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Abstract and Figures

This paper explores the assumptions underpinning de Broglie's concept of a wavepacket and the various conceptual questions and issues. It also explores how the alternative 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.
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De Broglie’s matter-wave: concept and issues
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
9 May 2020
Email: jeanlouisvanbelle@outlook.com
Summary
This paper explores the assumptions underpinning de Broglies concept of a wavepacket and the various
conceptual 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 Compton radius ....................................................................................... 1
De Broglie’s dissipating wavepacket ............................................................................................................. 3
De Broglie’s non-localized wavepacket ........................................................................................................ 5
The wavefunction, the wave equation and Heisenberg’s uncertainty ......................................................... 6
The wavefunction and (special) relativity ..................................................................................................... 9
The geometric interpretation of the de Broglie wavelength ...................................................................... 14
The real mystery of quantum mechanics ................................................................................................... 17
Annex: The wave equations for matter-particles in free space ................................................................. 19
Schrödinger’s wave equation in free space ............................................................................................ 21
The Klein-Gordon equation ..................................................................................................................... 22
1
De Broglie’s matter-wave: concept and issues
De Broglie’s wavelength and the Compton radius
De Broglie’s ideas on the matter-wave are oft-quoted and are usually expressed in de Broglie’s λ = h/p
relation. However, there is remarkably little geometric or physical interpretation of it
1
: what is that
wavelength, exactly? The relation itself is easy enough to read: λ goes to infinity as p goes to zero. In
contrast, for p = mv going to p = mc, this length becomes the Compton wavelength λ = h/p = h/mc. This
must mean something, obviously, but what exactly?
Mainstream theory does not answer this question because the Heisenberg interpretation of quantum
mechanics essentially refuses to look into a geometric or physical interpretation of de Broglie’s relation
and/or the underlying concept of the matter-wave or wavefunction which, lest we forget, must
somehow represent the particle itself. In contrast, we will request the reader to think of the
(elementary) wavefunction as representing a current ring.
To be precise, 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 dimensionless
2
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. This radius 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:





This easy derivation
3
already gives a more precise 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”
4
, but we may usefully further elaborate the details by visualizing the model (Figure
1
Wikipedia offers an overview of the mainstream view(s) in regard to a physical interpretation of the matter-wave
and/or the de Broglie wavelength by quoting from the papers by Erwin Schrödinger, Max Born and Werner
Heisenberg at the occasion of the 5th Solvay Conference (1927). These views are part of what is rather loosely
referred to as the Copenhagen interpretation of quantum mechanics.
2
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.
3
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!
4
Prof. Dr. Patrick LeClair, Introduction to Modern Physics, Course Notes (PH253), 3 February 2019, p. 10.
2
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.
5
Figure 1: The ring current model of an electron
Of course, the reader will, most likely, not be familiar with the ring current model or using the term
Erwin Schrödinger coined for it the Zitterbewegung model and we should, therefore, probably quote
an unlikely authority on it so as to establish some early credentials
6
:
“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)
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 (too) 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 note the momentum of the pointlike charge which we denote by p in the
illustration must be distinguished from the momentum of the electron as a whole. The momentum of
5
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 canique ondulatoire (1924) but the presentation helped him to secure the
necessary fame which would then lead to him getting the 1929 Nobel Prize for Physics.
6
For an overview of other eminent views, we refer to our paper on the 1921 and 1927 Solvay Conferences.
3
the pointlike charge will always be equal to p = mc.
7
The rest mass of the pointlike charge must,
therefore, be zero. However, its velocity give its an effective mass which one can calculate to be equal to
meff = me/2.
8
Let us now review de Broglie’s youthful intuitions.
9
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 the young Louis de Broglie that an electron must have a
frequency was, effectively, a stroke of genius. However, as the magnetic properties of an electron were,
by then, not well established and this may explain why Louis de Broglie is either not aware of it or
refuses to further build on it.
10
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 is, by now, well known: we
think of the particle as a wave packet composed of waves of slightly different frequencies νi.
11
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
7
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.
8
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, not self-evident at all!
9
We must quality 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! 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 connect the dots.
10
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.
11
We invite the reader to check our exposé against de Broglie’s original 1927 paper in the Solvay Conference
proceedings. We will try to stick closely to the symbols that are used in this paper, such as the nu (ν) symbol for the
frequency.
4
velocity β= v/c.
The assumption is then that we know how the phase frequencies νi are related to wavelengths λi. This is
modeled 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.
12
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 rather 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 rather inconvenient property in the ceremonial speech for the 1933 Nobel
Prize, which was awarded to Heisenberg for nothing less than the creation of quantum mechanics
13
:
“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 on is forced to the conclusion to conceive of matter as
not being durable, or that it can have 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 very familiar to you. However, it is, obviously, 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 slits
14
but not fundamentally so!
15
12
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):

13
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.
14
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.
15
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, move 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
5
We should let this problem rest. We will 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 Broglies
concept of a matter-wave packet: the problem of (non-)localization.
De Broglies 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 referred to this as an
Ungenauigkeit).
2. Our measurement might disturb the position and, as such, cause the information to get lost
and, as a result, introduce an uncertainty (Unbestimmtheit).
3. One may also think the uncertainty is inherent to Nature (Ungewissheit).
We think that despite all thought experiments and Bells No-Go Theorem
16
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?
17
However, the root cause of the uncertainty does
not matter. The point is this: 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. Let us, once again, quote Richard
Feynman here:
If an amplitude to find a particle at different places is given by ei(ω·tk·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 isit can be anywherethere 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
particle momentum.
18
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 quite see how the idea of a wave train or a wavepacket solves that problem. A composite
experiment for electrons: “Electrons always arrive in identical lumps.”
16
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 Bells (in)famous theorem. There is ample but, admittedly, non-
conclusive literature on that so we will let the interested reader google and study such metaphysics.
17
See our paper on the 1921 and 1927 Solvay Conferences in this regard. We translated Lorentz comment from
the original French, which reads as follows: Faut-il nécessairement ériger l’ indéterminisme en principe?
18
See: Probability wave amplitudes, in: Feynmans Lectures on Physics, Vol. III, Chapter 2, Section 1.
6
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 non-randomly Δx space.
We will let this matter rest too. Let us look at the concept of concepts: the wave equation.
The wavefunction, the wave equation and Heisenberg’s uncertainty
With the benefit of hindsight, we now know the 1927 and later Solvay Conferences pretty much settled
the battle for ideas in favor of the new physics. At the occasion of the 1948 Solvay Conference, it is only
Paul Dirac who seriously challenges the 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 as solutions dissipate away. Real-life electrons any real-life matter-particle, really do not do that.
In fact, we refer to them as being particle-like because of their integrityan integrity that is modeled 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 away
19
, 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
meaning.
20
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.
21
The other
part of the solution(s) is (are), effectively, bizarre oscillations which Dirac here refers to as ‘run-away
19
This corresponds to wavefunctions dissipating away. The matter-particles they purport to describe obviously do
not.
20
See pp. 282-283 of the report of the 1948 Solvay Conference, Discussion du rapport de Mr. Oppenheimer.
21
See the quote from Dirac’s 1933 Nobel Prize speech in this paper.
7
electrons’. With the benefit of hindsight, one wonders why Dirac did not see what we see now.
22
When discussing wave equations, it is always useful to try to imagine what they might be modeling.
Indeed, 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 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. Needless to say, we already have a set of wave equations here: those
that come out of Maxwell’s equations! Should we expect contradictions here?
We hope not, of coursebut then we cannot be sure. An obvious candidate for a wave equation for
matter-waves in free space is Schrödinger equation’s without the term for the electrostatic potential
around a positively charged nucleus
23
:



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 and so we can
effectively substitute 2meff for the mass of the electron m = me = 2meff.
24
So far, so good. The 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·(k(xωt).
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 equations
25
:
22
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.
23
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.
24
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.
25
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.
8
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 relation
26
:


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 assuming 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 traveling
27
:





 

We know need to think about the question we started out with: one wave or many component waves?
It is fairly 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 motivate or justify this? The electron mass or energy is known, isn’t it? This is where the uncertainty
comes in: the electron may have some (classical) velocity or momentum for which we may not have a
definite value. If so, we may assume different values for its (kinetic) energy and/or its (linear)
momentum may be possible. 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 physical sense as:





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.
In the next section, we will show how one relate the uncertainties in the (kinetic) energy and the (linear)
momentum of our particle using the relativistically correct energy-momentum relation and also taking
into account 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:
26
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.
27
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.
9
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
very 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.
28
The reason is
simple: Dirac’s equation comes with a nonsensical dispersion relation. Schrödinger’s original equation
does not, which is why it works so well for bound electrons too!
29
We refer the reader to the Annex to
this paper for a more detailed discussion on this.
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 frame
30
, we will have a position a mathematical point in space, that is
31
which is a function of time: x(t) = v·t. Let us now denote the position and time in the reference frame of
the particle itself by x’ and t’. Of course, the position of the particle in its own reference frame will be
equal to x’(t’) = 0 for all t’, and the position and time in the two reference frames will be related by
Lorentz’s equations
32
:
28
We offer a non-technical historical discussion in our paper on the metaphysics of modern physics.
29
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.
30
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.
31
We may, of course, also think of it as a position vector by relating this point to the chosen origin of the reference
frame: a point can, effectively, only be defined in terms of other points.
32
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 lecture on it (chapters 15, 16 and 17 of the first volume of Feynman’s Lectures on
10



Hence, if we denote the energy and the momentum of the electron in our reference frame as Ev and p =
m0v, then the argument of the (elementary) wavefunction a·ei can be re-written as follows
33
:



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 frame of the electron, the argument of the
wavefunction effectively reduces to E0t’/ħ in the reference frame of the electron itself.
Note that, in the process, we also demonstrated the relativistic invariance of the Planck-Einstein
relation! This is why we feel that the argument of the wavefunction (and the wavefunction itself) is
more real in a physical sense 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 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 will get some linear 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.
34
Physics).
33
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.
34
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).
11
Figure 2: An interpretation of the de Broglie wavelength?
However, that would be too easy: 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:



We recognize the inverse Lorentz factor here, which goes from 1 to 0 as v goes from 0 to c, as shown
below.
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:

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:
12


We introduced a different symbol here: the time in our reference frame (t) is the coordinate time, and
the time in the reference frame 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 gymnastic because we
want to introduce a not-so-intuitive but very important result: the Compton radius becomes a
wavelength when v goes to c.
35
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 are able to 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:



This is interesting because we can see these equations are not all that abstract: we effectively get an
explanation for relativistic time dilation out of them. An equally interesting question is this: what
happens to the radius of the oscillation for larger (classical) velocity of our particle? Does it change? It
must. In the moving reference frame, 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 = a2·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 frequency
36
, which we just write as m and E in the formula below:


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 below
37
: the radius of the circulatory motion must
effectively diminish as the electron gains speed.
38
35
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.
36
Again, the reader should note that both the formula for the Compton radius or wavelength as well as the Planck-
Einstein relation are relativistically invariant.
37
We thank Prof. Dr. Giorgio Vassallo and his publisher to let us re-use this diagram. It originally appeared in an
article by Francesco Celani, Giorgio Vassallo and Antonino Di Tommaso (Maxwell’s equations and Occam’s Razor,
November 2017).
38
Once again, however, we should warn the reader that he or she should imagine the plane of oscillation to rotate
or oscillate itself. He should not think of it of being static unless we think of the electron moving in a magnetic
13
Figure 4: The Compton radius must decrease with increasing velocity
Can the velocity go to c? In the limit, yes. This is very interesting because we can see that the
circumference of the oscillation effectively turns into a linear wavelength in the process!
39
This rather
remarkable geometric property related our zbw electron model with our photon model, which we will
not talk about here, however.
40
Let us quickly make some summary remarks here, before we proceed to
what we wanted to present here: a geometric interpretation of the de Broglie wavelength:
1. The center of the Zitterbewegung was plain nothingness and we must, therefore, assume some two-
dimensional oscillation makes the charge go round and round. This is, in fact, the biggest mystery of
the model and we will, therefore, 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 single mathematical point in space, with the
center of the oscillation as opposed to the rotating massless charge.
2. 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.
3. As Dirac duly notes, the model cannot be verified directly because of the extreme frequency (fe =
ωe/2π = E/h ≈ 0.123×1021 Hz) and the sub-atomic scale (a = rC = ħ/mc ≈ 386 × 1015 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. In
addition, it 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.
41
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.
39
We may, therefore, think of the Compton wavelength as a circular wavelength: it is the length of a
circumference rather than a linear feature!
40
We may refer the reader to our paper on Relativity, Light and Photons.
41
The two results that we gave also show we get the gyromagnetic factor (g = 2). We have also demonstrated that
14
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 has to 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:


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.
42
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.
43
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:



Hence, our Archimedes’ screw gets stretched, so to speak. Let us think about what happens here. We
get the following formula for the λ wavelength in Figure 2:


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!
44
We can now relate this classical velocity (v) to the equally
classical linear momentum of our particle and provide a geometric interpretation of the de Broglie
wavelength, which we’ll 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: the Compton
wavelength λC (which is a circumference, actually), that weird horizontal distance λ, and the de Broglie
wavelength λp. It is easy to make sense of them by relating all three. Let us first re-write the de Broglie
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).
42
Hence, the C subscript stands for the C of Compton, not for the speed of light (c).
43
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!
44
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 momentumregardless 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.
15
wavelength in terms of the Compton wavelength (λC = h/mc), its (relative) velocity β = v/c, and the
Lorentz factor γ:





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 actually 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.
Figure 5: The 1/γ, 1/β and 1/γβ graphs
This analysis yields the following:
1. The de Broglie wavelength will be equal to λC = h/mc for v = c:


2. We can now relate both Compton as well as de Broglie wavelengths to our new wavelength λ = β·λC
wavelengthwhich is that length between the crests or troughs of the wave.
45
We get the following two
rather remarkable results:


45
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 sameexcept for a phase factor equal to n·2π (n = 1, 2,…).
16


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 very interesting but not good enough yet: the formulas do not give us the easy geometric
interpretation of the de Broglie wavelength that we are looking for. We get such easy geometric
interpretation only when using natural units.
46
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:

We get equally simple formulas for the de Broglie wavelength and our λ wavelength:



This is quite deep: we have three lengths here defining all of 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:
This is nothing but the latus rectum formula for an ellipse, which is illustrated below.
47
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.
48
We
then have the usual distances along the major and minor axis (a and b). Now, one can show that the
46
Equating c to 1 gives us natural distance and time units, and equating h to 1 then also gives us a natural force
unitand, 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 motionbecause 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.
47
Source: Wikimedia Commons (By Ag2gaeh - Own work, CC BY-SA 4.0,
https://commons.wikimedia.org/w/index.php?curid=57428275).
48
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.
17
following formula has to be true:
a·p = b2
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 rather remarkable result. We must now proceed and offer some final remarks on a far
more difficult question.
The real mystery of quantum mechanics
We think we have sufficiently demonstrated the theoretical attractiveness of the historical ring current
model. This is why 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 electrons
49
, he wrote us this:
“I know many people who considered the electron as a toroidal photon
50
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. As far as we are concerned, 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!
49
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.
50
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 motionorbiting 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).
18
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 electronand 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. In the longer run, however, we should probably abandon an
explanation of the ring current model in terms of Maxwell’s equations in favor for what it is, effectively,
the more mysterious explanation of a two-dimensional oscillation. However, we have not advanced very
far in our thinking on these issues
51
and we, therefore, welcome any suggestion that the reader of this
paper might have in this regard.
Jean Louis Van Belle, 9 May 2020
51
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.
19
Annex: The wave equations for matter-particles in free space
We will not spend any time on Dirac’s wave equation for a very simple reason: it does not work. We
quoted Dirac himself on that so we will not even bother to present and explain it. 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 only
52
and, hence, we are then left
with two basic choices: Schrödinger’s wave equation in free space
53
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: 


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 his rather
shamelessly substitute the efficient mass (meff) for me rather than by me/2. They do so by noting,
without any explanation at all
54
, that the effective mass of an electron becomes the free-space mass of
an electron outside of the lattice.
We think this is totally unwarranted too. 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.
55
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:


52
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.
53
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).
54
See: Richard Feynman’s move from equation 16.12 to 16.13 in his Lecture on the dependence of amplitudes on
position.
55
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,
while its potential energy is field energy.
20
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 quantum-
mechanical waves’
56
:

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 this
57
:




For some reason we do not quite 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?
58
It is what it is. We are, in
any case, very fortunate that Feynman does go through the trouble of developing both Schrödinger’s as
well as the more popular Gordon-Klein wave equation for the propagation of quantum-mechanical
probability amplitudes.
59
Let us look at them in more detail now.
56
See: Richard Feynman, Waves in three dimensions, Lectures, Vol. I, Chapter 48.
57
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.”
58
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.
59
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.
21
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 rather 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.
60
We find this very ironic because we actually think Feynman’s derivation is essentially correct except for
the last-minute 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.
61
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 electronin a crystal lattice and in free
space.
62
As mentioned above, we think the argument he labels as being intuitive or heuristic himself is
essentially 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.
63
Hence, we just repeat the implied dispersion relation, which we derived in the body of
60
Lectures, Vol. III, Chapter 16, p. 16-4.
61
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 doesn’t 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.
62
See: Richard Feynman, 1963, Amplitudes on a line.
63
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.
22
our paper using the simple definition for equality of complex-valued numbers:


The Klein-Gordon equation
In contrast, the Klein-Gordon wave equation is based on a very different dispersion relation:

We know this sounds extremely arrogant but this dispersion relation results from a rather naïve
substation of the relativistic energy-momentum relationship:

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.
64
The gist of the matter is this:
1. There is no need for the Uncertainty Principle in the ring current model of an electron.
65
2. There is no need to assume we must represent a particle travels through space as a wave
packet: modeling charged particles as a simple two-dimensional oscillation in space does the
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. We are totally fine with the reader amateur or academic switching off here:
this is utter madness. Regardless, we do invite him or her to think about it. When everything is said and
done, truth is always personal: it arises when our mind has an explanation for our experience.
66
However, we would like to ask the reader this: why do we need a wave equation? Is this not some
rather desperate attempt to revive the idea of an aether?
64
See: Richard Feynman, 1963, Probability Amplitudes for Particles.
65
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.
66
This is why we do not want to write it all out here (we may do so in a future paper): we think the reader should
think for himself and, hence, go through the basic equations himself. As Daisetsu Teitaru Suzuki the man who
brought Zen to the West (he published his Essays in Zen Buddhism (1927), from which I am quoting here, around
the same time): “Zen does not rely on the intellect for the solution of its deepest problems. It is meant to get at the
fact at first hand and not through any intermediary. To point to the moon, a finger is needed, but woe to those
who take the finger for the moon.” We could not agree more: if you want to be enlightened, think for yourself!
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