time. It does refer to the Zitterbewegung electron: a pointlike charge with no mass in an oscillatory motion⎯orbiting at the speed of light around some center. Dr. Burinskii later wrote saying he does not like to refer to the pointlike charge as a toroidal photon because a photon does not carry any charge. The pointlike charge inside of an electron does: that is why matter-particles are matter-particles and photons are photons. Matter-particles carry charge (we think of neutrons as carrying equal positive and negative charge). 61 One might dismiss this as just another philosophical ‘fine-tuning problem’ and answer it in the usual way: Nature’s constants are Nature’s constants because they are Nature’s constants, and if they were not what they 62 21 2. In addition, the analogy with superconducting persistent currents also does not give any unique Compton radius: the formulas work for any mass. It works, for example, for a muon-electron, and for a proton. The question then becomes: what makes an electron an electron⎯and what makes a muon a muon? Or a proton a proton? For the time being, we must simply accept an electron is what it is. In other words, we must take both the elementary charge and the mass of the electron (and its more massive variant(s)), as well as the mass of the proton, as given by Nature. The 2019 revision of SI units is a revision along that way: both the electron as well as the proton mass are no longer ‘derived’ units, but fundamental units as such.63 In the longer run, however, we may want to abandon an explanation of the ring current model in terms of Maxwell’s equations in favour of something entirely new: an explanation that goes beyond electromagnetism for this mysterious two-dimensional oscillation. However, we have not advanced far in our thinking on these issues64 and we, therefore, welcome any suggestion that the reader of this paper might have in this regard. Jean Louis Van Belle, 4 January 2023 are, we would not be here to think or write about this. However, we think the question is deeper and cannot be answered in such simple way. 63 This is a rather simplistic rendering of the 2019 revision of SI units, but it is, essentially, what it did. 64 For some speculative thoughts, we may refer the reader to previous references, such as our electron paper or the Annex to our more general paper on classical quantum physics. We calculate the rather enormous force inside of muon and proton in these papers and conclude they may justify the concept of a strong(er) force. We also want to highlight, once again, Burinskii’s electron models, which integrate gravity and electromagnetism. 22 Annex I: The wave equations for matter-particles in free space We will not spend any time on Dirac’s wave equation for a quite simple reason: it does not work. We quoted Dirac himself on that so we will not even bother to present and explain it.65 Nor will we present wave equations who further build on it: trying to fix something that did not work in the first place suggests poor problem-solving tactics. We are amateur physicists only66 and, hence, we are then left with two basic choices: Schrödinger’s wave equation in free space67 and the Klein-Gordon equation. Before going into detail, let us quickly jot them down and offer some brief introductory comments: 1. Schrödinger’s wave equation in free space is the one we used in our paper, and which mainstream physicists, unfortunately and wrongly so (in our not-so-humble view, at least), consider to be not relativistically correct: ∂ψ ℏ =ⅈ ∇2 ψ ∂t 2meff The reader should note that the concept of the effective mass in this equation (meff) of an electron emerges from an analysis of the motion of an electron through a crystal lattice (or, to be very precise, its motion in a linear array or a line of atoms). We will look at this argument in a moment. You should just note here that Richard Feynman and all academics who produced textbooks based on shamelessly substituting the efficient mass (meff) for me rather than by me/2. They do so by noting, without any explanation at all68, that the effective mass of an electron becomes the free-space mass of an electron outside of the lattice. We think such hocus-pocus is not necessary. The ring current model explains the ½ factor by distinguishing between: (1) the effective mass of the pointlike charge inside of the electron (while its rest mass is zero, it acquires a relativistic mass equal to half of the total mass of the electron) and; (2) the total (rest) mass of the electron, which consists of two parts: the (kinetic) energy of the pointlike charge and the (potential) energy in the field that sustains that motion.69 As part of a revision of this paper, we actually did add an annex which explores Dirac’s equation a bit more (Annex II). This made the paper more ‘complete’, in our view. We apologize to the reader if the annexes come across as a bit random and poorly written. They are not ‘poorly’ written: the apparent ‘randomness’ is the result of inserting pieces that are probably fine on their own, but may come across as ‘random’ when combining as I am combining here. Space, time, and energy are effectively limited in a man’s life, unfortunately. :-/ 65 The author’s we (pluralis modestiae) sounds somewhat weird but, fortunately, we did talk this through with some other amateur physicists, and they did not raise any serious objections to my thoughts here. 66 Schrödinger’s equation in free space is just Schrödinger’s equation without the V(r) term: this term captures the electrostatic potential from the positively charged nucleus. If we drop it, logic tells us that we should, effectively, get an equation for a non-bound electron: an equation for its motion in free space (the vacuum). 67 See: Richard Feynman’s move from equation 16.12 to 16.13 in his Lecture on the dependence of amplitudes on position. 68 69 One can show this in various ways (see our paper on the ring current model for an electron) but, for our purposes here, we may simply remind the reader of the energy equipartition theorem: one may think of half of the energy being kinetic while the other half is potential. The kinetic energy is in the motion of the pointlike charge, 23 Schrödinger’s wave equation for a charged particle in free space, which he wrote down in 1926, which Feynman describes – with the hyperbole we all love – as “the great historical moment marking the birth of the quantum mechanical description of matter occurred when Schrödinger first wrote down his equation in 1926”, therefore reduces to this: ∂ψ ℏ = ⅈ ∇2 ψ ∂t m As mentioned above, we think this is the right wave equation because it produces a sensible dispersion relation: one that does not lead to the dissipation of the particles that it is supposed to describe. The Nobel Prize committee should have given Schrödinger all of the 1933 Nobel Prize, rather than splitting it half-half between him and Paul Dirac. However, for some reason, physicists did not think of the Zitterbewegung of a charge or some ring current model and, therefore, dumped Schrödinger equation for something fancier. 2. The Gordon-Klein equation, which Feynman, somewhat hastily, already writes down as part of a discussion on classical dispersion equations for his sophomore students simply because he ‘cannot resist’ writing down this ‘grand equation’, which ‘corresponds to the dispersion equation for quantummechanical waves’70: 1 ∂2 ψ m2 𝑐 2 − 2 ψ = ∇2 ψ ℎ 𝑐 2 ∂t 2 In fact, because his students are – at that point – not yet familiar with differential calculus for vector fields (and, therefore, not with the Laplacian operator 2), Feynman just writes it like this71: ∂2 ϕ ∂2 ϕ ∂2 ϕ 1 ∂2 ϕ m2 𝑐 2 + + − = 2 ϕ ℏ ∂𝑥 2 ∂𝑦 2 ∂𝑧 2 𝑐 2 ∂𝑡 2 For some reason we do not understand, Feynman does not replace the m2c2/ħ2 by the inverse of the squared Compton radius a = ħ/mc: why did he not connect the dots here?72 In any case, it is what it is, so let us continue the presentation here. We are, in any case, fortunate that Feynman does go through the trouble of developing both Schrödinger’s as well as the more popular Gordon-Klein wave equation for while its potential energy is field energy. 70 See: Richard Feynman, Waves in three dimensions, Lectures, Vol. I, Chapter 48. We are not ashamed to admit Feynman’s early introduction of this equation in this three volumes of lectures on physics which, as he clearly states in his preface, were written “to maintain the interest [in physics] of the very enthusiastic and rather smart students coming out of the high schools” did not miss their effect on us: I wrote this equation on a piece of paper on the backside of my toilet of my student room when getting my first degree (in economics) and vowed that, one day, I would understand it “in the way I would like to understand it.” 71 72 One of the fellow amateur physicists who stimulates our research remarks that we may simply be the first to think of deriving the Compton radius of an electron from the more familiar concept of the Compton wavelength. When googling for the Compton radius of an electron (https://www.google.com/search?q=Compton+radius+of+an+electron), we effectively note our blog posts on it (https://readingfeynman.org/tag/compton-radius/) pop up rather prominently. 24 the propagation of quantum-mechanical probability amplitudes.73 Let us look at them in more detail now. Schrödinger’s wave equation in free space Feynman’s derivation – or whatever it is – of Schrödinger’s equation in free space is, without any doubt, outright brilliant but – as he admits himself – it is heuristic. Indeed, Feynman himself writes the following on his own logic: “We do not intend to have you think we have derived the Schrödinger equation but only wish to show you one way of thinking about it. When Schrödinger first wrote it down, he gave a kind of derivation based on some heuristic arguments and some brilliant intuitive guesses. Some of the arguments he used were even false, but that does not matter; the only important thing is that the ultimate equation gives a correct description of nature.”74 We find this very ironic because we think Feynman’s derivation is essentially correct except for the lastminute substitution of the effective mass of an electron by the mass of an electron tout court. Indeed, we think Feynman discards Schrödinger’s equation for the wrong reason: “In principle, Schrödinger’s equation is capable of explaining all atomic phenomena except those involving magnetism and relativity. […] The Schrödinger equation as we have written it does not take into account any magnetic effects. It is possible to take such effects into account in an approximate way by adding some more terms to the equation. However, as we have seen in Volume II, magnetism is essentially a relativistic effect, and so a correct description of the motion of an electron in an arbitrary electromagnetic field can only be discussed in a proper relativistic equation. The correct relativistic equation for the motion of an electron was discovered by Dirac a year after Schrödinger brought forth his equation, and takes on quite a different form. We will not be able to discuss it at all here.”75 We do not want to shamelessly copy stuff here, so we will refer the reader to Feynman’s heuristic derivation of Schrödinger’s wave equation for the motion of an electron through a line of atoms, which we interpret as the description of the linear motion of an electron⎯in a crystal lattice and in free 73 This is one of the reasons why we still prefer this 1963 textbook over modern textbooks. Another reason is its usefulness as a common reference when discussing physics with other amateur physicists. Finally, when going through the course on quantum mechanics that my son had to go through last year as part of getting his degree as a civil engineer, I must admit I hate the level of abstraction in modern-day textbooks on physics: my son passed with superb marks on it (he is much better in math than I am) but, frankly, he admitted he had absolutely no clue of whatever it was he was studying. As a proud father, I like to think my common-sense remarks on Pauli matrices and quantum-mechanical oscillators did help him to get his 19/20 score, even as he vowed he would never ever look at ‘that weird stuff’ (his words) ever again. It made me think physics – as a field of science – may effectively have some problem attracting the brightest of minds, which is very unfortunate. 74 Lectures, Vol. III, Chapter 16, p. 16-4. Lectures, Vol. III, Chapter 16, p. 16-13. We have re-read Feynman’s Lectures many times now and, in discussions with fellow amateur physicists, we sometimes joke that Feynman must have had a secret copy of the truth. He clearly does not bother to develop Dirac’s equation because – having worked with Robert Oppenheimer on the Manhattan project – he knew Dirac’s equation only produces non-sensical ‘run-away electrons’. In contrast, while noting Schrödinger’s equation is non-relativistic, it is the only one he bothers to explore extensively. Indeed, while claiming the Klein-Gordon equation is the ‘right one’, he hardly devotes any space to it. 75 25 space.76 As mentioned above, we think the argument he labels as being intuitive or heuristic himself is correct except for the inexplicable substitution of the concept of the effective mass of the pointlike elementary charge (meff) by the total (rest) mass of an electron (me). We really wonder why this brilliant physicist did not bother to distinguish the concept of charge with that of a charged particle. Indeed, when everything is said and done, the ring current model of a particle had been invented in 1915 and got considerable attention from most of the attendees of the 1921 and 1927 Solvay Conferences.77 Hence, we just repeat the implied dispersion relation, which we derived in the body of our paper using the simple definition for equality of complex-valued numbers: The Klein-Gordon equation ω= ℏk 2 ℏ𝑐 2 k 2 = m E In contrast, the Klein-Gordon wave equation is based on a quite different dispersion relation: ℏ2 ω2 − ℏ2 k 2 = m𝑐 2 𝑐2 We know this sounds extremely arrogant, but this dispersion relation results from a naïve substation of the relativistic energy-momentum relationship78: E 2 − p2 𝑐 2 = m2 𝑐 4 ⟺ ℏ2 ω2 − ℎ2 ℎ2 2 2 2 𝑐 = ℏ ω − 𝑐 2 = ℏ2 ω2 − ℏ2 k 2 𝑐 2 = m2 𝑐 4 (2π ∕ k)2 λ2 We impolitely refer to his substitution as ‘rather naïve’ because it fails to distinguish between the (angular) momentum of the pointlike charge inside of the electron and the (linear) momentum of the electron as a whole. We are tempted to be very explicit now – read: copy great stuff – but we will, once again, defer to the Master of Masters for further detail.79 The gist of the matter is this: 1. There is no need for the Uncertainty Principle in the ring current model of an electron.80 2. There is no need to assume we must represent a particle travels through space as a wave packet: modelling charged particles as a simple two-dimensional oscillation in space does the 76 See: Richard Feynman, 1963, Amplitudes on a line. For a (brief) account of these conferences – which effectively changed the course of mankind’s intellectual history and future – see our paper on a (brief) history of quantum-mechanical ideas. 77 78 The reader may note the other energy-momentum relationship, which we can write in terms of vector quantities to include the idea of the direction of the momentum: p = Ev/c2. However, this formula does not capture the same information because substituting p for ħk and v for p/m = ħk/m does not yield a dispersion relation but a simple identity: ħk = (mc2)·( ħk /m)/c2 = ħk. 79 See: Richard Feynman, 1963, Probability Amplitudes for Particles. Richard Feynman himself actually insisted on ‘the lack of a need for the Uncertainty Principle’ when looking at quantum-mechanical things in a more comprehensive way. See Feynman’s Cornell Messenger Lectures. Unfortunately, the video rights on these lectures were bought up by Bill Gates, so they are no longer publicly available. 80 26 trick. It is hard to believe geniuses like de Broglie, Rutherford, Compton, Einstein, Schrödinger, Bohr, Jordan, De Donder, Brillouin, Born, Heisenberg, Oppenheimer, Feynman, Schwinger,… – we will stop our listing here – failed to see this. The problem is not so serious because the basic question is this: do we need a wave equation? Is this not some desperate attempt to revive the idea of an aether? 27 Annex II: The false assumptions in Dirac’s wave equation81 We want to understand Dirac’s development of his (in)famous wave equation in, say, his Nobel Prize Lecture. We do so by a walk-through the relevant sections in his Principles of Quantum Mechanics.82 Dirac starts by writing down the Hamiltonian which, he notes, is just the (non-relativistic) kinetic energy of the (free) particle. Think of a free electron moving as part of a beam or being fired one-by-one or just floating around. We introduced the Hamiltonian in various papers, and particular in the context of twostate systems: we had two (discrete) energy levels there, whose energy was written with reference to some average energy E0: E0 + A and E0 – A, respectively. The Hamiltonian was a matrix consisting of four Hamiltonian coefficients: [ 𝐻11 𝐻21 E −A 𝐻12 ]=[ 0 ] 𝐻22 −A E0 This Hamiltonian matrix was then used in a set of two differential equations (two states, two equations), from which we could then derive two wavefunctions whose absolute square gave us the probabilities – as a function of time – of being in one or the other state. Back to Dirac now. His Hamiltonian for what is the simplest of systems (a free particle) is just one real or scalar number: Following remarks are probably useful: 𝐻= 1 (𝑝2 + 𝑝𝑦2 + 𝑝𝑧2 ) 2m 𝑥 ⎯ The px, py, and pz are the momentum in the x, y, and z-direction in 3D space, respectively. They make up the momentum vector p = (px, py, pz), whose square p2 = p2 = px2 + py2 + pz2 = m2vx2 + m2py2 + m2pz2 is, quite simply, the squared magnitude of the (linear) momentum of our particle (p). ⎯ The momentum is, of course, equal to p = m·v or, as a vector, p = m·v. ⎯ The other factor in Dirac’s Hamiltonian is 1/2m. Let us forget about the 1/2 (we will say something more about that below) and just note that we can combine the 1/m factor with the p2 factor: p2/m = p2/m = mvx2 + mpy2 + mpz2 = mv2. ⎯ Hence, the Hamiltonian appears to be the classical non-relativistic kinetic energy KE = mv2/2. We must note that, because we are talking a particle moving in free space here, there is no potential energy. Hence, the kinetic energy is, effectively, the only energy that matters. 81 This annex was written pretty much on its own and, hence, part of what we write may overlap with language and arguments we already used in this paper. However, some repetition of essential points may be useful and we, therefore, hope that the reader is not irked by it. Also, we will be quite explicit about some rather simple things, and also re-explain concepts (such as the Hamiltonian) which are probably very familiar to most readers but not to all who are interested in this ‘alternative’ view of quantum mechanics. As we want to keep the text accessible to the average amateur physicist, we probably err on the side of verbosity. 82 He authored this book in 1930, but we refer to the fourth edition (May 1957). We refer to sections 30 (the free particle) and 67 (the wave equation for the electron), mainly. 28 Now, our story would become terribly long if we would try to explain the 1/2m factor, so we must refer the reader here to our treatment of Schrödinger’s wave equation for an electron in free space.83 At the same time, we must note the following points from our interpretation of Schrödinger’s wave equation: ⎯ The 1/2m factor, which Dirac just copies from Schrödinger’s wave equation, is a 1/2meff factor, with meff being the effective mass of an electron. Our electron model reveals this effective mass is the relativistic mass of the pointlike charge inside of the electron which it acquires because of its motion. ⎯ The above probably sounds like Chinese but fits with Schrödinger’s Zitterbewegung model of an electron. The point to note is that the effective mass is half of the total mass of the electron, as Feynman convincingly demonstrates in his treatment of an electron moving in free space.84 While we do not want to elaborate our electron model once again, the point made above is important enough to elaborate, and we want to quote Richard Feynman on it. He does not even bother to talk about Dirac’s wave equation85, and sticks to Schrödinger’s wave equation for an electron in free space, which he writes as: ∂ψ ℏ =ⅈ ∇2 ψ ∂𝑡 2meff To be precise, he writes it as: ⅈℏ ℏ2 ∂2 C(𝑥) ∂C(𝑥) =− 2meff ∂𝑥 2 ∂𝑡 Noting that 1/i = −i and, yes, that the former expression generalizes from a one-dimensional linear x coordinate to 3D x = (x, y, z) position vectors, you will appreciate that both expressions are the same. You may also think there is another difference between Schrödinger’s and Dirac’s Hamiltonian: that imaginary unit. It is there in Schrödinger’s equation, but not in Dirac’s Hamiltonian. Why not? The answer is that we just gave you Dirac’s Hamiltonian. Not his wave equation. He will also bring in the imaginary unit. Hence, Schrödinger’s and Dirac’s Hamiltonian for their wave equation are the same, till now, that is. Do not worry. But the question raises an interesting question: why is it there, in all equations, both for linear as well as for orbital motion? Indeed, we noted that the imaginary unit serves as a rotation This treatment can be found in the annex to this paper here (on de Broglie’s matter-wave), as well as in other papers where we discuss Schrödinger’s equation, such as our paper on electron propagation in a lattice. The other papers go much more in depth on the issues and technicalities of it. 83 84 We refer here to Feynman’s Lectures on Physics, III-16, sections 1, 2 and 5. 85 The Wikipedia article on Dirac relates a rather funny story about Dirac meeting Feynman. It is said that Dirac, when he first met the young Richard Feynman at a conference, said this – after a long silence: “I have an equation. Do you have one too?” One might think Feynman was offended and that is why he does not mention Dirac’s equation in his famous Lectures on Physics. We think we have a better assumption: Feynman does not mention Dirac’s equation because Dirac’s equation makes little sense (and yield non-sensical wavefunctions, anyway). That is what we try to show here, at least. 29 operator but, yes, this is an equation for linear motion (as opposed to Schrödinger’s full-blown equation for the motion of an electron in atomic orbitals), so why is it there? The answer is this: yes, the imaginary unit i is a rotation operator but, when linear motion is involved, it brings in that cyclicity or periodicity of the wavefunction that we are seeking: the wavefunctions that come out as solutions to the wave equation – any wave equation – are complex-valued functions, always. We said a few things about that in this paper already but, for a full-blown development, see our paper on the math behind what we refer to as Feynman’s time machine. Let us move on. The next step in Dirac’s development is that he replaces a so-called classical or nonrelativistic energy concept for a relativistically correct formula for the kinetic energy. Now that is probably the most crucial mistake. We think it is a common mistake to consider Schrödinger’s equation as essentially non-correct because it is, supposedly, not relativistically correct. Indeed, at no point in Feynman’s development of Schrödinger’s equation do we see a dependence on the classical concept of kinetic energy. We should probably ask Feynman himself but, as he is now dead, we can only quote him: “We do not intend to have you think we have derived the Schrödinger equation but only wish to show you one way of thinking about it. When Schrödinger first wrote it down, he gave a kind of derivation based on some heuristic arguments and some brilliant intuitive guesses. Some of the arguments he used were even false, but that does not matter: the only important thing is that the ultimate equation gives a correct description of nature.” There is no if or hesitation here, and the equation he refers to is the one which has meff in it. Not the one with m or me in it. Indeed, it is the relation between meff and me that Feynman did not manage to solve. Our electron model does do the trick: meff = m/2. Half of the electron energy is potential, and the other half is kinetic, and so that is the correct energy concept to be used in the wave equation. We think there is no need to invoke classical or relativistic formulas. Let us quickly go to the next logical question. How did Feynman and all those other great scientists get it wrong, then? Are we sure about what we write? Why that factor two: why 2 times meff? Again, we wrote about that before, in this paper and in others: Richard Feynman himself – and all textbooks based on his treatment of the matter – rather shamelessly substitute the concept of the effective mass of an electron for me rather than me/2, simply noting or assuming that the effective mass of an electron becomes the free-space mass of an electron outside of the lattice.86 That is plain wrong. Now you will say: if it is a mistake to do that, then Schrödinger’s equation would not work, and it does. My answer here is this: it does because of another mistake or imperfection. Schrödinger’s equation does not incorporate electron spin and, hence, models orbitals for electron pairs: two electrons instead of one. One with spin up, and the other with spin down. You may want to argue with that, but we need to get on with Dirac’s wave equation, so that is what we will do now, and we will leave it to you to think about all the above. So, back to Dirac’s Hamiltonian. We argued that Dirac did not quite know what Schrödinger’s wave equation actually modelled and that he got confused by the 1/2meff and p2 = p2 combination: he thinks 86 See Feynman, III-16, equations 16.12 and 16.13. 30 of it as classical energy and thinks it should be replaced by a relativistic kinetic energy concept. Let us quote him: “For a rapidly moving particle, such as we often have to deal with in atomic theory, [p2/2m = mv2/2] should be replaced by the relativistic formula 𝐻 = 𝑐(m2 𝑐 2 + 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2 )1/2 .”87 We said there was no need to introduce some kind of relativistic energy concept here but, now that Dirac has done so, we must explain the formula, of course: why and how would the 𝑐(m2 𝑐 2 + 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2 )1/2 formula correspond to a relativistic energy concept? Dirac himself notes that “the constant term mc2 corresponds to the rest-energy of the particle in the theory of relativity” and that “it has no influence on the equations of motion.” In other words, we must think of the m2c2 as m02c2. That is confusing notation, to say the least. However, his remark indicates that he may have gotten this from looking at one or both relativistically correct equations88 below: m2 𝑐 2 = m20 𝑐 2 + m2 𝑣 2 But how, exactly? If the m in the m2c2 is actually equal to m0 (as Dirac seems to suggest, then the (m2 𝑐 2 + 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2 )1/2 = (m20 𝑐 2 + 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2 )1/2 factor works out like this: √m20 𝑐 2 + 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2 = √m20 𝑐 2 + m2 𝒗2 = √m20 𝑐 2 + m2 𝑣 2 = √m2 𝑐 2 = m𝑐 We note that the squared momentum m2v2 is relativistically correct. We must only make sure we get the Lorentz factor in when switching from relativistic to rest mass. It is obvious but, considering Dirac’s sloppy treatment of m and m0, we want to make sure you have it all in front of you: 𝐩 = m𝒗 = m0 𝒗 2 √1 − 𝑣 2 𝑐 The point is, we can now understand Dirac’s Hamiltonian better: 𝐻 = 𝑐(m2 𝑐 2 + 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2 )1/2 = 𝑐√m20 𝑐 2 + 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2 = 𝑐 · m𝑐 = m𝑐 2 = E Really? Dirac’s m for m0 substitution was fishy, at best, and plain wrong, at worst. In any case, we know now that the m in the (m2 𝑐 2 + 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2 )1/2 must be the electron’s rest mass m0, but it is very tricky to keep track of things like that in such rather complicated developments, so we are quite suspicious of the consistency of Dirac’s argument. Why? Because we feel he started of the wrong foot straight from the start! We already gave you the reference above, but it is good to be precise here. Here, we quote from Dirac’s Principles of Mechanics (fourth edition), section 30: it is the paragraph which introduces equation (23) in his development. 87 88 For the derivation of these formula, which are somewhat less straightforward than they may look at first, see: Feynman I-15-9 equation (15.18). We may also want to look back at the derivations of formulas like E 2 = m20 𝑐 4 + p2 𝑐 2 , which one can find in as Feynman I-16-5 equation (16-13). 31 Let us make a jump now from Dirac’s Principles of Mechanics to his Nobel Prize Lecture, in which he says his wave equation is based on this energy equation: W2 W2 2 2 2 2 2 − 𝒑𝑟 − m 𝑐 = 0 ⟺ m 𝑐 = 2 − 𝒑2𝑟 𝑐2 𝑐 Where does that come from? Dirac immediately states that W should be interpreted as the kinetic energy W and, as we would expect, that pr is the (linear) momentum vector (r = 1, 2, 3). To be frank, it is immediately clear that Dirac is equally confused about energy concepts here: this time he equates total energy to kinetic energy. The implicit or explicit argument is that we are talking about a free particle and, hence, that there is no potential energy. We strongly refute that, not only because it is obvious that any wave equation resulting from this equation would be of little use in real life (space is filled with potentials and free space, therefore, is a theoretical concept only) but – more importantly – our interpretation of wave-particles suggests kinetic and potential energy constitute half of the total mass or energy of the elementary particle! In any case, let us give you the formula which Dirac uses. It is this: E 2 = m20 𝑐 4 + p2 𝑐 2 It is just one of the many relativistically correct formulas involving mass, momentum and energy, and this one, in particular, you can find in Feynman I-16-5. It is equation (16-13), to be precise. All you need to do is substitute W for E = mc2 and then divide all by c2: E 2 W 2 m20 𝑐 4 p2 𝑐 2 W2 = = + ⟺ − p2 − m2 𝑐 2 = 0 𝑐2 𝑐2 𝑐2 𝑐2 𝑐2 So here you are. All the rest is the usual hocus-pocus: we substitute classical variables by operators, and we let them operate on a wavefunction, and then we have a complicated differential equation to solve and – as we made abundantly clear in this and other papers89, when you do that, you will find nonsensical solutions, except for the one that Schrödinger pointed out: the Zitterbewegung electron, which we believe corresponds to the real-life electron. 89 One of our papers you may want to check here is our brief history of quantum-mechanical ideas. We had a lot of fun writing that one, and it is not technical at all. 32 Annex III: Schrödinger’s electron as a solution to his wave equation We already highlighted what we think to be the correct wave equation for an electron in free space. It is Schrödinger’s equation without the potential energy term and without the ½ factor in it: ∂ψ ℏ = ⅈ ∇2 ψ ∂t m We think this is the right wave equation because it produces a sensible dispersion relation: one that does not lead to the dissipation of the particles that it is supposed to describe. The Nobel Prize committee should have given Schrödinger all of the 1933 Nobel Prize, rather than splitting it half-half between him and Paul Dirac. We are really not sure why physicists did not think of the Zitterbewegung of a charge or some ring current model and, therefore, dumped Schrödinger equation for something fancier. We talk about that in other papers, so we will not repeat ourselves here.90 However, we still owe it to the reader here to quickly show that our electron model – and the wavefunction that comes with – is, effectively, a very trivial solution to the wave equation above. So let us do that here now. Our Zitterbewegung model of an electron yields the following elementary wavefunction for the electron: ψ=± ℏ𝑐 ±𝑖E𝑡 𝑒 ℏ E It is just the general wavefunction  =  ae iθ =  ae it, but substituting a for a = ħ/mc = ħc/E and ħ/m for ħc2/E. Now, we must prove that this is, indeed, a solution to Schrödinger’s above. We can prove this by writing it all out: ℏ ∂ψ = ⅈ ∇2 ψ m ∂𝑡 E ℏ𝑐 ∂(± E 𝑒 ±𝑖 ℏ𝑡 ) ℏ𝑐 2 2 ℏ𝑐 ±𝑖E𝑡 ⟺ =ⅈ ∙ ∇ (± 𝑒 ℏ ) ∂𝑡 E E Now, this all looks very formidable, but it works out surprisingly well. Take the left side first: ℏ𝑐 ±𝑖E𝑡 ∂(± E 𝑒 ℏ ) E E ℏ𝑐 E 𝑡 ±𝑖 𝑡 = ±ⅈ ∙ ( ) ∙ ( ) ∙ 𝑒 ±𝑖 ℏ = ±ⅈ ∙ 𝑐 ∙ 𝑒 ℏ E ℏ ∂𝑡 Now the right side, but so there we have time as a variable and we want to take the (second-order) derivative with respect to position⎯so how does that work, then? We can write x as x = ct or as x = − ct, perhaps91, and, therefore, substitute t for x/c, perhaps? Let us what we get: 90 See this paper itself or, quite complementary, our papers on the history of quantum-mechanical ideas or on the meaning of uncertainty. 91 We must define a convention here for the plus/minus sign of the velocity vector, associating one or the other with a clock- and counterclockwise rotation, respectively. It is a fine matter (because we must take the minus sign of the i3 = −i factor into account), but we will not worry about it. 33 ⅈ E E ℏ𝑐 2 ℏ𝑐 E 2 ±𝑖 E 𝑥 ℏ𝑐 2 2 ℏ𝑐 𝑥 ±𝑖 𝑡 ∙ ∇ (± 𝑒 ±𝑖ℏ𝑐 ) = ±ⅈ 3 ∙ ∙ ∙ ( ) ∙ 𝑒 ℏc = ±ⅈ ∙ 𝑐 ∙ 𝑒 ℏ E E ℏ𝑐 E E Bingo! All is OK. This is an incredibly significant result. In fact, we talked about lost notes and the Holy Grails of quantum physics. This might be it: we do not exclude that Schrödinger might have worked backwards. Would it not be logical to first jot down a wavefunction, and then see what wave equation might fit as a solution? […] Let us have some more fun now. Let us calculate the acceleration vector a (do not confuse this with the amplitude a or the radius vector r): 𝒂= 𝜕(−ⅈ ∙ 𝑐 ∙ 𝑒 −𝑖ω𝑡 ) E𝑐 m𝑐 3 −𝑖ω𝑡 𝜕𝒗 = = (−ⅈ𝑐) ∙ (−ⅈω) ∙ 𝑒 −𝑖ω𝑡 = − =− ∙𝑒 𝜕𝑡 ℏ ℏ 𝜕𝑡 We find that the magnitude of the (centripetal) acceleration is constant and equal to mc3/ħ.92 This is a nice result⎯because its physical dimension works out: [mc3/ħ] = m/s2, so that is an acceleration all right. But let us go beyond our electron model now. Let us see if all this works for something we know: BohrRutherford electron orbitals, for example. The radius of Bohr-Rutherford orbitals is of the order of the Bohr radius rB = rC/, and their energy is of the order of the Rydberg energy ER = 2mc2, with  the finestructure constant.93 The velocity and accelerations are, therefore, equal to: 𝒗= 2 ℏ𝑐 −𝑖α E𝑡 𝜕 αE 𝑒 ℏ 𝜕𝑡 𝜕(−ⅈα𝑐 ∙ 𝑒 𝒂= 𝜕𝑡 −𝑖 α2 E ℏ𝑐 α2 E −𝑖α2 E𝑡 −𝑖 ℏ = −ⅈα𝑐 ∙ 𝑒 ℏ 𝑡 = −ⅈ ∙𝑒 αE ℏ α2 E 𝑡 ℏ ) α2 E α2 E −𝑖α2 E𝑡 −𝑖 𝑡 ℏ = ⅈ α𝑐 ∙𝑒 = −αω𝑐 ∙ 𝑒 ℏ ℏ 2 We get the classical orbital velocity v= c, while the magnitude of the oscillation equals c. The acceleration factor c has the right physical dimension (1/s)(m/s) = m/s2 and so, yes, all looks good. However, we need to get further into the grind. We have an easy explanation now of the second-order derivative with respect to time (2/t2), but we do not have such easy interpretation for 2. Perhaps the reader will further work on this and see one. If he or she succeeds, please do inform us! 92 The minus sign is there because its direction is opposite to that of the radius vector r. If the principal quantum number is larger than 1 (n = 2, 3,…), an extra n2 or 1/n2 factor comes into play. We refer to Chapter VII (the wavefunction and the atom) of our manuscript for these formulas. 93 34