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A brief history of quantum-mechanical ideas
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
6 May 2020
Summary
In this paper, we pick some less well-known contributions of great minds to the history of ideas from the
proceedings of the Solvay Conferences. We hope to show there was nothing inevitable about the new
physics winning out. In fact, we suggest modern-day physicists may usefully go back to some of the old
ideas – most notably the idea that elementary particles do have some shape and size – and that they
should try somewhat harder to explain intrinsic properties of these particles – such as their angular
momentum and magnetic moment – in terms of classical physics. The contributions which we discuss
are those of Ernest Rutherford, Joseph Larmor, Hendrik Antoon Lorentz and Louis de Broglie.
Contents
The battle of ideas ........................................................................................................................................ 1
Rutherford’s idea of an electron ................................................................................................................... 6
Introduction .............................................................................................................................................. 6
Rutherford's contributions to the 1921 Solvay Conference ..................................................................... 8
Ernest Rutherford's idea of an electron in 1921 ..................................................................................... 10
Thinking backward and forward ............................................................................................................. 11
Joseph Larmor's idea of an electron ........................................................................................................... 12
The views of H.A. Lorentz on the new physics ............................................................................................ 14
Louis de Broglie’s mistake ........................................................................................................................... 17
1
A brief history of quantum-mechanical ideas
The battle of ideas
Dr. Oliver Consa traces all of the nonsense in modern physics back to the Shelter Island (1947), Pocono
(1948) and Oldstone (1949) Conferences.
1
However, the first Solvay Conference that was organized after
WW II (1948) – whose theme was ‘Elementary Particles’
2
– was also part of the watershed between
good and bad ideas, and Niels Bohr and Robert Oppenheimer – both on the side of the bad ideas,
unfortunately – pretty much dominated it. Bohr does so by providing the introductory lecture ‘On the
Notions of Causality and Complementarity’, while Oppenheimer’s paper on the ‘Electron Theory’ sets the
tone for subsequent Solvay Conferences—most notably the one that would consecrate quantum field
theory (QFT), which was held 13 years later (1961).
When going through the proceedings, it is quite obvious that Paul Dirac is pretty much the only one
asking Oppenheimer and the other new physics gurus critical questions.
3
Hence, while there are many
junctions in the history of ideas, this discussion between Oppenheimer and Dirac on the ‘Electron
Theory’ paper in 1948 is surely worth mentioning. We think both Oppenheimer and Dirac commit
historical blunders here.
Indeed, Oppenheimer’s use of perturbation theory to arrive at some kind of ‘new’ model of an electron
based on Schwinger’s version of quantum field theory is not only very confusing but also does not add
any value to classical approaches.
4
1
Oliver Consa, Something is rotten in the state of QED, February 2020.
2
The library of the Free University of Brussels (VUB/ULB) has digitized the proceedings of all Solvay Conferences
from 1911 to 1973. The reader who is fluent in French is, therefore, in a position to verify all references. We are
not aware of a freely available English translation of all of the papers and discussions, which add up to several
thousands of pages. However, you may be luckier than us when googling: if you find it, please do let us know.
3
Albert Einstein is often mentioned as an opponent of Bohr and Heisenberg in the context of these Solvay
Conferences but, truth be told, Einstein’s interventions in these Conferences are limited, and they do not suggest
much opposition to the new theories. If there were ‘heated discussions’ between Heisenberg and Einstein, or
‘ardent debate’ – or anything such as a ‘Bohr-Einstein controversy’ – then it is not documented in the formal
proceedings. Quotes attributed to either Einstein (“God does not play dice!”) or Heisenberg (“Stop telling God
what to do!”) may have been made during casual conversation – or, more probably, decades later – but surely not
during the formal proceedings. Einstein may have felt he should help H.A. Lorentz to come to mutually agreed
upon conclusions during those Conferences – rather than add to the discussions or debates – because, being the
very young scientist he was at the time, he would not have been invited to the first Solvay Conference without the
strong support of H.A. Lorentz. Einstein not interfering all that much may explain why he was such a consistent
presence: he was, effectively, a member of the Solvay scientific committee from the very first conference (1911) –
representing, in typical style, a country (Austria, not Germany) rather than an institution or just being a member in
some personal capacity – till 1948. He did not attend the 1921 Conference, however, to mark his disagreement
with the decision of the committee to not invite German scientists. He was also not a member of the 1951
scientific committee but the reason for that might well be age or a lack of interest, of course: Einstein was 72 years
in 1951, and would die four years later (1955).
4
The anomalous magnetic moment of an electron is not anomalous at all and can be explained using classical
physics (see p. 20 of our paper on classical quantum physics).
2
However, Dirac also chooses to disappear into the mist of history by continuing to defend his totally
undefendable electron equation.
5
Indeed, it is very intriguing that Dirac does not follow through on his
own conclusion:
“Only a small part of the wavefunction has a physical meaning. We now have the problem of
picking out that very small physical part of the exact solution of the wave equation.”
6
It is particularly weird because Dirac had already clearly stated his guts intuition in regard to what to
pick out 15 years before. Indeed, at the occasion of his Nobel Prize lecture (1933), he said this:
“The variables 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)
What Schrödinger and Dirac had stumbled upon is, quite simply, the physical ring current that generates
the magnetic moment of the electron. The British physicist and chemist Alfred Lauck Parson had already
proposed such ring current model for the electron back in 1915, when the magnetic properties of the
electron had not yet been experimentally proved. Arthur Compton had actually jumped on it and used it
to write a series of articles on the size and shape of an electron. He did so from 1917 to 1921
7
and, as we
will see in a moment, the ring current model of an electron was regularly referred to by other prominent
contemporaries, including Ernest Rutherford, H.A. Lorentz and other warriors in the battle of ideas then.
Unfortunately, all would be swept away by the abstract theories of the new generation of quantum
physicists led by the younger Heisenberg with strong support from the older Niels Bohr. In a rather
desperate attempt to save classical physics by destroying it
8
, they simply refused to accept an electron
might have some size and shape: they preferred to model the motion of electrons and other elementary
particles, and the atom itself, by thinking of it as a pointlike wavicle obeying abstract wave equations.
This development has, obviously, led nowhere: we are a hundred years later now, and physicists are still
utterly unable to explain the basic properties of elementary particles such as their charge radius, their
5
We are not quite sure what Dirac wanted to model with his wave equation but wave equations usually model the
properties of the medium, which is the vacuum here. If the vacuum has any properties, then they have been
modelled already by Maxwell’s equations.
6
Proceedings of the 8th Solvay Conference (1948), Discussion du rapport de M. Oppenheimer, p. 282.
7
The history here is well documented by the Wikipedia article on Parson’s model, which is also referred to as the
toroidal ring or magneton model.
8
When everything is said and done, one may say that Schrödinger, Heisenberg and other young wolves basically
adapted classical Hamiltonian mechanics by fitting quantum-mechanical operators and wavefunctions into it.
3
angular momentum, and their magnetic moment. Likewise, they are also unable to explain basic
diffraction and interference phenomena from first principles.
We are not joking here. Just as an example, we may mention that we were briefly in touch with the
PRad experimenters who put an end to the rather ridiculous 'proton radius puzzle' by re-confirming the
previously established 0.83-0.84 range for the effective charge radius of a proton.
9
We had sent them
our own classical back-of-the-envelope calculation of the Compton scattering radius of a proton based
on the ring current model, which is in agreement with these measurements and courteously asked what
alternative theories they were suggesting.
10
Their spokesman replied this:
"There is no any theoretical prediction in QCD. Lattice [theorists] are trying to come up [with
something] but that will take another decade before any reasonable number [may come] from
them."
One wonders if there is actually any real interest in solving these puzzles. The PRad team may have been
nominated for a Nobel Prize in Physics—we surely hope so because, in contrast to other Nobel Prize
laureates, the PRad team surely deserve one—but isn’t it rather incongruous to finally firmly establish
the size of a proton in some expensive experiment while, at the same time, admit that protons should
not have any size according to mainstream theory?
We should not be surprised, of course. Wave equations – linear differential equations – are useful if one
wants to model the properties of a medium in which physical waves (think of water or sound waves, for
example
11
) propagate. In fact, whether or not you believe there must be some medium in which
electromagnetic waves propagate – an aether, relativistic spacetime, or the vacuum
12
– we do have
9
See: https://www.jlab.org/prad/collaboration.html.
10
See p. 15-16 of our paper on classical physics. The email exchange was beginning of February 2020.
11
For a standard textbook derivation of the wave equation, see Chapter 47 of Richard Feynman’s Lectures on
Mechanics, Radiation and Heat.
12
The term suggest it cannot have any properties: if the vacuum is nothing, then it cannot have any properties,
right? However, Maxwell’s equations – and the discovery of the absolute velocity of light (light always travels at
lightspeed (299792458 m/s, exactly) – do actually model properties, including the property that an influence or a
signal, including light, cannot travel faster than the speed of light. Many scientists therefore continued to refer to
the concept of an aether long after Einstein’s relativity theory had already been well established (Joseph Larmor,
whom we will quote shortly, is one of them, and he does so to refer to these properties). We may usefully re-
quote Robert B. Laughlin here:
“The word ‘ether’ has extremely negative connotations in theoretical physics because of its past
association with opposition to relativity. This is unfortunate because, stripped of these connotations, it
rather nicely captures the way most physicists actually think about the vacuum. […] The modern concept
of the vacuum of space, confirmed every day by experiment, is a relativistic ether. But we do not call it
this because it is taboo.”
We may also usefully quote from Lorentz’ answer to Larmor as he mentions the concept in the discussions during
the 1921 Solvay Conference:
“As for the aether, even the physicists who still talk about it have stripped the concept of anything it
might have in common with matter. I was a believer in an immobile aether myself but I realize that,
because of relativity, we cannot talk about any force acting on the aether. However, I still think of the
aether as the seat of electromagnetic energy (“le siège de l’énergie électromagnétique”).”
4
wave equations for it: Maxwell’s equations, to be precise. However, when modeling simple or complex
systems that have some internal structure—atoms (think of Schrödinger’s wave equation here),
electrons (think of Dirac’s wave equation), or protons (which is what some others tried to do, but we will
let you do some googling here yourself), Maxwell’s equations have their limits. They are, after all, to be
used to model the properties of the medium, which is the vacuum here. That’s what they should be
used for. Hence, trying to use them to explain how an atomic system – consisting of electrons, protons
and neutrons that all interact
13
– is bound to lead to abstract generalizations that cannot possibly reflect
all of the degrees of freedom in the system that is being analyzed.
Let us go back to that 1948 Solvay Conference, where Dirac challenges the new approach based on
perturbation theory (as presented by Oppenheimer) by making 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. 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.”
Of course, Dirac is very much aware of another problem too: the wavefunctions that come out as
solutions dissipate away. Real-life electrons – or any real-life matter-particle, really – do not do that. In
fact, we refer to them as being particle-like because of their integrity⎯an integrity that, we believe, is
modeled by the Planck-Einstein relation.
14
Oppenheimer knew that too obviously (if Dirac’s theory
13
Think of the coupling of their magnetic moment, for example, which explains the hyperfine structure (including
the Lamb shift) of the spectrum of atoms. See our paper on the Lamb shift in this regard.
14
Prof. H. Pleijel, the Chairman of the Nobel Committee for Physics of the Royal Swedish Academy of Sciences in
1933, sums this rather particularly property of the new ‘matter waves’ rather well in his ceremonial speech for the
1933 Nobel Prize, which was awarded to Heisenberg for nothing less than “the creation of quantum mechanics”:
“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 familiar to you. The problem is this: this is everything but true! Real-life particles – electrons or
atoms traveling in space – do not dissipate: matter does not change form and extent in space! It is most
remarkable that nonsense like this has survived critical thought for almost a hundred years now!
As for using the Planck-Einstein relation in particle modeling, we refer the reader to our classical or realist
interpretation of quantum mechanics.
5
worked, there would have been no need for a new theory) and so Dirac immediately adds the following:
“We would not get infinities if we solve the wave equations without using a perturbation
method [but] 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
15
, 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.”
16
Again, 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.
17
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 we see now.
18
In any case, the gist of the matter is
that Dirac was rather easy prey in this 1948 discussion⎯no match at all for the brilliant Oppenheimer.
Dirac’s defense of his meaningless wave equation is, therefore, stubbornly foolish
19
: he clearly loved it
too much to jettison it.
“Kill your darlings” is a common piece of advice given to unexperienced writers. Dirac should have
thought about it, but then it is hard to criticize a theory for which one got a Nobel Prize, isn’t it?
20
Here
15
See our remarks in footnote 14: wavefunctions dissipate away. The matter-particles they purport to describe
obviously do not.
16
See pp. 282-283 of the report of the 1948 Solvay Conference, Discussion du rapport de Mr. Oppenheimer.
17
See the quote from Dirac’s 1933 Nobel Prize speech in this paper.
18
One of our correspondents wrote us this however: “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 prof in 1979 believed Schrödinger’s equation yielded the exact solution (electron orbitals)
for hydrogen.” Hence, we are probably too harsh in our judgment here.
19
I think it is significant that Paul A.M. Dirac was no longer invited nor even mentioned in the 1951 Solvay
Conference. Niels Bohr and Wolfgang Pauli made subsequent remarks that clearly show Paul Dirac was the only
one who actually dared to challenge what had, by then, become mainstream theory.
20
The Wikipedia article on Paul Dirac mentions Dirac boasting to the younger Feynman at a conference: “I have an
equation. Do you have one too?” We do not know if this story is true, nor are we aware of the context in which
this might have been said. Whatever the story, it is obvious from his Principles of Quantum Mechanics that Dirac
believed the task of physicists was to model what he referred to as the ‘equations of motion’ for elementary
particles. He showed no inclination to think about their possible shape, size or other fundamental properties. I am
grateful to a fellow amateur physicist who pointed out that, in 1962, Dirac did finally think about some electron
model. He got it published by the Royal Society but it attracted little attention, and rightly so because it hardly
explained anything. We may quote the abstract:
“It is proposed that the electron should be considered classically as a charged conducting surface, with a
surface tension to prevent it from flying apart under the repulsive forces of the charge. Such an electron
has a state of stable equilibrium with spherical symmetry, and if disturbed its shape and size oscillate. The
equations of motion are deduced from an action principle and a Hamiltonian formalism is obtained. The
energy of the first excited state with spherical symmetry is worked out according to the Bohr-Sommerfeld
method of quantization, and is found to be about 53 times the rest-energy of the electron. It is suggested
6
again, the Nobel Prize Committee for Physics may actually have consecrated bad theory that, as a result
of the award, became difficult – if not impossible – to criticize or backtrack on.
We think it is significant that Paul A.M. Dirac was no longer invited nor even mentioned in the 1951
Solvay Conference which, as a result, was much more consensual.
21
We will, therefore, not comment on
it⎯nor will we comment on the later conferences (not in this paper, at least). The battle for ideas was
over by then anyway. The question we should ask ourselves today is this: was this unavoidable? Was all
of this a logical historical evolution? Was there a choice, in other words?
22
We think there was⎯and we also think there still is a choice. We think the good ideas lost out and that,
after 100 years of a non-theory or a non-explanation
23
, we should go back to them. Let us, therefore,
effectively try to trace them back. Of course, the proceedings of the Solvay Conferences are several
thousands of pages and we will, therefore, limit ourselves to what we think of as crucial papers and
interventions. We will start with a fragment from Rutherford’s presentation on The Structure of the
Atom at the 1921 Solvay Conference.
Rutherford’s idea of an electron
Introduction
The New Zealand-born Ernest Rutherford came to be known as the father of nuclear physics. He was the
first to provide a reliable estimate of the order of magnitude of the size of the nucleus. To be precise, in
the 1921 paper – which is the one we will talk about here – he came up with an estimate of about 15 fm
for massive nuclei, which is pretty much the current estimate for the size of an uranium nucleus. In light
of the rather primitive equipment at the time, this is more than impressive!
Rutherford’s experiments also helped to significantly enhance the Bohr model of an atom, culminating -
just before WW I started - in the Bohr-Rutherford model of an atom.
24
This Bohr-Rutherford model of an
atom explained the (gross structure of the) hydrogen spectrum perfectly well, but it could not explain its
finer structure—read: the orbital sub-shells which, as we all know now (but not very well then), result
from the different states of angular momentum of an electron and the associated magnetic moment.
Let us quickly illustrate the issue by the two diagrams below, which we copied from Feynman's Lectures.
As you can see, the idea of subshells is not very relevant when looking at the gross structure of the
hydrogen spectrum because the energy levels of all subshells are (very nearly) the same. However, the
that this first excited state may be considered as a muon. The present theory has no electron spin, so it
cannot agree accurately with experiment.”
Frankly, this sounds like a bit of a joke to us: is this all what Dirac could come up with⎯in 1962?
21
Apart from Paul Dirac, Niels Bohr and Wolfgang Pauli commented on Oppenheimer’s paper, but their remarks
clearly show Paul Dirac was the only one who actually dared to challenge what had, by then, become mainstream
theory.
22
We request the reader to not try to connect the concepts of determinism and indeterminism at the atomic level
to the same concepts at the level of societal debates.
23
See our paper on the difference between a scientific theory and an explanation.
24
See: E. Rutherford, Phil. Mag. 27, 488.
7
diagram on the right-hand side shows that the Bohr model of an atom—which is nothing but an
exceedingly simple application of the E = h·f equation
25
—cannot explain the splitting of lines for a
lithium atom. Nor can it explain the splitting of spectral lines when we apply a stronger or weaker
magnetic field while exciting the atoms so as to induce emission of electromagnetic radiation.
Rutherford was well aware of this problem and, as we will see in a moment, by 1921 he probably
already had some ideas on how to fix it⎯because a significant part of his paper talks about the
magnetic moment of electrons, even if the evidence of such magnetic moment was rather rough at the
time!
However, using the new physics, Schrödinger would pre-empt him with his wave equation—which is
why Feynman and other modern physicists claim this equation is "the most dramatic success in the
history of the quantum mechanics" or, more modestly, a "key result in quantum mechanics" at least!
26
Such dramatic statements are wildly exaggerated. First, an even finer analysis of the emission spectrum
(of hydrogen or whatever other atom) reveals that Schrödinger's wave equation is also incomplete: the
hyperfine splitting, the Zeeman splitting (anomalous or not) in a magnetic field, or the (in)famous Lamb
shift 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.
27
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. Indeed, we
should mention once again that simple mathematical tools such as second-order differential equations
25
See p. 4-6 of our paper on classical quantum physics.
26
See Chapter 19 of Feynman’s Lectures on Quantum Mechanics, from which we also copied the two textbook
illustrations. The other quote is from the Wikipedia article on Schrödinger’s equation.
27
See our paper on the Lamb shift here.
8
are – quite simply – not sophisticated enough to capture the complexity of the atomic system.
Moreover, even the solution to the wave equation – the wavefunction – does not adequately represent
the physical reality of a spinning particle. Indeed, as we pointed out previously
28
, the current
convention in regard to the use of the imaginary unit (i) in the wavefunction does not capture the
spin direction and, therefore, makes abstraction of the direction of the magnetic moment too! The
wavefunction, therefore, models theoretical spin-zero particles which do not exist. In short, we cannot
hope to represent anything real with wave equations and wavefunctions.
Secondly, we would think the following question is even more important: what use is an 'explanation' in
terms of a wave equation or a wavefunction if we cannot explain what the wave equation or the
wavefunction actually represents? As Feynman famously writes: "Where did we get [Schrödinger’s
equation] from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind
of Schrödinger, invented in his struggle to find an understanding of the experimental observations of the
real world."
29
Our best guess is that it may, somehow, model the (local) diffusion of energy or mass densities as well as
the spherical and non-spherical geometries of the electron orbitals. We explored such interpretations in
our very first paper(s) on quantum mechanics.
30
However, the truth is that such interpretations are
probably incomplete or plain irrelevant because of what we mentioned a few times already: wave
equations are probably not the tools we need to describe simple or complex systems that have some
internal structure—atoms (think of Schrödinger's wave equation here), electrons (think of Dirac's wave
equation), or protons (which is what some others tried to do but, as mentioned before, we will let you
do some googling here yourself).
We will let you think about this because we need to move on. We need to get back to the matter at
hand here, which is Rutherford's idea of an electron back in 1921. What can we say about it?
Rutherford's contributions to the 1921 Solvay Conference
From what you know, from what we write above, and from the title of Rutherford’s 1921 paper (La
Structure de l’Atome) you will understand that Rutherford's research focus was not on electrons: his
prime interest was in explaining the atomic structure and in solving the mysteries of nuclear radiation—
most notably the emission of alpha- and beta-particles as well as highly energetic gamma-rays by
unstable or radioactive nuclei. In short, the nature of the electron was not his prime interest.
However, this intellectual giant was, of course, very much interested in whatever experiment or
whatever theory that might contribute to his thinking, and that explains why, in his contribution to the
1921 Solvay Conference—which materialized as an update of his seminal 1914 paper on The Structure of
the Atom—he devotes considerable attention to Arthur Compton's work on the scattering of light from
electrons which, at the time (1921), had not even been published yet (Compton's seminal article on
(Compton) scattering
31
was published in 1923 only).
28
See, for example, our paper on the difference between a scientific theory, a calculation and an explanation.
29
Chapter 16 of Feynman’s Lectures on Quantum Mechanics.
30
See, for example, our paper on a possible geometric explanation of Schrödinger’s equation.
31
Arthur Compton, A Quantum Theory of the Scattering of X-rays by Light Elements, Phys. Rev. 21, 483 (1 May
9
It is also very interesting that, in the very same 1921 paper—whose 30 pages are more than a multiple
of his 1914 article
32
and later revisions of it (see, for example, the 1920 version of the same article,
which actually has wider circulation on the Internet)—Rutherford also offers some short reflections on
the magnetic properties of electrons while referring to Parson's ring current model which, in French, he
refers to as "l’électron annulaire de Parson."
It is, of course, somewhat strange that we should translate Rutherford's paper for the 1921 Solvay
Conference back in English—as we are sure the original paper must have been translated from English to
French rather than the other way around. However, it is what it is, and so here we will do what we feel
we should do: translate some of Rutherford's remarks during the 1921 Solvay Conference on the state of
research regarding the electron at that time.
These remarks on the electron are, in fact, part of a larger piece on the emission of β particles by
radioactive nuclei which, as it turns out, are nothing but high-energy electrons (or their anti-matter
counterpart—positrons). In fact, we should—before we proceed—draw attention to the fact that the
physicists at the time had no clear notion of the concepts of protons and neutrons. This is, indeed,
another remarkable historical contribution of the 1921 Solvay Conference because, as far as we know,
this is the first time Rutherford talks about the neutron hypothesis, so let us quickly say something about
it.
First, we should note that it is quite remarkable that Rutherford does not advance the neutron
hypothesis to explain the atomic mass of atoms combining what we know think of as protons and
neutrons (Rutherford regularly talks of a mix of 'positive and negative electrons' in the nucleus—neither
the term proton or neutron was in use at the time) but as part of a possible explanation of nuclear
fusion reactions in stars or stellar nebulae. This is, indeed, his response to a question during the
discussions on Rutherford's paper on the possibility of nuclear synthesis in stars or nebulae from the
French physicist Jean Baptiste Perrin who, independently from the American chemist William Draper
Harkins, had proposed the possibility of hydrogen fusion just the year before (1919):
"We can, indeed, think of enormous energies being released from hydrogen nuclei merging to
form helium—much larger energies than what can come from the Kelvin-Helmholtz
mechanism.
33
I have been thinking that the hydrogen in the nebulae might come from particles
which we may refer to as 'neutrons': these would consist of a positive nucleus with an electron
at an exceedingly small distance ("un noyau positif avec un électron à toute petite distance").
These would mediate the assembly of the nuclei of more massive elements. It is, otherwise,
difficult to understand how the positively charged particles could come together against the
repulsive force that pushes them apart—unless we would envisage they are driven by enormous
velocities."
Rutherford is immediately requested to elaborate his point by the Danish physicist Martin Knudsen, who
asks him this follow-up question: "What's the difference between a hydrogen atom and this neutron?"
1923)
32
Ernest Rutherford, The Structure of the Atom, Phil. Mag. 27, 488 (1914).
33
The reader can google what this is about.
10
Rutherford’s answers is this: "In a neutron, the electron would be very much closer to the nucleus."
In light of the fact that it was only in 1932 that James Chadwick would experimentally prove the
existence of neutrons (and positively charged protons), we are, once again, deeply impressed by the
foresight of Rutherford and the other pioneers here: the predictive power of their theories and ideas is,
effectively, truly amazing by any standard—including today's. We should, perhaps, also add that we fully
subscribe to Rutherford's intuition that a neutron should be a composite particle consisting of a proton
and an electron
34
—but that is a different discussion altogether.
We must come back to the matter at hand, and that is the electron. Before we proceed, however, we
should highlight one other contextual piece of information here: at the time, very little was known
about the nature of α and β particles. We now know that beta-particles are electrons, and that alpha-
particles combine two protons and two neutrons (as such, they are nothing but a helium nucleus). That
was not known in the 1920s, however: Rutherford and his associates could basically only see positive or
negative particles coming out of these radioactive processes.
35
Hence, it would seem Rutherford is not
quite sure if the beta-particle is, effectively, an electron. This further underscores how good his
intuitions were, and how much knowledge he was able to gain from rather limited sets of data.
Ernest Rutherford's idea of an electron in 1921
We thank the reader for patiently waiting. So here is the translation of some of Rutherford’s remarks on
the electron as part of his paper on atomic structure. Needless to say, the italics, boldface and additions
between [brackets] are not Rutherford's but ours, of course.
"We may think the same laws should apply in regard to the scattering ["diffusion"] of α and β
particles.
36
However, we see marked differences. Anyone who has carefully studied the
photographs from the Wilson cloud chamber of beta-particles will note the trajectories show a
regular curvature. Such curved trajectories are even more obvious when they are illuminated by
X-rays. Indeed, A.H. Compton noted that these trajectories seem to end in a converging helical
path turning right or left. To explain this, Compton assumes the electron acts like a magnetic
dipole whose axis is more or less fixed, and that the curvature of its path is caused by the
magnetic field [from the (paramagnetic) materials that are used].
Further examination would be needed to make sure this curvature is not some coincidence, but
the general impression is that the hypothesis may be quite right. We also see similar curvature
and helicity with α particles in the last millimeters of their trajectories.
37
34
See our paper on the possible nature of protons and neutrons.
35
Needless to say, they could infer quite a lot from these trajectories.
36
Earlier in his paper, Rutherford noted that, based on the scattering patterns and other evidence, the force
around the nucleus must respect the inverse square law near the nucleus—moreover, it must also do so very near
to it. To be precise, Rutherford mentions the experiments confirm the inverse square law remains valid to as close
as 40 fm (4×10−12 cm) from the pointlike nucleus.
37
We should note here that α-particles are, obviously, also charged particles and, hence, they may also have some
magnetic moment. However, we think Rutherford's remark in regard to α particles also following a curved or
helical path in the “last millimeters” of their path must be exaggerated or over-enthusiastic: the order of
magnitude of the magnetic moment of protons and neutrons is much smaller and, in any case, they tend to cancel
11
The idea that an electron has magnetic properties is still sketchy and we would need new and
more conclusive experiments before accepting it as a scientific fact. However, it would surely be
natural to assume its magnetic properties would result from a rotation of the electron.
Parson's ring electron model ["électron annulaire"] was specifically imagined to incorporate such
magnetic polarity ["polarité magnétique"].
A very interesting question here would be to wonder whether such rotation would be some
intrinsic property of the electron or if it would just result from the rotation of the electron in its
atomic orbital around the nucleus. Indeed, James Jeans usefully reminded me any asymmetry in
an electron should result in it rotating around its own axis at the same frequency of its orbital
rotation.
38
We should also wonder if an electron might acquire some rotational motion from being
accelerated in an electric field and if such rotation, once acquired, would persist when
decelerating in an(other) electric field or when passing through matter. If so, some of the
properties of electrons would, to some extent, depend on their past." (Ernest Rutherford, 1921
(3rd Solvay Conference), italics and boldface added).
Each and every sentence in these very brief remarks is wonderfully consistent with the ring current
model of an electron and/or more modern-day modelling of electron behavior
39
⎯modern non-
mainstream modeling of electrons we should say, of course, but then the addition is superfluous
because mainstream physicists stubbornly continue to pretend electrons have no internal structure nor
physical dimension, so they do not have any electron model. While, in light of the
numerous experimental measurements of the effective charge radius as well as of the dimensions of the
physical space in which photons effectively interfere with electrons, such mainstream assumptions seem
completely ridiculous, this is unfortunately the rather sad state of physics today.
Thinking backward and forward
We think that Rutherford and others would have been able to adapt their model of an atom to better
incorporate the magnetic properties not only of electrons but also of the nucleus and its constituents
(protons and neutrons) if they would have had the time and space. Unfortunately, scientists at the time
seem to have been swept away by the new ideas and charisma of Bohr, Heisenberg and others, as well
as by the mathematical brilliance of the likes of Sommerfeld, Dirac, and Pauli, of course! They, therefore,
decided to take a new road instead. We now know that new road has not led us very far. We concur
with Oliver Consa's scathing but essentially correct appraisal of the current sorry state of physics:
"QED should be the quantized version of Maxwell’s laws, but it is not that at all. QED is a simple
addition to quantum mechanics that attempts to justify two experimental discrepancies in the
Dirac equation: the Lamb shift and the anomalous magnetic moment of the electron. The reality
each other out. Also, because of the rather enormous mass of α particles (read: helium nuclei) as compared to
electrons, the effect would probably not be visible in a Wilson cloud chamber.
38
The reader can easily imagine or verify this for himself: think of an asymmetric object going around in a circle
and returning to its original position. In order to return to the same orientation, it must rotate around its own axis
one time too!
39
Think, for example, of the Dirac-Kerr-Newman electron model of Alexander Burinskii.
12
is that QED is a bunch of fudge factors, numerology, ignored infinities, hocus-pocus,
manipulated calculations, illegitimate mathematics, incomprehensible theories, hidden data,
biased experiments, miscalculations, suspicious coincidences, lies, arbitrary substitutions of
infinite values and budgets of 600 million dollars to continue the game. Maybe it is time to
consider alternative proposals. Winter is coming."
40
We suggest we should just go back where we went wrong: it may be warmer there, and thinking both
backward as well as forward must be a much more powerful problem-solving technique than relying
only on expert guessing on what linear differential equation(s) might give us the S-matrix!
41
Let us get back to the Solvay Conference. Rutherford was an intellectual giant, but he was just one of the
many presenters at the 1921 Solvay Conference. What about the others? Again, we cannot possibly give
a complete overview of the Conference but we may single out one or more lesser known scientists who
were present there. Joseph Larmor – whom we know from the Larmor frequency of precession of
atomic or elementary magnetic dipoles – was one of them, so let us see what he had to say back in
1921.
Joseph Larmor's idea of an electron
Joseph Larmor is surely not among the more famous participants in the Solvay Conferences. He only
joined the 1921 Conference, together with Charles Glover Barkla and others, and his one and only
substantial intervention there is limited to some remarks and questions following a presentation by H.A.
Lorentz on the Theory of Electrons, during which Lorentz highlights all of the issues in regard to what
was then supposed to be the understanding of what an electron actually is (which, in my not-so-humble-
view, is still pretty much the state of our current understanding of it). We find his one intervention – and
Lorentz' reply to it – very interesting though
42
:
"I understand that Mr. Lorentz was given the task to give an overview of how electrons behave
inside of an atom. That requires an overview of all possible theories of the electron. That is a
highly worthwhile endeavor which, in itself, would already justify the holding of this Conference.
However, Mr. Lorentz might have paid more attention to the viewpoint that the electron has
some structure, and that its representation as a simple distribution of electric charge can only
40
Oliver Consa, Something is rotten in the state of QCD, February 2020.
41
An S-matrix (scattering matrix) relates all likely or possible initial and final states of some system or process: it
shows us what happens, but it does not tell us how things happen. As such, we do not think much of it: it is a
calculation instead of an explanation. See our paper on the difference between a theory, a calculation and an
explanation.
42
We should, again, note that Joseph Larmor’s intervention was probably in English and it is, therefore, somewhat
strange that we should translate back from the French to English. As for H.A. Lorentz, he was fluent in French,
English, and German (and his native language, of course⎯Dutch), so he may also have replied in English. We may
mention, in this regard, that Marcel de Broglie had already mobilized his younger brother, Louis de Broglie, for the
very first Solvay Conferences⎯if only to help with translation. As the historian Jagdish Mehra notes, he was only
nineteen years old when attending the first Solvay Conference and it made a deep impression on him. He would
later write: “With all the ardor of my youth, I was swept away by my enthusiasm for the problems discussed and I
resolved to devote all my efforts to understanding the true nature of the mysterious quanta that Max Planck had
introduced ten years earlier.”
13
be provisional: electrons explain electricity, but electricity does not explain electrons. However,
the description of an electron in terms of a charge distribution is, for the time being, all we can
imagine. In the past, we thought of the atom as an indivisible unit – a fundamental building
block – and we imagined it as a swirling ring. That idea is gone now, and the electron has now
taken the place of the atom as an indestructible unit. All we can know about it, is how it
influences other bodies. If this influence is transmitted all across the aether, we need to be able
to express the relations between the electron and the aether
43
, or its force field in the space
that surrounds it. It may have other properties, of course, but physics is the science that should
analyze the influence or force of one body upon others.
The question we should raise here is whether or not an electron formed by a perfectly uniform
current ring can grab onto the aether in a physical sense, and how it does so if its configuration
does not change." (Joseph Larmor, 1921, italics added)
Larmor then talks about the (possible) use of the energy-momentum tensor to address the latter
question, which is a very technical discussion which is of no concern to us here. Indeed, the question on
how to use tensors to model how an electron would interact with other charges or how it would create
an electromagnetic field is, effectively, a rather standard textbook topic now.
44
What grabbed my
attention here was, effectively, not the technicality of the question in regard to the exact machinery of
the electromagnetic force or field. It was Larmor's description of the electron as a perpetual or
persistent current ring (the French reference to it is this: un electron formé par un courant annulaire
parfaitement uniforme), and his language on it, which indicates he thought of it as a rather obvious and
natural idea!
In other words, his intervention strongly suggests that Parson's 1915 toroidal ring model – the precursor
to Schrödinger's Zitterbewegung model and modern-day ring current models – was apparently pretty
well established at the time! In fact, Rutherford's lecture on the Structure of the Atom at the 1921
Conference further confirms this, as he also talks about Parson's électron annulaire (ring electron) and
the apparent magnetic properties of the electron. Of course, Larmor's belief that the electron was not
pointlike should, in fact, not surprise us in light of his rather famous work on the quantum-mechanical
precession of the magnetic moment of an electron, but we actually were not aware of Joseph Larmor's
own views in regard to its possible reality. In fact, we are only guessing here but his rather strong views
on its reality may explain why the scientific committee − which became increasingly dominated by
scientists in favor of the Bohr-Heisenberg interpretation of physical reality (basically saying we will never
be able to understand it) − did not extend an invitation to Larmor to attend the all-important Solvay
conferences that would follow the 1921 Conference and, most notably, the 1927 Conference that split
physicists between realists and... Well... Non-realists, we should say, I guess. :-)
Lorentz' immediate reaction to Larmor mentioning the idea of a swirling ring (in French: un anneau
tourbillon), which is part of his reply to Larmor's remarks, is equally interesting:
43
See footnote 12 for some remarks on the idea of the aether and the vacuum. We also mention Lorentz’ response
to this topic there.
44
In case you'd be interested, you can check my blog on it or, else, (re-)read Chapters 25, 26 and 27 of
Feynman's Lectures on electromagnetism.
14
"There is a lot to be said for your view that electrons are discontinuities in the aether. [...] The
energy-momentum formulas that I have developed should apply to all particles, with or without
structure. The idea of a rotating ring [in French: anneau tournant] has a great advantage when
trying to explain some issues [in the theory of an electron]: it would not emit any
electromagnetic radiation. It would only produce a magnetic field in the immediate space that
surrounds it. [...]" (H.A. Lorentz, 1921, boldface and italics added)
Isn't that just great? Lorentz' answer to Larmor's question surely does not solve all of the problems
relating to the interpretation of the electron as a current ring, but it sure answers that very basic
question which proponents of modern quantum mechanics usually advance when talking about the so-
called failure of classical physics: electrons in some electron orbital in an atom should radiate their
energy out, but so they do not. Let us actually quote from
45
Feynman's Lectures on Quantum Mechanics
here:
"Classically, the electrons would radiate light and spiral in until they settle down right on top of
the nucleus. That cannot be right."
Surely You're Joking, Mr. Feynman! Here is the answer of the classical quantum theorists:
superconducting rings of electric current do not radiate their energy out either, do they?
Let us continue, however. Next, we would like to present the views of the scientist who took most of the
burden of actually organizing the Solvay Conferences on his shoulders and who, among all of these
intellectual giants, may well be the ‘giant of giants’: Hendrik Antoon Lorentz. We actually already let him
talk in 1921 above, so let us move forward in time to see what he had to say at the even more
(in)famous Solvay Conference of 1927.
The views of H.A. Lorentz on the new physics
It is part of scientific lore that the 1927 Solvay Conference was the real battlefield on new physics, and
that the generals on the two sides were Heisenberg and Einstein, respectively. However, while
Heisenberg is, effectively, all over the place⎯presenting, together with Max Born, the main paper
(simply titled ‘Quantum Mechanics’
46
) at the Conference as well as actively participating in discussions
on this and the other papers
47
⎯the proceedings of the conference reveal that Einstein hardly
intervened.
The same reports of the proceedings also reveal that 'battlefield stories' such as Heisenberg telling
Einstein to "stop telling God what to do" or - vice versa - Einstein declaring "God doesn't play dice" are
what they are: plain gossip or popular hear-say. Neither Heisenberg nor Einstein ever said that—or not
at the occasion of the 1927 Solvay Conference, at least! Instead, we see very nuanced and very deep
45
See: Chapter 2, Section 4 of Feynman's Lectures on Quantum Mechanics
46
Max Born and Werner Heisenberg, La Mécanique des Quanta, 5th Solvay Conference (1927).
47
The other key players are Louis de Broglie, who presents his views on the matter-wave, and Erwin Schrödinger,
who presents a paper on wave mechanics. Niels Bohr then sums everything in the last intervention (The Quantum
Postulate and the New Atomistic Theory, in French: Le postulat des quanta et le nouveau développement de
l’atomistique), which elicits the response from H.A. Lorentz that we have translated in this paper. The papers of
W.L. Bragg and Arthur Compton are interesting as well, but clearly did not do much to stop the new movement.
15
philosophical statements—on both sides of the so-called 'divide' or 'schism'. We think the intervention
of the Dutch scientist Hendrik Antoon Lorentz stands out among of these. In fact, his intervention is
quite unlike his other remarks and interventions. It stands out because it is somewhat emotional and,
with the benefit of hindsight once more, because it is also very historical: H.A. Lorentz – who was,
without any doubt, clearly the driving force behind all of the pre-WW II Solvay Conferences till 1927 –
would, effectively, die a few months later.
48
In fact, the 1927 conference proceedings have both the sad
announcement of his demise as well his interventions—such was the practice of actually physically
printing stuff at the time.
GENERAL DISCUSSION OF THE NEW IDEAS ON CAUSALITY, DETERMINISM AND PROBABILITY
49
M. LORENTZ: "I would like to draw your attention to the difficulties in these theories. We are trying to
represent phenomena. We try to form an image of them in our mind. Till now, we always tried to do
using the ordinary notions of space and time. These notions may be innate; they result, in any case,
from our personal experience, from our daily observations. To me, these notions are clear, and I admit I
am not able to have any idea about physics without those notions. The image I want to have when
thinking physical phenomena has to be clear and well defined, and it seems to me that cannot be done
without these notions of a system defined in space and in time.
To me, the electron is a particle which, at any moment, must be at some specific point in space, and if I
think it should be somewhere else at the next moment, then I need to be able to think of its trajectory,
which is a line in space. And if that electron meets an atom and penetrates it and if it, after several
adventures, leaves that atom, then I need to have some theory in which that electron conserves its
individuality. In other words, I actually think of a trajectory of the same electron within the atom.
Now, it may be difficult to develop such theory but, a priori, this should not be impossible. Hence, I
would think that, in your new theories, you would still have electrons. It is possible, of course, that these
electrons may be subject to transformations. I am willing to think of electrons as some kind of cloud, but
even then I would look for the event that produces these transformations. If one would like to tell me
that such (re)search is not allowed by invoking some principle, then I would be much annoyed. I would
probably reply that, what we cannot do now, we may be able to do at a later point. Even we abandon
our old ideas, we should still be able to translate things back to those old ideas. Hence, I would like to
conserve the old ideal: to talk about the things that happen in this world in clear and well-defined
images. I am willing to accept new theories, but only on the condition that they should allow me to
translate things back to these clear and well-defined images.
I am not so well acquainted with the ideas that have just been revealed, but I would think of them like
this. Let us effectively take the case of an electron leaving an atom, and let us assume there has been an
emission of a photon.
50
We should then consider the wave system that corresponds to the atom and the
48
Lorentz died of a very trivial cause: erysipelas, commonly known as St Anthony's fire. Hence, this may well have
been one of his very last public statements⎯if not the last.
49
This general discussion follows all of the presentations by Heisenberg, Schrödinger, Bohr, etcetera. See the
proceedings of the Conference itself: even if the reader is not familiar with French, he will be able to understand
the titles of the papers and recognize their authors.
50
Lorentz uses the terms photon and a quantum of light interchangeably.
16
electron, respectively. We will have a new system before and after the emission of a photon. This wave
system may then be described by some ψ function in a (mathematical) space with multiple dimensions,
and it would probably satisfy some differential equation. The new wave mechanics should then
effectively describe this function before and after the emission of the light quantum, and we can
imagine other experiences, such as a beam of particles inside a Faraday cylinder, for example, and [but]
we would again have to take into account the individuality of the electrons, and also of the photons.
Now, I can understand that the ψψ* expression
51
would effectively give us the probability of the
electron or the photon being in some clearly defined space and, perhaps, that we would have to content
ourselves with this information.
However, the examples that have been given by Mr. Heisenberg tell me that I should not hope to be
able to learn anything more⎯that this probability is all that I can possibly hope to learn from the
experiment. However, I would think such notion of probability would come at the end, and as a
conclusion, to the theoretical considerations of any experiment⎯not as an a priori axiom.
Hence, while I am prepared to admit that the conditions of an experiment are such that, from a practical
point of view, we would have indeterminism, I would still keep my deterministic belief in fundamental
phenomena, of which we cannot talk, perhaps. Perhaps a deeper mind could know the movements of
these electrons? Can we not keep determinism as an object of faith? Why do we have to elevate
indeterminism to a philosophical principle?”
[…]
As my translation of the latter paragraph involves some interpretation, we would like to quote the same
in the original French language:
“Je pense que cette notion de probabilité [in the new theory] serait à mettre à la fin, et comme
conclusion, des considérations théoriques, et non pas comme axiome a priori, quoique je veuille
bien admettre que cette indétermination correspond aux possibilités expérimentales. Je
pourrais toujours garder ma foi déterministe pour les phénomènes fondamentaux, dont je n’ai
pas parlé. Est-ce qu’un esprit plus profond ne pourrait pas se rendre compte des mouvements
de ces électrons. Ne pourrait-on pas garder le déterminisme en en faisant l’objet d’une
croyance? Faut-il nécessairement ériger l’ indéterminisme en principe?"
What a beautiful statement: why should we elevate indeterminism to a philosophical principle?
Let us move to the next: we think we should, perhaps, say a few words about the man who invented the
matter-wave: Comte Louis de Broglie.
52
We all know that Louis de Broglie was the first to postulate the
wave nature of matter – and of the electron in particular – in his 1924 PhD thesis, for which he was
praised by Albert Einstein. His 1927 paper for the Solvay Conference is, therefore, highly significant. So
where did he go wrong?
51
Lorentz refers to the standard interpretation of the absolute square of the wavefunction.
52
Louis de Broglie would become the 7th Duc de Broglie when his older brother Maurice de Broglie – the 6th Duc de
Broglie – passed away. Maurice de Broglie was an active participant in the earlier Solvay Conferences and was,
therefore, in a good position to introduce his younger brother to physics and all of the great minds of the time.
17
Louis de Broglie’s mistake
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 Louis de Broglie that an electron must have a frequency was,
effectively, a stroke of genius. However, here too we wonder why he did not consider to further build on
Parson’s ring current model of an electron, especially in light of that nasty property of a wave packet: it
dissipates away. Real-life electrons stay together.
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.
53
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
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.
54
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. Now, we
already mentioned we need a wave equation only to model the properties of the medium – which,
supposedly, is the vacuum here – through which these waves travel.
We can, for example, use the Schrödinger equation without the Coulomb term
55
:
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.
56
The
53
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.
54
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):
55
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.
56
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.
18
question now is: are we talking one wave or many waves? Let us first make the analysis for one wave
only, assuming that we can write ψ as some elementary wavefunction ω = a·eiθ = a·ei·(k(x−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 equations
57
:
• 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
58
:
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
59
:
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:
We refer the reader to previous papers on how one can now relate the uncertainties in the (kinetic)
57
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.
58
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.
59
We apologize if this sounds slightly ironic but we are actually astonished Louis de Broglie and so many others
shamelessly assume superluminal speeds for wave velocities, even if it is a phase rather than a group velocity.
19
energy and the (linear) momentum of our particle using the relativistically correct energy-momentum
relation and also taking into account 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.
60
We come to the following conclusions:
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 may now refer the reader to de Broglie’s 1924 thesis, his 1927 Solvay paper (even the original would
have been in French, unfortunately) or whatever other English-language article or paper from de Broglie
that the reader is able to google. He or she can then compare the logic above with de Broglie’s logic, and
understand where Louis de Broglie disappears into the mist. Indeed, in physics – perhaps even more
than in other sciences – Aristoteles’ maxim applies too: a small mistake in the beginning is a big one in
the end.
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It is, therefore, quite essential to go back where de Broglie went wrong, rather than try to fix the
problem later⎯as Louis de Broglie would later try to do himself by introducing the concept of pilot
waves and other metaphysical nonsense. Here again, we think the Nobel Prize Committee could have
used some common sense before awarding him the Nobel Prize two years later (in 1929, to be precise):
wavefunctions may dissipate, but real-life particles definitely do not.
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Hence, a matter-wave that
quickly dissipates in space cannot possibly represent anything real. At best, it can model our own
uncertainty in regard to reality, but not reality itself.
We may, in this regard, also usefully note here that Heisenberg initially preferred to use the German
term Ungenauigkeit to describe the apparent randomness in our mathematical description of Nature.
Ungenauigkeit translates as imprecision and it is a concept that is valid in classical mechanics as well.
Indeed, it is just what it is: an imprecision, as opposed to the weird metaphysical quality which
Heisenberg would later claim it to be and which, without any precise definition, physicists now refer to
as ‘uncertainty’. We may, therefore, use Lorentz’s question as an affirmative statement: there is no
need whatsoever to elevate indeterminism to a philosophical principle.
In fact, without the assumption of determinism at the most elementary level of analysis, science would
basically not be science: we would relegate it to the realm of beliefs.
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See pp. 72-74 and 94-97 of our popular book (The Emperor Has No Clothes: A Realist Interpretation of Quantum
Mechanics). We have more detailed mathematical analysis in our other papers.
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This is usually quoted from Aquinas De Ente et Essentia (Quia parvus error in principio magnus est in fine) but
Thomas Aquinas actually quotes ‘The Philosopher’, which is Aristoteles.
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One should note that small initial uncertainties quickly become much larger ones: this is a common, in fact, the
foundation of chaos theory.