The Meaning of Uncertainty and
the Geometry of the Wavefunction
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
4 November 2020 (corrected on 15 September 2022
Uncertainty may result from (1) an impossibility to measure what we want to measure, i.e. an
impossibility to observe the system, (2) the limited precision of our measurement, (3) the measurement
fundamentally disturbing the system and, as such, causing the information to be unreliable, (4) an
uncertainty that is inherent to Nature. The latter position is referred to as the Copenhagen
interpretation of quantum mechanics. We agree with Lorentz’s and Einstein’s viewpoint that there is no
need to elevate indeterminism to a philosophical principle. The more important question is: how does
quantum physics model it? How does it deal with it?
This paper offers some thoughts on that and, in the process, highlights some contradictions which
support Lorentz’s (and Einstein’s) position: we only have statistical indeterminism in quantum physics
and, as such, quantum physics is not a radical departure from classical physics. Statistical indeterminism
is, effectively, the fifth interpretation of uncertainty which can be added to the list above, and we think
it is the right one. We illustrate our position with a detailed discussion of the wavefunction(s) in the
context of Schrödinger’s wave equation for the hydrogen atom. The same example also further explores
the question in regard to the (possible) physical dimension of the real and imaginary part of the
wavefunction. To paraphrase Feynman, we wonder what could be ‘sloshing back and forth’ between the
real and imaginary part of the wavefunction? We think it is kinetic and potential energy. We, therefore,
briefly present our two-dimensional oscillator model again, but using the metaphor of a multi-piston
radial engine as a metaphor this time, and augmented by an analysis of the quantum-mechanical energy
Introduction ................................................................................................................................................................... 1
Functions and physical dimensions ............................................................................................................................... 2
What does it all mean? .................................................................................................................................................. 5
The oscillator model ...................................................................................................................................................... 9
The meaning of the wavefunction ............................................................................................................................... 11
Conclusions .................................................................................................................................................................. 16
Annex: Spin and the sign of the imaginary unit in the wavefunction .......................................................................... 17
We found an error in the dE/dt formula on page 10. The correction solves the puzzle which was mentioned in
previous versions of this paper. We also found the reference to the paper that inspired part of that section (the
math of simple relativistic oscillators) in this paper and, hence, we have dutifully added it. It is a good piece.
The Meaning of Uncertainty and
the Geometry of the Wavefunction
Quantum mechanics combines Maxwell’s equations and the Planck-Einstein relation. The Planck-
Einstein relation gives us Planck’s quantum of action, which models an elementary oscillation: an
electron is an oscillating charge, a photon, a ring current in a superconductor is an oscillation too, an
atomic or molecular orbital obeys the same law, an oscillation in a two-state system, etcetera.
Understanding quantum is difficult because the mathematical formalism abstracts away from such
specifics. We talk of quantum-mechanical states, but we abstract away from the physical reality
underneath: we think of them as energy states only, but they must represent the system as a whole. The
wavefunction must have all of the information on position and momentum (linear or angular): otherwise
we would not be able to apply the relevant operators and get (average) values (or probabilities) for all of
the observables (or measurables) out of it.
The main difference between classical physics and quantum physics is that, in quantum physics, we have
only limited knowledge of the state of the system: there is uncertainty. The exact nature of this
uncertainty is the subject of philosophical discussion. Uncertainty may result from:
1. An impossibility to measure what we want to measure, or an impossibility to observe the
system: we might, perhaps, refer to this as an Ungewissheit.
2. The limited precision of our measurement: this is what Heisenberg originally referred to as an
Ungenauigkeit, i.e. before it became some metaphysical or epistemological principle.
3. The measurement might fundamentally disturb the system and, as such, cause the information
to be unreliable.
4. The uncertainty is, perhaps, inherent to Nature. This philosophical position is referred to as the
Copenhagen interpretation of quantum mechanics, and Heisenberg referred to it as the
Bell’s theorem is supposed to prove the latter position but a theorem depends on its assumptions – and
these assumptions may be challenged. We basically agree with the remarks of the Dutch physicist H.A.
Lorentz at the occasion of the 1927 Solvay Conference: there is no need to elevate indeterminism to a
The more important question is: how does quantum physics model it? How
Physicists prefer the term observable: a physical quantity that can be measured. This definition shows we could
also refer to it as a measurable. Both nouns have the same meaning.
We did not check with the philosophers here, so our terminology suggestions are just what they are: suggestions.
Words do not matter, but the distinctions might.
The full quote is this : “Je pense que cette notion de probabilité [Heisenberg-Bohr] 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?"
does it deal with it? This paper wants to offer some thoughts on that and, in the process, highlights
some contradictions which support Lorentz’s (and Einstein’s) position: we only have statistical
indeterminism here and, hence, quantum physics is not a radical departure from classical physics.
Hence, we will argue that quantum-mechanical uncertainty is nothing but statistical indeterminism. This
is, effectively, a fifth interpretation which can be added to the list above, and we think it is the right one.
The more interesting – but related – question is whether or not we can show that quantum-mechanical
amplitudes and the wavefunction
– think of Schrödinger’s equation and the solutions to it – have
physical meaning. We think we can.
Functions and physical dimensions
A dimensional analysis is always a good place to start when trying to understand the equations
describing a physical situation, but what equations should we use? Feynman’s canonical examples
include the maser (the ammonia molecule as a two-state system), an electron moving in a lattice (n-
state system modeling position), electron orbitals (Schrödinger’s equation in a central field
), and many
others. So where exactly should we start? We will probably want to start from the simplest and let us,
therefore, analyze the two-state system. In fact, our short list already triggers an obvious remark: the
formalism of quantum mechanics talks about the states of system but, in practice, the state is often
reduced to one aspect only: the position state, the momentum state, the energy state, etcetera. Using
Dirac’s bra-ket notation, we may formally write this as:
⎯ x = n − 1, x = n , x = n + 1, etc. (position states in an n-state system
⎯ mom = p (momentum state
⎯ E = −ER/n2 (energy states
Hence, we should be cautious and, at each stage, clearly identify what exactly we are talking about.
These states will all be represented by a complex-valued function (the wavefunction) or a complex
number (a quantum-mechanical amplitude) but, a priori, we should expect that the interpretation of
what the real and imaginary part of the wavefunction or amplitude might actually be, might depend on
the situation at hand and, while developing the argument, we should carefully watch out to not widen
or narrow the meaning of the symbols we are using.
As we are talking terminology here, we should warn the reader for another potentially confusing thing:
the term amplitude may refer to (i) the complex number as a whole (let us, as per the convention
The two are not necessarily the same, and their meaning may also depend on the situation that is being
A central field depends on r only: the distance from the pointlike charge which, in the case of electron orbitals, is
the nucleus (the proton inside of the hydrogen atom).
Think of a lattice on a line (a linear array of atoms or molecules).
The mom abbreviation is Feynman’s, and the example here is linear momentum. If we are interested in the
direction, we should probably write the momentum as a vector: p. We could also have given an example of an
angular momentum state, in which case we should also distinguish between the magnitude and the direction of
spin. Linear momentum is a polar vector (aka a true vector). Angular momentum is an axial vector (aka a
pseudovector). Both are equally real – in a physical sense, that is.
The energies here are the energy levels of the nth orbital. ER is the Rydberg energy (ionization energy).
The use of a plus or a minus sign for the phase (+θ or −θ) in the complex exponential – hence, writing eiθ or e−iθ –
is a matter of mathematical convention. In our papers, we have consistently argued the two mathematical
write it as r = a·e−iθ) or (ii) to the coefficient in front of it (a only). Because the reader may doubt this
statement, we will immediately give an example out of one of the more advanced models
wavefunctions for the state with an angular dependence to Schrödinger’s equation for the hydrogen
atom. These wavefunctions are written as
These wavefunctions are, in fact, only the coefficient of the actual wavefunction because the whole
derivation is based on a separation of the time-dependent and the spatial part of the wavefunction.
Somewhat confusingly, the same symbol (psi) is used to denote both, so the difference is only obvious
when one writes the argument (independent variables) of the function in full:
This all looks rather monstrous – because it is ! – so let us break it down piece by piece. You should first
note the switch from Cartesian coordinates r = (x, y, z) to polar (or spherical
) coordinates r = ,
because that is easier when talking circular or orbital motion.
In addition, the distance from the center
(the radial coordinate r) is now measured in a natural unit that goes with the system – the Bohr radius
rB, to be precise
possibilities may represent two different states: if, for some reason, the wavefunction would actually represent a
physical rotation (of charge or whatever), then the two possibilities obviously represent opposite spin directions.
So we will not start with the simplest of models (the two-state system), then. We think we have analyzed the
two-state system – and why and how probabilities (and, therefore, amplitudes) ‘slosh back and forth’ (as Feynman
puts it) between two states – ad nauseam already. See, for example, our rewrite of Feynman’s theory of
We follow the notation from Feynman’s Lectures, from which we borrow a lot of the material. We trust that the
reader will be able to look up the original Lectures and distinguish between Feynman’s formulas and text and our
presentation and interpretation of it.
Polar coordinates usually refer to a two-dimensional coordinate system, so a spherical coordinate system is then
its three-dimensional version.
We still need to prove we are actually talking circular or orbital motion of some charge here, but we think the
circumstantial evidence is fairly convincing.
We wrote the Bohr radius as a fraction of the Compton radius here. The reader can verify the substitutions,
including Feynman’s use of e2 (the squared charge of an electron divided by 4πε0), by substituting the fine-
structure constant (α) for its definition:
Talking natural units, as part of solving the (Schrödinger wave) equation(s), Feynman also writes energies E in
terms of the Rydberg energy: E = ER·ϵ, with
. Hence, ϵ is like ρ, but it is
used to measure energy.
As we are talking natural units, we may also note that, as per the Planck-Einstein relation (E = ħ·ω ω =
E/ ħ), the time-dependent part of the wavefunction (e−ω·t) may be thought of as a clock ticking at the
natural frequency of this oscillation.
The (other) functions and symbols may be briefly explained as
⎯ The Fn,l(ρ) function is a (finite) power series and is, obviously, just some real-valued function of
the radial distance ρ.
⎯ The Plm(cosθ) functions are known as the ‘associated Legendre polynomials’ (or functions). They
are usually written in terms of derivatives of ordinary Legendre polynomials. We must refer the
reader to readily accessible material here
The Yl,m(θ, Φ) functions as a whole are known as the spherical harmonics (beautiful name, isn’t it?
they are a function of the polar and azimuthal angles θ and Φ.
You should note that the ψn,l,m
amplitude (the coefficient of the actual wavefunction, really) would be real-valued, always, if we would
not have that eimΦ factor, which is equal to 1 (and, therefore, equally real-valued) if m = 0. And, of
course, if we would multiply it through with the time-dependent part of the wavefunction (e−i·(E/ħ)·t):
e−i·(E/ħ)·t·ei·m·Φ = e−i·[(E/ħ)·t + m·Φ] = e−i·(ω·t + m·Φ)
Hence, this factor is just a phase shift and, therefore, should not matter at all in terms of the physics of
the situation (it is just a matter of choosing our t = 0 point). So let us quickly look at that quantum
number: what does it stand for? It is the magnetic quantum number, and it is usually denoted as mz and
referred to as the z-component of the angular momentum. This sounds very mysterious, and it is: it is
related to the weird 720-degree symmetry of the wavefunction of spin-1/2 particles which, in turn,
results from mainstream academics not using the plus or minus sign of the imaginary unit to distinguish
between the direction of spin.
[…] You should read the latter phrase again, slowly. And because you may not understand what we are
talking about here, we added an annex to this paper which briefly talks about spin and the mathematical
convention(s) in regard to the sign of the imaginary unit in the wavefunction. So here we will only
We will let the reader think this through, and just remind him of the obvious formula for the cycle time (T): ω =
2π·f T = 1/f = 2π/ω. This shows the cycle time T is equal to T = ω/2π = E/2πħ = E/h. The natural (angular)
frequency is nothing but the natural time measured in radians: ω = 2π/T. It is a somewhat weird idea to measure
time in radians but, on the unit circle, the radian may be thought of as a natural distance as well as a natural time
unit. It helps to literally think of an old-fashioned clock (with a hand for the seconds) ticking time away, with the tip
of the hand doing the (circular) distance. Another, more abstract way, of thinking is this: we count the time in
terms of the cycle of this oscillation (1, 2,…, n,…) but, if we would want to subdivide these cycles any further, we
would divide them 2π (radians) rather than 12 (hours) or 60 (minutes or seconds).
The superscript m is an order number here: it is not an exponential. It is not a power of Pl, in other words. We
used the Wikipedia article on these mathematical functions for more detail.
The remark is not cynical. One of my early blog pieces is titled Music and Math, and it is one of the blog pieces I
still like: simple, logical and, therefore, beautiful.
We use Feynman’s notation here, and so he uses θ (theta) instead of some other letter (e.g. , phi) for the polar
angle, which is slightly confusing because, in physics, θ is also used to denote the phase of the wavefunction, like in
ψ = e−θ = e−ωt. Wikipedia says the mathematical convention is to use θ (theta) and (phi) for the polar and
azimuthal angle respectively. Our phi (Φ) for the azimuthal angle is the capital letter phi. We may, therefore, use
the lowercase phi () if we would need to denote a phase, which is what we might do. As long as we know what
we are talking about, it is all good, right?
remind you of what you know already: m is a number between −l and +l ( −l mz +l) and it gives us the
(possible) orientations of the subshell. As a further reminder of the basics, we should quickly add that l is
the quantum number that gives us the subshell within a given energy state n. This n is the principal
quantum number, and l = 0, 1, 2,… n − 1. Hence, if we have one energy state only, then we have only
state: l = 0.
What is the point? The point is that, when thinking about the physics of the situation, we can forget
about that that eimΦ factor. Think of it as being part of the time-dependent part of the wavefunction: we
just shift the origin of time. That amounts to looking at the system – the oscillation, that is – a tiny bit
earlier or later, and that does not matter because it is a perfectly regular oscillation. What we are
interested in the shape of the physical orbitals, their energies, and other physical variables. Hence, for
all practical purposes, we should think of the coefficient of our wavefunction – or the amplitude sensu
stricto, or the spatial (position-dependent) part of the wavefunction, or whatever you want to call it – as
a real number !
Is that important? Yes, it is. Knowing that a wavefunction – any wavefunction, really – can always be
written as the product of a time-dependent and a spatial or time-independent function is huge, and it is
equally huge to know that the time-dependent part will always look like e−i·ω·t + , and that the here is
just some random phase shift which does not matter because we can always shift the t = 0 point
however we would want to shift it: the physics of the situation won’t change ! This is reflected in the
fact that the absolute square
of a complex exponential (when its coefficient a is 1, of course) is always
equal to 1
Let us continue our search of some physical meaning of the real and imaginary parts of the
wavefunction by continuing our example.
What does it all mean?
Below we copy table 19.1 out of Feynman’s Lectures, which gives us the functional form of those
spherical harmonics: they combine sine and cosine functions. Now, we are interested in the probability
to find the electron at point x = (x, y, z)
, and quantum mechanics tells us we can calculate these
probabilities by taking the absolute square of the ψ(x) wavefunction. To be precise, the theory of
We should refer to standard textbooks here, but we think our own presentation in our classical explanation of
the Lamb shift has the advantage of (1) being succinct and (2) relating it to what we said on these weird 720-
degree symmetries vanishing if one would use the sign in front of the imaginary unit to incorporate the two
possible spin directions in the analysis straight from the start.
This term is a (slightly confusing, perhaps) shorthand for the square of the absolute value of a (complex- or real-
valued) number. It is also referred to as the square of the modulus of the complex sum (sum of the real and
imaginary part of the number).
We apologize for writing such simple things but it is, perhaps, good to remind ourselves of what a complex
number really is (the vector sum of a sine and a cosine) and, hence, that they are nothing but just one of the many
logical expressions of Pythagoras’s Theorem.
We have a bad habit of switching from r to x, or vice versa, for no reason whatsoever – except that you will find
x is more common than r in the literature. A bold letter is a vector, in any case, and you may think r suggests we
are working in polar rather than Cartesian coordinates, and vice versa.
operators – and of the position operator, in particular – tells us the probability density P(x) will be equal
to P(x) = ψ(x)2 = ψ(x)·ψ*(x) = ψ*(x)·ψ(x), with ψ*(x) the complex conjugate of ψ(x).
Figure 1: Spherical harmonics (source: Feynman III-19-3)
That gives us these wonderful polar graphs which, literally, depict the shape of those electron orbitals.
We may note here that we are taking the square of the absolute value of a real-valued amplitude here.
Hence, what matters is the magnitude only: positive or negative amplitudes give the same probability.
Take, for example, the p-orbital (l = 1) for m = 0. The spherical harmonic is a simple cosθ function and,
yes, cosθ2 = cos2θ = cos(−θ)2 = cos2(−θ).
So, yes, interpreting the math is not all that difficult. We are effectively talking the physical orbitals of
the pointlike electron charge here, and the uncertainty is a mere statistical indeterminism. So it is really
just like the propeller of that airplane: we do not know where it is, exactly, but we know it is always
somewhere, at any given point in time. Please note this is not your usual crackpot interpretation of
quantum physics. We may usefully quote Richard Feynman here:
“The wave function ψ(r) for an electron in an atom does not describe a smeared-out electron
with a smooth charge density. The electron is either here, or there, or somewhere else, but
wherever it is, it is a point charge.” (Feynman’s Lectures, III-21-4)
The extension of quantum-mechanical ideas and formulas from one-dimensional space (a line) to three
dimensions is not always as straightforward (Feynman, III-20-4) but, in this case, it surely is!
To show we do google other textbooks from time to time, we refer the reader to a chapter of a course (in
physical chemistry) at the University of Michigan: instructive, no hocus-pocus and good graphs.
Figure 2: Where is the propeller, exactly?
The point is this: because of the high velocity
, we are not able to precisely define the position (x, y, z)
at time t, because we are not able to precisely define the initial (x0, y0, z0, t0) condition(s). Hence, we can
only talk about cycles, averages, and probabilities. This rather primitive comparison with the physics of
an airplane propeller triggers two more useful associations. One is the metaphor of an old-fashioned
radial airplane engine, in which linear and circular motion come together (we will come back to this).
The other is an analogy with the synchronization gear that was used in WW I for machineguns firing
their bullets through the propeller: if there was no synchronization gear, some of the bullets would
actually hit and considerably damage the propeller: the analogy with light (consisting of photons) going
through a three-dimensional lattice with electrons in all kinds of orbitals readily comes to mind. We
invite readers to also google scatter plots of electron position measurements for hydrogen and other
However, these reflections do not solve the question we started out with: what is the physical meaning
of the real and imaginary parts of the wavefunction? Would they have a physical dimension, like a field –
something like newton per coulomb (N/C), like the electric field, for example? In addition, we should,
perhaps also raise some other interpretational issues: Schrödinger’s orbitals imply the electron spends
most of its time right on top of the proton, so how should we think of that? We could, perhaps, imagine
some short-range repulsive force here but such solution would inject entirely new dynamics and,
therefore, looks pretty unacceptable: assuming the electron, somehow, does go straight through the
center or, else, bounces back – fully elastically, because momentum and energy should be conserved – is
the only solution but raises other questions (which we will try to examine later
). Back to the question
of a physical dimension for the wavefunction.
I downloaded this image from a website selling Christmas presents long time ago, and I have not been able to
trace back from where I have got it. If someone recognizes this as their picture, please let us know and we will
acknowledge the source or remove it.
The classical analysis (Bohr orbitals) tells us the velocity of the electron in an atomic orbital is of the order of v =
α·c, with α the fine-structure constant (approximately 1/137 or 0.73%) and c the speed of light. Velocities further
decrease as a fraction of this velocity in outer orbitals. To be precise, v = (α/n)·c with n = 1, 2, 3,… the (main) orbital
The above-mentioned basic physical chemistry course of the University of Michigan offers one, but here is
another one from Chemistry LibreTexts.
Our more speculative papers – such as the one on what protons and neutrons might actually be – made a start in
exploring these, and we also added a few notes on this in the Annex to this paper as part of our discussion of spin
versus orbital angular momentum.
Should it have one? The argument is time and position – simple numbers, right? – so the wavefunction
might just project these numbers onto a two-dimensional mathematical space only, right? Maybe.
Maybe not. Perhaps the operators can give us a clue? Unfortunately not. Their physical dimensions are
⎯ The energy operator
comes with the
dimension, so that is the physical dimension of energy alright.
⎯ Likewise, the position operator x or and the momentum operator
come with the
physical dimension of distance (m) and momentum (
⎯ Finally, the angular momentum operator
comes with the
dimension, so that is, effectively the same as that of Plank’s quantum of
action itself (in reduced or non-reduced form).
So there is nothing lacking here: there seems to be no need to associate a physical dimension with the
real and imaginary part of the wavefunction. However, we need to be able explain these probabilities in
terms of the physics, right? Right. So let us soldier on. Can we think of a physical dimension that would
suit the P(x) = ψ(x)2 equation? Thinking of our airplane propeller again, we may think probabilities or –
to be precise – probability densities – should match energy or mass densities, right? Hence, we are
talking kg/m3 or N·m/m3 = N/m2, and we can now take a square root or something, right?
Correct, but note that the wavefunction here does not have the time-dependent part.
In fact, this
wavefunction – the wavefunction for Schrödinger’s electron orbitals – is a real-valued wavefunction: it is
the amplitude sensu stricto and, hence, talking of the meaning of the real or imaginary part of this
wavefunction makes no sense: there is only a real part to it. If we want to talk about the whole thing,
then we should put the time-dependent part (the complex-valued function that gives the whole its real
and imaginary mathematical dimension) back in.
So, again, what are we talking about, really?
The energy operator – and the others as well, perhaps – depend on the problem at hand. The one here is
derived from Schrödinger’s wave equation for electron orbitals, so we basically continue the analysis for the very
same problem at hand. Note that the symbols used for operators vary (with or without hat or special script). Ours
are probably too simple.
Note that we can often switch from energy to mass units and vice versa without too much trouble, but units
matter here, and kg/m3 or J/m3 are different units. The physical dimension of the c2 in the mass-energy
equivalence relation (E = mc2) matters here. It is not just some constant. Converting kg to N·s2/m units yields the
kg/m3 = N·s2/m4 unit. We have no idea what we could possibly do with that. In contrast, the N/m2 is much more
natural: force per unit surface. Easy, right?
The reader should also carefully check on what the listed operators are operating on: as mentioned, physicists
often conveniently forget about the time-dependent when doing their math. It is usually not a problem but when
trying to carefully interpret what is what – as we are trying to do here – it is.
The oscillator model
We have been thinking about these things for a while now, and we have no definite answer. However,
the interplay between the real and imaginary part of the wavefunction does remind one of these
probabilities ‘sloshing back and forth’, as Feynman would say, as a function of time in a simple two-state
system. So what would slosh back and forth between the real and imaginary part of the wavefunction in
an n-state system, or in a more complicated analysis such as this one (Schrödinger orbitals)? We see
only one obvious candidate and that is kinetic and potential energy.
Here we need to revive, perhaps,
our two-dimensional oscillator model, but extend it from circular orbitals to orbitals with fancier
geometric shapes, such as those in Schrödinger’s model of an atom, indeed!
Let us briefly recap the metaphorical idea.
If we combine two oscillators in a 90-degree angle – think of
two springs or two pistons attached to some crankshaft
– then we get some perpetuum mobile which
stores twice the energy of a single oscillator, and the motion of the pistons will reflect that of a mass on
a spring: it is described by a sinusoidal function, with the zero point at the center of each cylinder. We
detailed the math elsewhere
and only note the model is relativistically correct. Indeed, the
relativistically correct force equation for one oscillator is:
F = dp/dt = F = –kx with p = mvv = γm0v
The energy conservation equation can be derived from multiplying both sides with v = dx/dt. One can
then verify the following
For the potential energy, one gets the same kx2/2 formula one gets for the non-relativistic oscillator. No
surprises: potential energy depends on position only, not on velocity, and there is nothing relative about
However, the (½)m0v2 term that we would get when using the non-relativistic formulation of
To be truthful: we do not see any other candidates. The reader may suggest other suitable complementary
variables (our use of complementary here has nothing to do with Bohr’s concept of complementary or conjugate
pairs, of course), but so we do not have any.
These ideas will probably intrigue us for the rest of our life, and we are not sure if we will ever get beyond
metaphorical ideas only in regard to these deep questions.
Academics seem to prefer springs, but I like engines. In fact, the metaphor was inspired by a discussion with my
son on the efficiency of a Ducati engine, which effectively has a 90-degree bank angle. The 90° angle of the V-2
makes it possible to perfectly balance the counterweight and the pistons, ensuring smooth travel always. With
permanently closed valves, the air inside the cylinder compresses and decompresses as the pistons move up and
down. It provides, therefore, a restoring force. As such, it will store potential energy, just like a spring.
See: The Wavefunction as an Energy Propagation Mechanism.
We are grateful to a bright BA student, Dylan B. Clark, for making things so simple. Most papers on relativistic
oscillators are very hard to read, but his BA thesis on relativistic springs (2011) is a true delight. In case you do not
have access to the arxiv.org papers, he repackaged it in a paper on academia.edu (2012), which you can find here.
You may want to think about this because the distance between two points can vary because of relativistic
Newton’s Law is now replaced by the γmc2 = γ2m0c2 term
, which incorporates that velocity impacts
kinetic energy (and momentum) in two ways: directly, and through the relativistic mass effect.
Both energies vary – with position and with velocity respectively – but the equation above tells us their
sum is some constant. Hence, the game with two oscillators working in tandem should work here too.
In addition, the analogy can be extended to include two pairs of springs or pistons, in which case the
springs or pistons in each pair would help drive each other. Even more interestingly, we may imagine a
multi-piston radial engine (Figure 3)
Figure 3: The metaphor of the radial engine (source: Wikipedia)
The point is this: somehow, in this beautiful interplay between linear and circular motion, energy is
borrowed from one place and then returns to the other, cycle after cycle. While transferring kinetic
energy from one piston to the other(s), the crankshaft will rotate with a constant angular velocity: linear
motion becomes circular motion, and vice versa. Most importantly, we can add the total energy of all of
the oscillators to get the total energy of the whole system to get the E = ma2ω2 formula. The only thing
that remains to be done then, is to substitute for the tangential velocity vt = aω. In fact, substituting aω
for c = aω gives us Einstein’s mass-energy equivalence relation (E = mc2) is what inspired our mass
without mass model of an electron.
This is obvious and not so-obvious. Before we move on, let us briefly consider the one question that
puzzles us too and that we have not satisfactorily answered as yet: we were switching between a
classical (non-relativistic) oscillator and a relativistic energy formula. We should make a choice, right?
Maybe. Maybe not. The classical oscillator gives us an energy (kinetic and potential) that adds up to E =
m·a2·ω2/2. Because of the 1/2 factor, we think of two oscillators working in tandem. The ½ factor then
disappears, and Einstein’s mass-energy equivalence relation gets a physical meaning, in combination
with the Planck-Einstein relation:
E = mc2 = m·a2·ω2 c = a·ω
This tells us the energy of the elementary oscillation is proportional to the square of its amplitude and
the square of its (angular) frequency, and m is the proportionality coefficient. The Planck-Einstein
We note that previous versions of this paper had an error: they did not have the Lorentz factor. We apologize. It
is the main reason we revised this paper.
The analogy can be extended to include two pairs of springs or pistons, in which case the springs or pistons in
each pair would help drive each other.
We did not google references here, but the Wikipedia article on radial engines looks like a good start.
relation then tells us the argument of the wavefunction (in its own frame of reference) θ = ω·t is equal
This gives us, for the electron itself (which we think of as a pointlike charge with zero rest mass
OK. This was a long digression. Back to the question: how can the oscillator metaphor shed any light on
The meaning of the wavefunction
So we have this general idea that the oscillations of the real and imaginary part of the wavefunction,
somehow, incorporate the energy conservation law. This interpretation is quite consistent with
Feynman’s characterization of the wave equation as an energy diffusion equation, of course. Let us
quote him once more:
“We can think of Schrödinger’s equation as describing the diffusion of the probability amplitude
from one point to the next. […] But the imaginary coefficient in front of the derivative makes the
behavior completely different from the ordinary diffusion such as you would have for a gas
spreading out along a thin tube. Ordinary diffusion gives rise to real exponential solutions,
whereas the solutions of Schrödinger’s equation are complex waves.”
So, yes, we get this, sort of: the ‘complex waves’ are just local cyclical things – like circular or elliptical or
other regular non-linear waves. Stuff that goes around and around or, when it starts moving linearly,
combines linear and circular motion.
For linear waves – think of sound waves, water waves, radio
waves or whatever wave that moves from here to there in space – we have real-valued wave equations,
but for this circular stuff we have complex-valued wave equations because… Well… Because Euler
invented complex numbers and they magically fit the bill when trying to model all of this. So that is clear
and obvious enough, but is this interpretation compatible with all of the formalism of quantum
mechanics, and with operator theory in particular? It should be: if we know the potential and kinetic
energy – at any point in time – we should be able to derive position, momentum, and all other relevant
physical observables from it, isn’t it?
The mass of the electron as a whole is the equivalent mass of the inertia of the energy in this oscillation: we have
a very practical interpretation of Wheelers’ ‘mass without mass’ model.
Feynman further formalizes this in his Lecture on Superconductivity (Feynman, III-21-2), in which he refers to
Schrödinger’s equation as the “equation for continuity of probabilities”. However, the analysis here is really
centered on the local conservation of energy, which confirms the interpretation of Schrödinger’s equation as an
energy diffusion equation.
If there is one other paper of ours that we would recommend reading, it is the one that attracts the most
attention on ResearchGate – for the right reasons, we think: De Broglie’s Matter-Wave: Concept and Issues. We
describe the (possible) geometry of the matter-wave in full detail there, including a geometric interpretation of the
de Broglie wavelength.
Of course, we admit we should formally show this by reexamining the textbook derivations of operators
so as to prove the point. So how can we proceed then? We know we can extract the real and imaginary
part using the general Re(z) = (z+z*)/2 and Im(z) = (z-z*)/2i for a complex-valued number (and, hence,
for a function as well) and, hence, we could use this operators and then try to see whether we find
anything more interesting than what we already wrote above. Let us quickly do the first step, continuing
the electron orbital example. The ψ(r) function is the ψn,l,m(ρ, θ, Φ) function without the complex
exponential and is, therefore, the real-valued spatial (time-independent) part of the wavefunction. We,
therefore, just get the obvious result that we started out with
This just shows, once again, that the real and imaginary part of our wavefunction (r, t) – yes, we are
talking this very complicated functional form which combines power series and derivatives of Legendre
polynomials! – varies as a simple sine and cosine at any point r in space. A sine and cosine function of
what? Time. So what do we have here? It is a clock, once more, but this time it is a clock with a hand
whose length varies as a function of the position. An elliptical clock, perhaps?
What is the formula for
an ellipse again?
That is too simple, obviously! This equation is not going to get us anywhere: our x and y here, so to
speak, are the cos(ω·t + )·ψ(r) and sin(ω·t + )·ψ(r) functions and they are very different beasts! Real-
valued functions, yes, but complicated functions: just look at those polar graphs once more, or the
wonderful shapes of those subshells in 3D illustrations!
However, jotting the functional form for an
ellipse down usefully reminds of what a function actually is: a (mathematical) constraint on a set of
variables. So what constraints do we have here?
Well… The wavefunction is a solution for a definite energy state, right? Hence, we should get the energy
out the wavefunction and then we get an equation E = En with ψ(r) in it, and then… Well… Then what?
We took the eimΦ factor out of the and replaced the m·Φ term by an arbitrary phase shift .
We are talking the shape of the clock as carved out in space by the tip of the hand of the clock, of course.
That is the reason why we keep putting the factor in: it is just a phase shift, but we need the quantum number
m also for our derivatives (as an order number) of the Legendre polynomials: we can neatly separate out the time-
dependent part but – for the time being, at least – we cannot simply forget about it!
We should just apply our energy operator
, right? And we should just get
Schrödinger’s wave equation again – which we started out with, right? Let us check, though, just to
make sure we are not finding anything new or doing something wrong here. In fact, let us recap where
those formulas for the energy operator come from, so we know what is what – not approximately, but
exactly? These operators are actually used to calculate average or expected values
. So we are not
assuming anything about the value for the energy, and we just take the value for the average energy of
the system. This means we are going to start off by not assuming that the system (read: the state of our
electron in its orbital – whatever that may be) should be in a definite energy state. The formula
integral, taken over the whole volume of the atom:
Are we allowed to write that Hamiltonian in front of the H expression? Good question. It is, but you
should double-check: note that ψ is, once again, the ψ(r) function only: it does not include the time-
dependent part. How should we think of this? You will want to think we have averaged the energy over
a cycle of the oscillation. Sorry for mixing high-class math with simple illustrations once again, but
inserting an easy reminder of how potential and kinetic energy vary and add up over a full cycle of an
oscillation might help here (Figure 4).
Figure 4: Kinetic (K) and potential energy (U) of an oscillator
So what do we have here? The absolute square of the wavefunction is the probability of finding our
electron at x, so when integrating ψ(r)2 over the volume, we get 1, right? All probabilities add up to 1:
ψ(r)2dVol = 1. Yes. For normalized wavefunctions. If we do not normalize our wavefunction, we should
use this formula for the energy:
We use both as synonyms. To be precise, the expected value is the average value which a variable will take
when an experiment (so that is a measurement) is repeated a large or (theoretically) an infinite number of times
even, and so the mean (or weighted average) of all the values is calculated along the way.
In case the reader would like to check the formulas we are using (or our consistency in terms of definitions), we
refer to Feynman’s treatment of operators and more in particular, Feynman’s Lectures, section III-20-3 (average
energy of an atom).
You will find this diagram in many texts, but we took this one from the https://phys.libretexts.org/ site—
excellent hub for open-access textbooks.
But what are we talking about here? How would we go about normalizing, anyway? The ψ(r)2dVol = 1
condition amounts to:
And what is all that talk about averaging energy if we are talking definite energy states and we know we
are averaging energy over a full cycle of the oscillation? Because that is what we are doing when
separating out the time-dependent part of the wavefunction, right? Right. So we can just write this:
So what are we doing here? We are applying our energy operators – total energy (H), kinetic energy (T)
and potential energy (V)
– to the probability P(x) = ψ(x)2. But the energy of what? It must be the
energy of our pointlike electron if and when it would happen to be at x, right? And then we multiply that
value with the probability of the electron being there. So what we are doing is this: we do sum all of the
energy densities – a sum of an infinite number of infinitesimally small volume elements (I am just
reminding you of the definition of a 3D integral) – and, no surprise, we get the total energy E which – in
turn – is used to normalize the probabilities that we are using. We can illustrate this physical
normalization condition by writing:
Is this a circular argument? It is, but we think it is a useful one (in the sense that it helps us
understanding what is what here
). So where are we now? We now understand how potential and
kinetic energy slosh back and forth in this system, always adding up to some constant, but we forgot
about the original question: the real and imaginary parts of the wavefunction. We abstracted away from
that by looking at the spatial part of the wavefunction only. So let us look at the whole thing by plugging
the time-dependence back in. So we have a wavefunction which we can not only split into a time bit and
a space bit – a simple scalar product of both, to be precise:
ψ(r, t) = ψ(r)·e−i·[(E/ħ)·t + ]
Now, the time-dependent thing is the simplest of complex exponentials and allow us to also nicely
separate everything out into a real and an imaginary bit:
We apologize once again for not using fancier hat or script notation. We think it is not necessary: the meaning of
the symbols is clear from the context.
We hope it helps the understanding of the nature of the wavefunction for the reader too, but that is for him or
her to judge, of course.
These are two orthogonal vectors in the complex plane
and we can, therefore, apply Pythagoras’s
However, this just reminds us of the fact that the square of the modulus of a real number (the absolute
square) is just the squared number itself. And taking the square root back allows for positive or negative
(but always real-valued) amplitudes (spatial bit only), of course. But this does not add anything to our
interpretation of the wavefunction. Can we add anything to the interpretation by trying to find some
latus rectum formula? It might be possible, but we do not think so.
So that is it, then. We have:
1. Kinetic and potential energy sloshing back and forth and, obviously, adding up to the total
2. The sum of squares of the real and imaginary part adding up to give us the energy density (non-
normalized wavefunction) at each point in space or, after normalization, a probability P(r) =
ψ(r)2 to find the electron as a function of the position vector r.
In short, the wavefunction is the pendant to the Planck-Einstein relation. To be precise, the example we
explored showed how Schrödinger’s orbitals incorporate a Planck-Einstein cycle or, we should say,
Planck’s quantum of action tout court: the energy, the frequency, the (linear and circular) momentum,…
All comes out of the E = h·f = p·λ equation (or its reduced form
) combined with Maxwell’s equations
written in terms of the scalar and vector potential.
We should note that the indeterminacy in regard to the position is statistical only: it arises because of
the high velocity of the pointlike charge, which makes it impossible to accurately determine its position
at any point in time. It would disappear if we would be able to do so. In that case, we would be able to
define a precise (x0, y0, z0, t0) point – or, in polar coordinates – a (ρ0, θ0, Φ0, t0) point – with the pointlike
charge actually crossing position (x0, y0, z0) at time t = t0. In other words, we would be able to determine
A complex space is usually associated with complex-valued coordinates or may have some other meaning. The
complex plane is just two-dimensional Cartesian space, with the x-axis representing the real part (the axis with the
cosine values) and the y-axis representing the imaginary part (the axis with the sine values).
The latus rectum formula is a·p = b2, with a, p and b the lengths as depicted below.
The latus rectum formula popped up quite naturally in our geometric interpretation of the de Broglie wavelength,
which was quite surprising and very interesting. However, our earlier attempt to interpret Schrödinger’s orbitals in
terms of elliptical orbitals failed. We, therefore, regret this early paper remains popular, even if it gave us early
ideas on the nature of Schrödinger’s wave equation (not wavefunction) as an energy diffusion equation.
The Compton wavelength is a linear concept, and the Compton radius of an electron is just its reduced form: rC =
λC/2π. The fine-structure constant relates the various radii of the electron: radius of the pointlike charge (re = α·rC),
radius of the free electron (rC = α·rB), and the Bohr radius (rB). It, therefore, makes sense that the fine-structure
constant is not one of Nature’s (independent) constants but a combination of them: the electron charge,
lightspeed, and Planck’s quantum of action.
the initial condition of the system which, in turn, would allow us to go from indefinite integrals to
definite integrals, and so we would have a completely defined system.
Feynman once wrote this: “These philosophers are always with us, struggling in the periphery to try to
tell us something, but they never really understand the subtleties and depths of the problem.” (Lectures
on Physics, Vol. I, Ch. 19). I always found this remark rather disparaging, but he was right.
We think we found an awful lot of meaning but, yes, the question remains: do we really get this?
Maybe. Maybe not. Can we do better – any more explaining that what we have done already? We do
not think so but, of course, we invite the reader to think this through for him- or herself, and to check
whether or not the bottom line is really this: the real and imaginary part of the wavefunction(s) – i.e. the
solution(s) to the wave equation that applies to the situation at hand – combines not only the energy
conservation law (potential and kinetic adding up to the (constant) total) but all of physics, plus
Pythagoras’s (complex number theory, that is), operator theory and all of the math in-between.
It may look like some miracle that, somehow, all laws of physics – and all of geometry, of course! –
combine into Euler's function, but so that is it then: there is no further explanation, and we should just
marvel at the fact that we sort of intuitively get this. And that is all of the mystery of quantum
mechanics, then. No weird metaphysical uncertainty. And there is also no need for ‘hidden variables’ –
because all is determined and any indeterminism that is there is of a statistical nature only: think of the
airplane propeller again! And, yes, of course we should thank and remember Leonhard Euler for
inventing the ei·θ = cosθ + sinθ formula
, without which physicists would have had an awful lot of
trouble to concisely model and represent Nature’s fundamental cycle which – in turn – is represented by
the Planck-Einstein relation.
So what equations should we show a visiting alien as part of the earliest discussions when trying to
The equations of modern physics, of course: Maxwell’s equations (preferably in four-
vector notation), Schrödinger’s wave (probably the more general one for an electron in an
), and the Planck-Einstein relation. But I think we should scribble a few math
formulas in the margin too, perhaps. Which ones? Pythagoras’s Theorem or – closely related – Euler’s
Jean Louis Van Belle, 4 November 2020
We may usefully quote one of the other great polymaths of history here, Pierre-Simon Laplace, who is said to
have said: "Read Euler, read Euler, he is the master of us all!" While he stood on the shoulders of other giants (the
Wikipedia article on complex numbers offers a useful short historical introduction), such as Descartes and de
Moivre, Euler’s formula remains Euler’s formula: “the most remarkable formula in mathematics”, according to
Feynman, that is.
We refer to Feynman’s story about the Martian, in the context of his very insightful discussion of (a)symmetries
and matter-antimatter in the last chapter of his first volume of lectures. Needless to say, before talking, we should
make sure he or she or it does not feel threatened, so it is not tempted to blow us away – literally !
As mentioned in other papers, we think Schrödinger’s wave equation might be relativistically correct, because
the ½ factor does not refer to a (non-relativistic) concept of kinetic energy. The factor is there because we are
basically modelling the motion of two electrons with opposite spin. And what Schrödinger wave equation or
Hamiltonian should we show our Martian? The complete one:
Annex: Spin and the sign of the imaginary unit in the wavefunction
When thinking of spin as physical angular momentum, one can easily integrate the concept of spin in the
elementary wavefunction by thinking about the direction of motion, as illustrated below (Figure 5): we
can go from the +1 to the −1 position on the unit circle taking opposite directions.
Figure 5: e+iπ e−iπ
Hence, combining the + and − sign for the imaginary unit with the direction of travel, we get four
mutually exclusive structures for our electron wavefunction (see Table 1).
Spin and direction of travel
Spin up (J = +ħ/2)
Spin down (J = −ħ/2)
* = exp[−i(kx−t)] = exp[i(t−kx)]
χ = exp[−i(kx+t)] = exp[i(t−kx)]
χ* = exp[i(kx+t)]
Table 1: Occam’s Razor: mathematical possibilities versus physical realities
Unfortunately, the mainstream interpretation of quantum mechanics does not integrate the concept of
particle spin from the outset because the + or − sign in front of the imaginary unit (i) in the elementary
wavefunction (a·e−i· or a·e+i·) is thought as a mathematical convention only. This non-used degree of
freedom in the mathematical description then leads to the false argument that the wavefunction of
spin-½ particles has a 720-degree symmetry. Indeed, physicists treat −1 as a common phase factor in the
argument of the wavefunction.
However, we should think of −1 as a complex number itself: the phase
factor may be +π or, alternatively, −π: when going from +1 to −1 (or vice versa), it matters how you get
there⎯as illustrated above.
What are the implications? Physicists should go about their calculations more carefully, drag a sign
along, and inverse it when appropriate. And they should carefully think about the physics when getting
rid of a n·π factor: the concept of parity is important, and should be integrated in the analysis from the
Mainstream physicists therefore think one can just multiply a set of amplitudes – let us say two amplitudes, to
focus our mind (think of a beam splitter or alternative paths here) – with −1 and get the same physical states.
The quantum-mechanical argument is technical, and so I am not going to reproduce it here. I do encourage the
reader to glance through it, though. See: Euler’s Wavefunction: The Double Life of – 1. Note that the e+iπ e−iπ
expression may look like horror to a mathematician! However, if he or she has a bit of a sense for geometry and
the difference between identity and equivalence relations, there should be no surprise. If you are an amateur
physicist, you should be excited: it is, effectively, the secret key to unlocking the so-called mystery of quantum
mechanics. Remember Aquinas’ warning: quia parvus error in principio magnus est in fine. A small error in the
beginning can lead to great errors in the conclusions, and we think of this as a rather serious error in the
Spin versus orbital angular momentum
Spin is angular momentum. When analyzing a free electron or, to be precise, a pointlike charge as
, one should think of its spin angular momentum: the charge of the electron spins around its own
axis, thereby generating a magnetic moment. However, Schrödinger’s wave equation – in the context of
electron orbitals – does not take this into account: we think the 1/2 factor may be explained because it
actually models an electron pair – two electrons with opposite spin, effectively lowering their joint
energy as a pair. There is no good reason to assume such electron pair should be essentially different
from, say, a Cooper pair in perpetual currents in superconductors, when there is no thermal motion or
radiation to prevent such pairs from forming. Schrödinger’s wave equation does, however, take orbital
angular momentum into account. In fact, that is why the spin number l comes out when solving it for
definite energy states.
These subshell solutions are, in essence, Schrödinger’s great new addition to the Rutherford-Bohr
model. In fact, Sommerfeld had already made major steps in this direction.
wave equation is not complete as an explanation either: while it explains the gross and fine structure of
a hydrogen atom, it does not explain the hyperfine splitting of spectral lines, which results from the
coupling of the spin angular momentum of the electron and the spin angular momentum of the nucleus,
which is just one proton in the case of the hydrogen atom.
It, therefore, continues to model the
electron as a pointlike charge only, with no substructure of its own. That is fair enough because it would
be difficult to integrate such substructure in these ‘equations of motion’ (we are just using Dirac’s
characterization of wavefunctions here): when everything is said and done, one should not expect
models to incorporate each and every detail, right? However, the fact remains such incomplete models
trigger their own set of questions. One is this: Schrödinger’s orbitals imply the electron spends most of
its time right on top of the proton, so how should we think of that?
We could, perhaps, imagine some short-range repulsive force here but such solution would inject
entirely new dynamics and, therefore, looks pretty unacceptable: assuming the electron, somehow,
does go straight through the center or, else, bounces back – fully elastically, because momentum and
energy should be conserved – is the only solution but, of course, raises the obvious question: how does
that work, exactly? We have no answer to that, but the assumption that the electron and the proton
should have a substructure that allows them to go right through each other makes more sense than one
might expect at first. Personally, we have no difficulty whatsoever to accept the radius of the electron in
its orbital motion is nothing but the Compton radius, and that the classical electron radius is just the
radius of the oscillating pointlike charge inside of Compton’s electron. To explain the point, we copy –
once again – the illustration of the spiral-like motion of the Zitterbewegung electron (Figure 6).
We should qualify this statement somewhat. To explain the spin angular moment of a free electron - and the
anomaly in the magnetic moment – one must assume the free electron has a substructure. We elaborated the
model in several places, but a concise presentation of the argument can be found in our classical explanation of
the Lamb shift. The point is this: if the free electron has a substructure (a point charge within the point charge),
one might – once again – distinguish between spin and orbital angular momentum. Oliver Consa suggests such
fractal structure in his helical solenoid model of an electron.
See the notes on the Bohr-Sommerfeld theory in the Wikipedia article on ‘old quantum theory’, which we think
is not old (in the sense of irrelevant) at all!
We refer once again to the above-mentioned paper on the hyperfine structure and the Lamb shift.
Figure 6: The Compton radius must decrease with increasing velocity
The idea is simple enough: a free electron combines a pointlike charge with zero rest mass orbiting
around some center at lightspeed
. This motion generate the (rest) mass of the electron
as well as its
magnetic moment. However, the radius of this orbital motion must decrease when adding linear motion
because the velocity of the pointlike charge cannot exceed the speed of light, which is shown in the
We can then assume the magnetic moment of the electron and the magnetic moment of the proton will
then line up as these two particles approach each other at each and every passage through the center of
the atom and this must, somehow, avoid the collision between the charges themselves. Indeed, it is
hard to avoid the conclusion that electrons and protons do not engage in mutual annihilation – as
opposed to electrons and positrons – because the electron and the proton have very different
geometries. We must refer our reader to yet another paper with speculative thoughts on the nature of
because we have not done much thinking on this, and so this is all what we can
reasonably say about these questions right now.
The velocity must be lightspeed because its rest mass is zero. This pointlike Zitterbewegung charge is, therefore,
photon-like. However, unlike a photon, it does carry charge: the elementary charge, to be precise – which is why
electrons are electrons and photons are photons.
It is a mass without mass model: the mass is just the equivalent energy of motion – so it is a measure of inertia
only. As mentioned above, we know this may all sound rather fantastical to our reader and so we do not want to
dwell on this: we refer to our other papers for detail.
The illustration is used – with permission – from Vassallo, G., Di Tommaso, A. O., and Celani, F, The
Zitterbewegung interpretation of quantum mechanics as theoretical framework for ultra-dense deuterium and low
energy nuclear reactions, in: Journal of Condensed Matter Nuclear Science, 2017, Vol 24, pp. 32-41. The reader
should not worry about the rather weird distance scale (110−6 eV−1). Time and distance can be expressed in
inverse energy units when using so-called natural units (c = ħ = 1). We are not very fond of this because we think it
does not necessarily clarify or simplify relations. Just note that 110−9 eV−1 = 1 GeV−1 0.197510−15 m. As you can
see, the zbw radius is of the order of 210−6 eV−1 in the diagram, so that is about 0.410−12 m, which is what we
calculated: a 0.38610−12 m.
See: The Ring Current Model for Antimatter and Other Questions.