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Wavefunction math and dimensional analysis

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Very basic re-explanation of basic geometry and mathematical concepts (K-12 level).
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Wavefunctions and dimensional analysis
Jean Louis Van Belle, Drs, MAEc, BAEc, BPhil
8 April 2021 (revised on 1 October 2022
Introduction .................................................................................................................................................. 1
Cyclical functions and complex numbers ...................................................................................................... 1
Derivations of cyclical functions ................................................................................................................... 3
Dimensional analysis ..................................................................................................................................... 5
Real-valued wave equations ......................................................................................................................... 7
Quantum-mechanical wave equations ......................................................................................................... 9
The quantum-mechanical wavefunction .................................................................................................. 9
Dimensional analysis of Schrödinger’s wave equation ........................................................................... 12
The trivial solution to Schrödinger’s wave equation: Schrödinger’s electron ........................................ 14
The acceleration vector .......................................................................................................................... 16
Conclusion ................................................................................................................................................... 17
Annex: musing and exercises ...................................................................................................................... 18
Schrödinger’s electron ............................................................................................................................ 18
Radians and measurement units ............................................................................................................ 19
The arrow of time ................................................................................................................................... 20
Solving the dimensions of the wave equation once again ..................................................................... 21
What is relative and absolute? ............................................................................................................... 23
The nuclear oscillation and quaternion math ......................................................................................... 24
We took this paper offline for a while. Its style is very loose and what we write in it overlaps with lots of other
papers. However, we think our dimensional analysis of Schrödinger’s wave equation and the wavefunction itself is
rather solid and a good introduction to some of the more advanced analysis we do in papers such as, say, our
presentation of quaternion math or our paper on scattering matrices and other more high-brow quantum-
mathematical topics. We changed the title of the original paper (The Language of Math), which was rather
pretentious. The new title is a better flag to cover the load. Still, we feel the flow of this paper is quite sloppy. Also,
there is definitely some repetition and overlap between various sections. However, we think some repetition is not
bad, and we also do not have sufficient time or energy to substantially rewrite it.
Wavefunctions and dimensional analysis
In the epilogue to his Lectures, Feynman writes the following:
“The main purpose of my teaching has not been to prepare you for some examination—it was
not even to prepare you to serve industry or the military. I wanted most to give you some
appreciation of the wonderful world and the physicist’s way of looking at it, which, I believe, is a
major part of the true culture of modern times. (There are probably professors of other subjects
who would object, but they are completely wrong.) Perhaps you will not only have some
appreciation of this culture; it is even possible that you may want to join in the greatest
adventure that the human mind has ever begun.”
This paper which aims to offer a very basic introduction to the mathematical concepts that you will
need, and how they relate to (quantum) physics may or may not encourage you to effectively start
exploring things yourself, so it can become part of your culture too!
The audience of this paper is the smart K-12 level student. As such, this paper is probably most
representative of what I refer to as my K-12 level physics project.
Cyclical functions and complex numbers
Where would I start when explaining the basics of math to a K-12 level student? I would probably
remind him of the basic geometry of a circle. The Pythagorean theorem and all that. A K-12 level student
knows a sine from a cosine, but do they appreciate these two cyclical functions are, in fact, the
sameexcept for a phase shift of 90 degrees (/2 radians)? I would ask him to use a graphing tool (e.g.,
Desmos) to make sure he sees and, thereby, understands why the two functions are basically the same.
We write:
cosθ = sin(θ + /2)
And then they are not the same, of course: a phase shift is a phase shift. So, I would ask him to draw a
circle and point out the sine and cosine of an angle. I would then try to make him appreciate the sine is
just a rotation over 90 degrees of the cosine, and tell him we can represent a physical rotation of some
point by a mathematical operator. That operator is referred to as the imaginary unit, but is everything
but imaginary. The imaginary unit (i) is, effectively an operator that does what it does: it rotates axes, or
vectorsanything that has a direction. So, yes, we can think of cosθ as a vector too (a vector with length
cosθ pointing in the positive or negative x-direction, and rotate it by 90 degrees to get sinθ. We write
vectors in boldface, so let us do that here:
sinθ = icosθ
You will probably not have seen a sinθ and cosθ written in boldface before, but think of it: we can
effectively think of them as point or line vectors, like a or b. And, yes, think about the power of that little
symbol in front of cosθ: i times something (i). It is not a multiplication of, say, three times five. That is
flat one-dimensional logic. Line logic. We travel left and right of one axis when we add, subtract,
multiply or divide real numbers. The imaginary unit takes us from the line to a two-dimensional plane.
Now, we might think of dividing i into smaller units: radians. The imaginary unit (i) corresponds to /2
radians, right? So, 1 rad would, then, corresponds to 2i/ or whatever else we use to measure an angle.
But, no, then we would not need this new imaginary unit. We do need it, however. We cannot
substitute it for /2 or some other angular unit. Why not?
The student can only appreciate this by thinking some more about the idea of an operation. At this
point, I would probably ask him to explain to me the difference between F and F, and ask him how one
calculates the magnitude of a vector. He should remember this from one of his K-12 classes, and it
would help him to understand the geometry of the Pythagorean theorem and use the sine and cosine
function to calculate lengths. And then I would ask him what this weird number, or /2, actually
means, and how it differs from i or 2i. Can we get rid of or /2 by substituting them for 2i or i?
We can and probably should get rid of the old degrees when talking about angles, because that is of
no use, really. It goes back to the base-60 numerical system of the Mesopotomians, which also informs
our system of dividing hours into 60 minutes and minutes into 60 seconds. But what about ? Can we
get rid of it by defining some other system of numbers? I would let him think about that, and I would
hope he would find the answer for himself: is a natural unit for expressing angles because of the 2r
and r2 formulas for the circumference and surface area of a circle, respectively. In other words, we do
need it and we cannot get rid of it.
We cannot redefine the unit for angles in terms of a fraction or multiple of i: +i, −2i, i/100 or, as
mentioned above, 2i/. Why not? The 2r and r2 formulas would then be written as 4ir and 4ir2, and
the reduced form of Planck’s quantum of action h = ħ/2 would then be written as ħ/4i. That is OK, isn’t
No. It is not OK. We get in trouble because 4 rotations by i brings us back to the zero point and, hence,
we would have weird identities such as 4i = 0 and, therefore, 4ir = 0, and a lot of our calculations would
stop making sense. Hence, it is preferable to keep to denote a length and i to denote a rotation: we
cannot get rid of these two ‘numbers’: the i and symbols both serve a purpose: is an arc length, and i
is a rotational operator. The two symbols cannot be mixed or substituted for each other.
Now we must go one step further. I must try to explain Euler’s function, and the mathematical
properties of Euler’s number (e), and how that number also relates to the circle, and talk about how
weird that all is: we have two so-called irrational numbers ( and e), both numbers with an infinite
number of decimals (that is why we call them irrational: we cannot reduce them to a ratio), but they are
used in expressions which relate very finite distances, surfaces, and when introducing rotations in the
two other planes that make up 3D space
volumes to each other.
Let us make our first deep philosophical or ontological remark here: this π and i symbolism is rooted in Occam’s
Razor Principle. That principle says each and every symbol must correspond to a physical reality, but in the most
parsimonious way possible.
If we have an xy-plane, then we must think of the yz- and xz-planes too. That is why the 19th century
mathematician invented quaternion algebra. We think his own intuition about it that, as he put it, “we must
admit, in some sense, a fourth dimension of space for the purpose of calculating with triples is not to the point:
On to the next. One of the most easy and difficult things to understand is this: the ei = 1 and e+i = 1
expressions do not have the same meaning. When going from 1 to 1 in the two-dimensional number
world, it matters how you get thereas illustrated below.
Indeed, complex numbers are, basically,
two-dimensional numbers, so we should also write 1 to 1 as vectors or complex numbers: 1 = (1, 0) and
1 = (1, 0). We think some physicists made big mistakes because they did not appreciate the multi-
dimensional nature of the problem that they were looking at! 
Figure 1: e+iπ eiπ
And then I would have to start talking about derivatives and integrals, and I would probably introduce
the concept of linear and local or circular waves (linear and orbital oscillations), and talk about motion,
and frequencies, and how we could measure both time as well as distance in radians or other natural
units. And I would show how each and every mathematical concept can be grounded in our intuitive
understanding of right/left, up/down, back/front, and our intuitive understanding of time going in one
direction only.
Would I have lost them by then? Maybe. Maybe not. Did I lose you, just now? 
Derivations of cyclical functions
Cyclical functions have a property that is very handy in both classical as well as quantum mechanics:
their derivative is a cyclical function too. In fact, after two or more derivations, one may or may not get
the same function again. Let us show this first for the sine and cosine components of the wavefunction,
first. Because a lot of physics is really about oscillations, we will immediately introduce the wavefunction
argument, which is usually written as θ = ω·t.
θ is usually referred to as the phase and while the clock ticks the phase goes around and around goes
with time: θ = ω·t. Imagine it going from 0 to π/2, and then to π and then back to where it started: 2π.
we still talk 3D space (that is all we can imagine), and the three imaginary units which he introduced (i, j and k)
reflect 3D space rather than some four-dimensional reality. We will say some more about this in the Annex to this
paper. To be concise but complete, we should, of course, mention time: you will know we use time as a sort of
fourth dimension in four-vector algebra. But it is and remains a separate beast altogether.
This may seem self-obvious, but it sounds like horror to many mathematicians (and too many physicists too). The
quantum-mechanical argument is technical, and so I will not reproduce it here. I encourage the reader to glance
through it, though. See: Euler’s Wavefunction: The Double Life of – 1 and Feynman’s Time Machine. 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
beginning of many standard physics textbooks! It gives rise to so-called 720-degree symmetries, which do not exist
in real (physical) space. Of course, we can define mathematical spaces in which everything is possible. A
mathematical object which has 720-degree symmetry is likely to be a rotation within a rotation. Jason Hise
visualizes such objects nicely.
And then it goes around for another cycle: 4π, 6π, 8π, etcetera. That is why we write the frequency as an
angular or radial frequency. Indeed, f is the frequency that you are used to: it is the inverse of the cycle
time T = 1/f.
We might say that the use of the angular frequency is a way to express our time unit in
radians: ω = 2π·f and, hence, θ = ω·t = 2π·f·t. Note that this expression shows you that the phase θ has
no physical dimension: the second in the time variable and the 1/s dimension of f cancel each other.
We should not dwell on this. Here are the derivatives of the sine and cosine functions of time:
 󰇛󰇜
 󰇛󰇜
 󰇛󰇜
 󰇛󰇜
How does this work for Euler’s function? We vaguely introduced Euler’s function
above already, but let
us do it explicitly now:
Figure 2: Euler’s formula
We will need to take the derivative of Euler’s function. That is not difficult. The rather special thing we
should note is that the natural exponential function ex is its own derivative: d(ex)/dx = ex. When we
move from the real function ex to Euler’s function, the imaginary unit works just like any other
coefficient in front of a function: d(eix)/dx = eix·d(i·x)/dx = i·eix. You can work it out:
 󰇛󰇜
 󰇛󰇜
 󰇛󰇜󰇛󰇜
 󰇛󰇜
 󰇛󰇜
 󰇟󰇛󰇜󰇟󰇛󰇜󰇠󰇛󰇜󰇛󰇜
You can google other ways to get that derivation
but, again, we cannot dwell on this. We must move
Do not be afraid to understand this with easy examples: if the cycle time is 3 seconds, for example, then the
frequency will be equal to f = 1/T = 1/3 herz. Herz (Hz) is just a very honorable term for 1/s. A frequency is a
number expressed per second. The frequency is, obviously, inversely proportional to the cycle time: high
frequencies make for very short cycle times.
The literature will talk about Euler’s formula rather than Euler’s function. Euler invented many great things and,
hence, Euler’s function often refers to one of his other formulas. We will let the reader google this.
See the Wikipedia article or Khan Academy on it, for example.
Dimensional analysis
Preliminary note (1 October 2022): When rereading the section that follows, we feel it does not read easily. We
should probably have taken another example of equations to play with dimensional analysis. In fact, the
dimensional analysis of Schrödinger’s wave equation is the more interesting bit in this paper, and that comes only
later. However, it is what it is, and we will not rewrite or restructure this paper. If the reader is bored, he can skip
and go straight to the next section (real-valued wavefunctions).
Dimensional analysis is probably one of the easiest ways to get an intuitive understanding of equations,
and also a very easy way to quickly check if some new high-brow equation in some letter to a journal (or
in an article) makes sense. [In case you wonder if non-sensical equations ever make it to high-brow
journals, it is, sadly, the case: the mathematization of physics has, unfortunately, led to an ‘anything
goes’ attitude and a desire to grab attention no matter what it takes.]
Let us give an example. Below, we have two equations which model an electromagnetic and a nuclear
oscillation, respectively. To be precise, the equation gives us the (orbital) energy per unit mass. Do not
worry if you do not understand the terms of the equation: one is related to the kinetic energy, and the
other to the potential energybut, as mentioned, do not worry about that, right now. Just check the
physical dimensions.
The C and N subscripts stand for Coulomb and nuclear, respectively, so the equations above imply there
is such thing as Coulomb and nuclear mass, respectively. But let us focus on the dimensional analysis:
Energy is expressed in joule, which is newton-meter. Velocity is something in meter per second. Mass is
the inertia to a change in motion caused by a force, so Newton’s force law
(F = ma) tells us that 1 kg =
1 Ns2/m. So, for E/m, we have (Nm)/(Ns2/m) = m2/s2. That is fine, and you should note that it is the
dimension not only of v2/2, but also of c2: if Einstein’s mass-energy equivalence relation (E = mc2) is
correct, then E/m should, somehow, be equal to c2. We might come back to that.
Let us, as for now,
The relativistically correct view of Newton’s force law defines a force as that what causes a change in
momentum, which is the product of velocity and mass, so we should write: F = dp/dt, but the dimensions work out
the same.
E/m is not equal to c2 when considering gravitational orbitals. The orbital energy equation follows from Kepler’s
laws for the motion of the planets:
The kinetic and potential energy (per unit mass) add up to zero here, instead of c2 (nuclear and electromagnetic
orbitals), which is why a geometric approach to gravity makes eminent sense: massive objects simply follow a
geodesic in space, and there is no (gravitational) force in such geometric approach. We can compare force
magnitudes by defining a standard parameter. In practice, this means using the same mass and charge! in the
equations (we take the electron in the equation below) and, when considering the nuclear force, equating r to a:
just look at the physical dimension of the second term. The physical dimension of the Coulomb constant
(ke = 1/40
) is Nm2C2, so we combine that with the C2 from the qe2 factor, the Ns2/m for the mass
and, let us not forget, the m1 for the 1/r factor:
You can now see why we need a range parameter (a) for the nuclear force: a 1/r2 potential may or may
not exist (a hot topic for discussion in physics), but without a range parameter (expressed as a distance),
the equation would not make sense at all!
The nuclear potential is weird because, at first, it does not seem to respect the so-called inverse-square
law. This force, therefore, does not seem to respect the energy conservation principle. We can fix this by
adding a unit vector n (same direction of the force but with magnitude 1) in the nuclear potential
formulaso we should probably write something like this:
The vector dot product na = nacosθ = acosθ (the cosθ factor should be positive so n must be suitable
defined so as to ensure /2 < θ < −/2
) introduces a spatial asymmetry (think of an oblate spheroid
instead of a sphere here), which should ensure energy is conserved in the absence of an inverse-square
Is this an ad hoc solution? Yes, so you might want to think about a better theory.
Perhaps we should use
a vector cross-product na = nnasinθ = nansinθ? In light of our sinθ = icosθ equation, that should
amount to the same (think of nsinθ as a vector, with sinθ modulating the magnitude of the vector n), but
Hence, the force of gravity if considered a force is about 1042 weaker than the two forces we know
(electromagnetic and nuclear). What if we compare the electromagnetic and nuclear force? We get this:
We will let you think about this result. The nature of the two forces is very different, of course. However, because
we defined the range parameter here as the distance r = a for which the magnitude of the two forces (whose
direction is opposite) is the same, we get unity for their ratio.
Check your old physics course or just google. 0 is, of course, the electric constant.
This is why Yukawa’s nuclear potential function (a 1/r instead of a 1/r2 potential) does not make sense. It is just
one example of a historical blunder: no scientist ever wondered why the physical dimensions of Yukawa’s equation
did not make sense. Why? Probably because no one likes to challenge a Nobel Prize award.
Defining a such that it broadly points in the same direction of the line along which we want to measure the force
F should take care of this. Or perhaps we should introduce a cosθ or cos2θ factor. The point is this: we need to
integrate over a volume and ensure that the nuclear potential respects the energy conservation law.
We think we offer one in the Annex to this paper.
you may want to practice your newly acquired operator skills here and, for example, think about the sinθ
= icosθ isinθ = (1/i)sinθ = iicosθ = i2cosθ = cosθ identity. 
Real-valued wave equations
An exception to the general principle that both sides of a physical equation must be expressed in the
same SI units (or need to reduce to the same (combination of) SI units), may be wave equations. Wave
equations are mathematical conditions: they tell us what a wave shape (the wavefunction) must have in
order to be possible at all. In short, it boils down to this: the wave is that what perpetuates itself in
space, and the wave equation is a clever way of expressing all of the physical laws that apply to it.
The derivation of a wave equation is a lot of work, but you usually get a delightfully elegant resultbut
elegant is not necessarily immediately intelligible. On the contrary! It takes a lot of past knowledge and
practice to appreciate what elegant equations (including equations such E = mc2 or ħ = ET = p) actually
Take a look at Feynman’s derivation of the wave equation for sound waves, for example. The
derivation itself consists of two dense pages, but it takes a lot of previously acquired knowledge to
understand each and every step of it. Anyway, the result is this wave equation:
Feynman uses a different symbol for the wavefunction, but we intentionally use the same symbol (psi)
as the one that is (mostly) used for the quantum-mechanical wavefunction, even if we do not have any
complex numbers here. So, this wave equation is very beautiful, but looks completely mysterious, at
first, that is. Let us try to demystify it. First, note the equation relates a time derivative (a second-order
derivative, to be precise) to a derivative with respect to a spatial direction (the second-order derivative
with respect to x, to be precise). So, we have the 1/s2 dimension of the 2/t2 operator on the left side,
and the 1/m2 dimension of the 2/x2 operator on the right side. And then we have a physical
proportionality constant () and, last but not least, the physical dimension of the wavefunction itself.
A physical proportionality constant is, basically, a mathematical proportionality coefficient but, unlike a
purely mathematically proportionality constant, a physical proportionality constant has a physical
dimension (some combination of SI units) which makes the dimensional analysis come out all right.
The wave equation, and its physical proportionality coefficient in particular, usually describes the
relevant properties of the medium: that what makes wave propagation possible. In this case, the
derivation shows that must be equal to:
The second (set of) equation(s) is the Planck-Einstein law (E = hf = h/T), which you may want to think as the law
that expresses the quantization of Nature. Planck’s quantum of action – whose physical dimension is force times
distance times time can, effectively, be expressed as either (i) energy (E) times a (cycle) time (T), or momentum
(p) times a (wave)length (). The wavelength may be linear or non-linear (think of circular/elliptical orbitals here).
Both the ħ = ET and ħ = p relations imply a small, non-finite space over which the physical action is expended.
Planck's quantum of (physical) action, therefore, effectively quantizes spaceand energy, momentum, and
whatever other related physical variables. So, the quantization of space (or spacetime, if you wantbut no one
really knows what the latter term actually means) is a 'variable geometry' (I am using a term coined by a French
President here).
What is this? P is the pressure, and is the mass density of the gas (think of sound propagating in air or
some other gas), so dP/d tells us how the pressure of the gas changes when its mass density changes.
We cannot dwell on this, but you will probably accept that pressure is measured as force per unit area
(N/m2), and that mass density must be measured in kg/m3, which is equivalent to (Ns2/m)/m3 = N
s2/m4. Hence, the physical dimension of might be (something like) this
Strange, we get the physical dimension of a squared velocity once more. It must be coincidence, of
course.  Or not? Of course not! There is no such thing as coincidence in physics. One can easily
the speed of wave propagation is equal to the square root of :
Let us see what is left to explain by writing this:
We are fine! We do not need to associate a physical dimension to the wavefunction . We could, if we
would wish to do so (what about the dirac, or the einstein
), but we do not have to, and so we will not!
Note that the wavefunction is a pure mathematical function. It has no physical dimension: it projects the
position and time variables x and t (which do have a physical dimension, of course: meter and second,
respectively) onto a purely mathematical space.
On to the next. The quantum-mechanical wave equations. Note that I use a plural (equations) because
there are several candidates (Schrödinger, Dirac, Klein-Gordon), and these candidates also look different
depending on what it is that we are trying to model (electron orbitals, nuclear oscillations, two-state
systems, etcetera).
Do not worry about it. We will try to guide you through and, remember, we are only talking about doing
dimensional analysis right now, so you do not need to worry too much about what the equations
actually represent. Nobody really knows anyway because Schrödinger did not leave any notes on his
derivation. Feynman writes this about the origin of Schrödinger’s wave equation:
“Where did we get that? Nowhere. It is not possible to derive it from anything you know. It
Feynman refers to atm or bar, but you should always convert to SI units.
Squared brackets can mean many things, but here we use them as an instruction (think of it as another
operator): take the physical dimension of the thing between the brackets.
See the reference above (Feynman, Vol. I, Chapter 47).
The einstein actually exists: just google it. The einstein is defined as a one mole (6.022×1023) of photons As for
the dirac, we initially thought there might be a separate nuclear chargesomething different from the electric
charge: a nucleon charge. We did a (s)crap paper on that. You may want to read it if you are interested in how trial
and error might help you to make sense of things.
came out of the mind of Schrödinger, invented in his struggle to find an understanding of the
experimental observations of the real world.” (Feynman, III-16-5)
We are sure the notes must be somewherein some unexplored archive, perhaps. If there are Holy
Grails to be found in the history of physics, then these notes are surely one of them.
Quantum-mechanical wave equations
We introduced a real wave above: the sound wave, and we found that we could represent it by (or
associate it with
) a simple real-valued wavefunction which does not necessarily have to have any
physical dimension. The wavefunction may be a mathematical function only. Of course, the function
does depend on physical variables: position (x) and time (t), respectively. So, the soundwave function
projects physical variables to a purely mathematical space only: we associate each x and t with a purely
mathematical value (x, t). Let us not think about this too much, and just move on.
Now, I have a theory about the quantum-mechanical wavefunction: I think it does have some physical
dimension! And so I want to test it by doing a dimensional analysis of wave equations. If the dimensions
come out all right, I might be right, right?  So let me present my theory first, and then I will present
one or more wave equations and see if my theory makes sense. So let us do a sub-section on my theory,
and then another one with the dimensional analysis of some wave equation.
The quantum-mechanical wavefunction
So I think the quantum-mechanical wavefunction may describe both the position (r = a = eiθ = cosθ +
isinθ) of a pointlike charge on its orbit in terms of its coordinates x = (x, 0) = (cosθ, 0) on the real axis
and y = (0, y) = (0, sinθ) on the imaginary axis (y = ix) or, alternatively, in terms of the force F = Fx + Fy
which keeps the pointlike charge in place (see Figure 3). The force is a centripetal force, so it must be
proportional to r.
Of course, the orbit may not be perfectly circular. In fact, it most likely is not. It can be elliptical, or have
some other strange form. Perhaps it is chaotic, but then it must have some regularity because otherwise
we would not be able to associate a regular frequency with it. In short, the amplitude a will itself be a
function of x and t too! But, in a first approach, we will consider a to be some constant. To be precise,
our ring current model tells us a must be equal to the Compton radius of the particle: a = ħ/mc.
MIT published about everything they have about Feynman. Perhaps it is somewhere there. There is a book
about a mysterious woman, who might have inspired Schrödinger, but I have not read it: it is on my to-read list,
but that list is too long.
You can imagine philosophers spend quite some time debating such statements. We will not amuse ourselves
with that. We are just trying to enlighten you a bit about the language of physics (and math). We do not want to
get into ontological discussions. We do that in (some of) our more advanced papers (not K-12 level, that is).
Figure 3: The ring current model of an elementary matter-particle
Of course, this is quite a mouthful, and we do not expect you to understand much at the moment. Here
we are interested in physical dimensions only: we just want to give you a feeling of what keeps
physicists busy (or what keeps me busy, at least). If I say the wavefunction describes a position vector,
then its physical dimension must be expressed in distance units: meter, that is. If I say it is a force, then
its dimension must be newton or newton per unit area, perhaps (N/m2).
So, what is it? I do not know. Perhaps it is either, but the second possibility is appealing. Why? Wave
equations usually also incorporate the energy conservation law; so I note that, if I would express the
force as a force per unit area (I find it hard to imagine a force grabbing onto an infinitesimally small
point), then this force per unit area dimension equals an energy density (energy/volume):
Now, you can see that I also put a momentum vector p = mc in Figure 3. The initial point of the position
vector r = eiθ is the zero point of the reference frame, while the initial point of the momentum vector p
(i.e., its point of application) coincides with the (moving) terminal point of the position vector. Denoting
vectors in the negative x- and y-direction as x and y respectively, we can now easily relate the two
components of the momentum vector to the x and y components of the position vector:
px = iy and py = ix
We can, therefore, effectively consider the wavefunction to describe the position r of the pointlike
This model is also referred to as the Zitterbewegung model. Erwin Schrödinger stumbled upon it, and identified
it as a trivial solution to Dirac’s wave equation. Zitter refers to a rapid trembling or shaking motion in German.
Dirac highlighted the significance of Schrödinger’s model at the occasion of his Nobel Prize lecture:
“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)
charge, while its time derivative describes the momentum vector. We, therefore, write:
r = eiθ
p = ieiθ
Note that it is tempting to write the imaginary unit as vector quantity too: it has a magnitude (90
degrees or /2 radians) and, as a rotation, a direction too (clockwise or counterclockwise). However, its
direction depends on the plane of oscillation and we, therefore, write it in lowercase (i instead of i). That
is also the reason we wrote sinθ = icosθ instead of sinθ = icosθ: we would have a vector on the left,
and a scalar on the right (a vector dot product yields a scalar).
Any case, we are pushing our theory already. Let us just calculate derivatives. They may or may not
come in handy later, and they will give you some firsthand experience of calculating derivatives of
complex-valued functions (which will be useful when we will be discussing quantum-mechanical
operators, which we may or may not do in this paper).
If we stick to the ‘position interpretation’ of the wavefunction, then we can take the derivative of the
position vector r = aeit with respect to time to get the velocity vector v = c:
 
So, what about the dimensional analysis? The dimension of c is, of course, the velocity dimension m/s,
so that is all right.
Now, we also have the imaginary unit here, of course, which is nothing but a rotation operator. It also
rotates coordinate axes: the x-axis rotates onto the y-axis, and the y-axis becomes the new x-axis. So, if
the x-axis is position (m) and the y-axis is time (s), then we might associate i with the s/m dimension.
This sounds rather fuzzy, of course, but think about the directions of the electric and magnetic field
vectors: we can write the magnetic field vector B as B = iE/c or iE/c (depending on your orientation
vis-à-vis these fields
), and the physical dimension of B is (N/C)(s/m): the dimension of E multiplied by
s/m. Hence, if we think of a multiplication with the imaginary unit as a multiplication by s/m, then the
m/s and s/m dimension cancel out! Just like the sound wave. We should be happy, right?
Maybe. Maybe not. We need a m/s dimension for a velocity, and so this does not quite cut it. What if we
go back to the force idea and associate the N/m2 dimension with that eit wavefunction? As we pointed
out already, it is quite appealing because the N/m2 also amounts to an energy density
, and Feynman
talked about wave equations as modeling energy diffusion.
But let us just leave this idea with you, and
The plus or minus sign of i determines whether or not you have to change the direction of one of the two axes.
We leave considerations of plus or minus signs for energies out for the time being. Those have to do with
conventions (or perspectives, cf. left- or right-hand rules) and, when talking energies, the point of reference for the
U = 0 point, which we can choose to be infinity or, preferably, the center of the reference frame. When considering
multiple charges orbiting around each other (e.g. when building a neutron model (n = p + e) or a deuteron model
(d = p + p + e), the center-of-mass (barycenter) of the various oscillations becomes an important point of reference.
See Feynman’s Lectures, Vol. III, Chapter 16: “We can think of the [wave equation] as describing the diffusion of
a probability amplitude from one point to the next along the line. That is, if an electron has a certain amplitude to
be at one point, it will, a little time later, have some amplitude to be at neighboring points. In fact, the equation
talk about something else again.
Let us think about the v = c identity. How does that work? If we write the circumference of the orbital as
and the cycle time as T, then it is rather obvious that will be equal to cT. Now, the Planck-Einstein
relation tells us the cycle time will be equal to T = ħ/E.
Now, E = mc2: all of the mass of our particle
(that is what the elementary wavefunction represents) is in the oscillation of the pointlike charge.
we can write:
ħ = ET = mc2T ħ/m = c2T = ccT = c
This gives us a m2/s dimension for the ħ/m factor in Schrödinger’s equation: [ħ/m] = [c] = (m/s)m =
m2/s. So, what is the deal here? We have more than just a dimensional analysis here: because we now
used the Planck-Einstein and mass-energy equivalence relations, we know why the dimension of ħ/m
must be m2/s. We write:
We are getting a bit ahead of ourselves here: we introduced the ħ/m factor the physical
proportionality coefficient which we see in Schrödinger’s equation – but we did not introduce
Schrödinger’s wave equation yet! Let us do that now. It will be Schrödinger’s wave equation in the so-
called free space. Free space means we have no potential (electromagnetic or nuclear). Hence, our
pointlike charge (and the particleor should we call it a wavicle?) can just move freely around. How
exactly does that work? Why does it do so? That is what the wave equation should tell us.
Dimensional analysis of Schrödinger’s wave equation
Schrödinger’s wave equation in free space is this
The 2 term looks frightening, but it is just the same as the 2/x2 derivative in our soundwave
equation: it is the second-order derivative with respect to position. The only difference is that we are
applying it to a position vector x or r = (x, y, z).
looks something like the diffusion equations which we have used in Volume I. But there is one main difference: the
imaginary coefficient in front of the time 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 [the wave equation] are complex waves.
Because we are using the reduced Planck constant here, time is measured in radians here. We would usually
write this: E = hf = h/T T = h/E. We might have used a different symbol for a cycle time expressed in radians,
such as , but that symbol is usually reserved to denote the lifetime of non-stable particles (transients).
For electron orbitals in an atom, the energy to be used will be of the order of the Rydberg energy.
You will usually it with a ½ factor, but that has to do with the enigmatic concept of effective mass. We will not
go into that. If you want to know, see our paper on the matter-wave.
We could also use spherical coordinates (r, θ, φ) instead of Cartesian coordinates. In fact, that is what is usually
done when working with wave equations because it makes the calculations easier. However, we do not want to
confuse the reader too much here.
So, what about the dimensional analysis? Let us do it for the two sides of the equation
󰇟󰇠 
This looks good: we have the 1/s dimension on both sides, but what is that square of the physical
dimension of in the second equation? Good question. It does not make sense
󰇟󰇠 
So, it is all OK. Of course, you may wonder, do we not need some functional form here? We do not have
one, yetwe only have that theory about circular motion, but so that is not a proof. In fact, the
equation above suggests our wavefunction can have any physical dimension: whatever it is,
Schrödinger’s equation will make sense, physically, that is. Its functional form and its physical dimension
can be whatever, and our ‘bracket operator’ – [X] will work: the /t and 2/x2 will bring out the 1/s
and 1/m2 dimension respectively, but whatever other dimension is in (N, J, C, etcetera, or any
combination thereofliterally whatever!) will still be there.
So, in short, we can just write this::
What about [i] or [1/i]?
What should we do with that? We are not sure. Perhaps you will find the
deeper meaning of this one day once you also understand how the amplitude a varies as a function of x
(or r)and t! All vectors v, a, p, F, etcetera then all become variables, and all depend on each other, in
a very complicated set of equations: we are talking about a system here, in other words. If you do, let us
know.  All we wanted to do here is to explore our interpretation of the wavefunction
mathematicallyin a very first and, therefore, rather rough approach, and so we do that with a
dimensional analysis.
Let us do something else with second-order derivatives. Let us calculate the second-order derivative
with respect to time: this should give us an acceleration. Think of it as a warm-up for further thinking on
that second-order derivative with respect to x or x.
Note that we use the square brackets as a sort of operator too here. They say this: take the physical dimension
of what is between the brackets. It can be quite confusing because square brackets are used as… Well… Just plain
square brackets in a few other places of this paper. We hope we do not confuse the reader too much! 
This is one of the many places where we thought we should rewrite and restructure the paper a bit, but we do
want you to think everything through for yourself, and so we will not explain you where we went wrong and why
and how we corrected ourselves: you should really work out this dimensional analysis for yourself! 
It has nothing to do with Dirac’s ‘bra-ket’ operators, of course!
We can, effectively, move i to the other side of the equation, but it should not matter: 1/i = i and, hence, 1/i
must do the same to the physical dimensions: it amounts to a multiplication by second/meter (s/m) which,
incidentally, is the physical dimension of 1/c.
The trivial solution to Schrödinger’s wave equation: Schrödinger’s electron
If you google proper textbooks (we think of Feynman’s Lectures here
), then you will that
Schrödinger’s wave equation for an electron in free space (i.e., in the absence of any potential energy
term) has an extra 1/2 factor. It is written like this:
We must make a few remarks here about things which we believe to be the case. However, you must
make up your own mind about our remarks below:
Mainstream physicists consider this equation to be not relativistically correct. We think that is
unfortunately and unjustified.
The reader should also note that the concept of the effective mass in this equation (meff) of an
electron emerges from an analysis of the motion of an electron through a crystal lattice (or, to be
very precise, its motion in a linear array or a line of atoms). However, Richard Feynman and all
academics who produced textbooks based on his, then rather shamelessly substitute the efficient
mass (meff) for me rather than by me/2. They do so by noting, without any explanation at all
, that
the effective mass of an electron becomes the free-space mass of an electron outside of the lattice.
We think this is unwarranted too.
The ring current model explains the ½ factor by distinguishing between (1) the effective mass of the
pointlike charge inside of the electron (while its rest mass is zero, it acquires a relativistic mass equal
to half of the total mass of the electron) and (2) the total (rest) mass of the electron, which consists
of two parts: the (kinetic) energy of the pointlike charge and the (potential) energy in the field that
sustains that motion.
Of course, now you will say that Schrödinger’s equation works in the context of electron orbitals for
the hydrogen atom. Hence, that factor ½ or 2 times m must be correct, right? Wrong. Schrödinger’s
wave equation for electron orbitals with a potential term basically models the orbitals of two
electrons rather than just one: it abstracts away from spin and we, therefore, think the orbitals are
orbitals of electron pairs. That is why the factor 2 pops up and, yes, it is correct and must be there in
the context of that model (electron orbitals in a hydrogen atom). We think it is not correct to use it
in Schrödinger’s equation for electron motion in free space.
In short, in our not-so-humble view, Schrödinger’s wave equation for a charged particle in free
space, which he wrote down in 1926, which Feynman describes with the hyperbole we all love as
“the great historical moment marking the birth of the quantum mechanical description of matter
occurred when Schrödinger first wrote down his equation in 1926”, effectively reduces to this:
We must be precise: we refer to Feynman’s derivation of Schrödinger’s wave equation in a lattice.
See: Richard Feynman’s move from equation 16.12 to 16.13 in his Lecture on the dependence of amplitudes on
We make this point quite forcefully because it is one of the key differences between our interpretation of
Schrödinger’s Zitterbewegung electron and that of an author like David Hestenes.
We think this is the right wave equation because it produces a sensible dispersion relation: one that
does not lead to the dissipation of the particles that it is supposed to describe. The Nobel Prize
committee should have given Schrödinger all of the 1933 Nobel Prize, rather than splitting it half-half
between him and Paul Dirac. We are really not sure why physicists did not think of the Zitterbewegung
of a charge or some ring current model and, therefore, dumped Schrödinger equation for something
fancier. We talk about that in other papers, so we will not repeat ourselves here.
However, we still owe it to you to show that our electron model and the wavefunction that comes
with is, effectively, a very trivial solution to the wave equation:
So let us do that here now.
Our Zitterbewegung model of an electron yields the following elementary
wavefunction for the electron:
It is just the general wavefunction = ae iθ = ae it, but substituting a for a = ħ/mc = ħc/E and ħ/m
for ħc2/E. Now, we must prove that this is, indeed, a solution to Schrödinger’s above. We can prove this
by writing it all out: 
 
Now, this all looks very formidable, but it works out surprisingly well. Take the left side first:
 
Now the right side, but so there we have time as a variable and we want to take the (second-order)
derivative with respect to positionso how does that work, then? We can write x as x = ct or as x =
ct, perhaps
, and, therefore, substitute t for x/c, perhaps? Let us what we get:
See our paper on de Broglie’s matter-wave and, quite complementary, our papers on the history of quantum-
mechanical ideas or on the meaning of uncertainty.
This section got inserted in the October revision of this paper. We basically copied it from a section that comes
much later in this paper. We thought the reader might not get there and, hence, it is probably better to put this
key result here.
We must define a convention here for the plus/minus sign of the velocity vector, associating one or the other
with a clock- and counterclockwise rotation, respectively. It is a fine matter (because we must take the minus sign
of the i3 = i factor into account), but we will not worry about it.
Bingo! All is OK. This is a very significant result. In fact, we talked about lost notes and the Holy Grails of
quantum physics. This might be it: we do not exclude that Schrödinger might have worked backwards.
Would it not be logical to first jot down a wavefunction, and then see in what wave equation it might fit
as a solution?
The acceleration vector
Let us have some more fun now. Let us calculate the acceleration vector a (do not confuse this with the
amplitude a or the radius vector r
 󰇛󰇜󰇛󰇜
We find that the magnitude of the (centripetal) acceleration is constant and equal to mc3/ħ.
This is a
nice resultbecause its physical dimension works out: [mc3/ħ] = m/s2, so that is an acceleration all right.
Let us go beyond our electron model now. Let us see if all this works for something we know: Bohr-
Rutherford electron orbitals, for example. The radius of Bohr-Rutherford orbitals is of the order of the
Bohr radius rB = rC/, and their energy is of the order of the Rydberg energy ER = 2mc2, with the fine-
The velocity and accelerations are, therefore, equal to:
 
 
We get the classical orbital velocity v= c, while the magnitude of the oscillation equals c. The
acceleration factor c has the right physical dimension (1/s)(m/s) = m/s2 and so, yes, all looks good.
However, we need to get further into the grind. We have an easy explanation now of the second-order
derivative with respect to time (2/t2), but we do not have such easy interpretation for 2. Do you
see one?
It is and remains all very mysterious but, at the very least, you can already appreciate that there is
nothing magical or mysterious about quantum-mechanical operators: /t and 2/t2 are quantum-
mechanical operators too! 
It is like using m for mass and for meter. Textbooks will usually take care to differentiate symbols, but we think
that does not make you any smarter.
The minus sign is there because its direction is opposite to that of the radius vector r.
If the principal quantum number is larger than 1 (n = 2, 3,…), an extra n2 or 1/n2 factor comes into play. We refer
to Chapter VII (the wavefunction and the atom) of our manuscript for these formulas.
We hope we have succeeded in giving you a feel for the real mystery of quantum physics: the wave
equation. We are sure some higher mind will, one day, be able to reconstruct Schrödinger’s derivation
of his wave equation, in very much the same way as Feynman gave us a derivation of the soundwave
equation. Perhaps it will be you! 
Brussels, 8 April 2021
Annex: musing and exercises
Schrödinger’s electron
We gave you the solution to Schrödinger’s wave equation in free space. It is a complex exponential with
a (real-valued) coefficient
, so we should not write as r = eiθ but as r = aeiθ, and this coefficient is,
effectively, the Compton radius of our particle. For an electron, we can easily calculate it as follows:
Paraphrasing Prof. Dr. Patrick LeClair
, we understand this distance as “the scale above which the
electron can be localized in a particle-like sense, and it also clarifies what Dirac referred to as the law of
(elastic or inelastic) scattering of light by an electron”Compton’s law, in other words.
So, you can check the physical dimension of a:
Now, when we derive the wavefunction, we can treat this coefficient like a constant
and the
dimensional analysis of Schrödinger’s equation of the left and right side respectively can then be written
Physics textbooks will tell you the coefficient may be complex-valued, but when you multiply everything through
in practical examples, you will see you can always write the whole thing as a real-valued coefficient times a
complex exponential.
See:, p. 10.
Compton scattering may be explained conceptually by accepting the incoming and outgoing photon are
different photons (they have different wavelengths so it should not be too difficult to accept this as a logical
statement: the wavelength pretty much defines the photonso if it is different, you have a different photon). This,
then, leads us to think of an excited electron state, which briefly combines the energy of the stationary electron
and the photon it has just absorbed. The electron then returns to its equilibrium state by emitting a new photon.
The energy difference between the incoming and outgoing photon then gets added to the kinetic energy of the
electron through Compton’s law: 󰆒
This physical law can be easily derived from first principles (see, for example, Patrick R. Le Clair, 2019): the energy
and momentum conservation laws, to be precise. More importantly, however, it has been confirmed
We can only do that because we model circular orbitals here. For elliptical orbits (or whatever other complicated
shape), the coefficient itself will vary as a function of the position (r) and time (t), so we must apply the product
rule and the derivation then becomes quite complicated, so generalizing the model to encompass all possible
orbitals is quite complicated.
In the second equation (righthand side of Schrödinger’s equation), we make use of the 󰇣
result, which we derived above (the rotation operator swaps axes and, incidentally, has the same dimension as the
1/c operator. We wrote a bit about that in a very early paper of ours, in which we explore Feynman’s suggestion to
think of it all as some energy diffusion or energy propagation mechanism. That paper has visual illustrations which
you might want to explore.
So, we are goodonce more! We now have a m/s dimension, but it is OK because it is the same on both
sides of the equation. And, yes, now that we have a functional form for the wavefunction itself (), we
could do an even better job at writing it all out, but you can work that out for yourself, right? Try it as an
exercise and, no worries, we will do it for you at the end of this paper.
Radians and measurement units
Let us get back to pure math and explain some more about radians. There is something very special
about them. We can not only measure distance in radians but time as well. Read this again: we can
measure time in radians (a distance unit) rather than seconds. Because Schrödinger’s wave equation has
the reduced Planck constant in it (ħ, not h), this is sort of logical.
But it may come across as a mystery
to you. In fact, if there is any mystery in quantum physics, then this might be it: I do not see any other
The point to be appreciated here is that (circular) motion is associated with a change in the phase angle
θ, which we will write as a differential Δθ. Now there are two viewpoints:
1. For a given (small) interval in time (Δt), the distance traveled (Δx) will be equal to the radius
vector times Δθ (approximately, at least
), so we can write: Δx rΔθ.
2. For a given (small) interval in space (Δx or, in 3D space, Δx), the time needed to cover that
distance (Δt) will be equal to the same: Δt rΔθ.
This shows we can use the radian as a unit of time as well as a unit of distance, as illustrated below
(Figure 4).
Figure 4: The radian as unit of distance and of time
The use of the radian as an equivalent time and distance unit can also be illustrated by playing with the
associated derivatives, and taking their ratios
The reduced quantum (ħ) is equal to h/2. This division by 2 distinguishes a cycle time measured in cycles from
time measured in radians, which corresponds to the difference between frequency (f = 1/T) and angular frequency
( = 1/). The latter symbol () is, unfortunately, not commonly used.
We have two approximations here: (1) the length of the hypotenuse and the adjacent side of the triangle are
equated to the radius and the arc length; (2) we use the small-angle approximation to equate sin(Δθ) to Δθ.
We must make an important remark here: our playing with differentials here assumes a normalized concept of
velocity: we can only use the radian simultaneously as a time and distance unit when defining time and distance
 
 
 
 
This is as far as we can go in terms of understanding the nature of space and timephilosophically,
mathematically, and physically. Or… Perhaps not. Let us say something more about the so-called
arrow of time.
The arrow of time
Spacetime trajectories or, to put it more simply, motion need to be described by well-defined
functions. That means that for every value of t (time), we should have one, and only one, value of x
The reverse, of course, is not true: a particle can travel back to where it was (or, if there is no
motion, just stay where it is).
This is illustrated below: a pointlike particle which moves like what is shown on the right-hand side
cannot exist because there are a few occasions here where the particle occupies multiple positions in
space at the same point in time. Now, some physicists may believe that should actually be possible, but
we do not want to entertain such ideas, really.
Figure 5: A well- and a not-well behaved trajectory in spacetime
This shows that time must go in one direction only. We can play a movie backwards, but we cannot
reverse time. Think of this: a movie in which two like charges (say, two electronsor two protons)
would attract rather than repel each other does not make sense. We would, therefore, know this is a
movie which was being played backwards, and we would say it is impossible: time cannot be reversed.
This intuition contrasts with the erroneous suggestion of Richard Feynman that we should, perhaps,
think of antimatter-particles as particles that travel back in time. It is nonsense: the plus/minus sign of
the argument of the wavefunction gives us the spin direction of a particle. It has nothing to do with time
going in this or that direction. As for antimatter, we believe it can be modelled by the plus/minus sign of
units such that v = / = 1. The in this equation is the circumference of the circle (think of it as a circular
wavelength), and = T/2 is the (reduced) cycle time. See footnote xlvii.
We can generalize to two- or three-dimensional space, of course. The x in the illustration then becomes a vector
in a three-dimensional vector space: x = (x, y, z). We should note there is no such thing as four-dimensional
physical space. Mathematical spaces may have any number of dimensions, but the notion of physical space is a
category of our mind, and it is three-dimensional: left or right, up or down, front or back. You can try to invent
something else, but it will always be some combination of these innate notions. Time and space are surely related
(through special and general relativity theory, to be precise) but they are not the same. Nor are they similar. We
do, therefore, not think that some ‘kind of union of the two’ will replace the separate concepts of space and time
any time soon, despite Minkowski’s stated expectations in this regard back in 1908.
the coefficient of the wavefunction.
Solving the dimensions of the wave equation once again
We are now ready to tell you how we make sense of the world. We measure distance and time as arc
lengths. The full wavelength corresponds to the circumference of the circle, and the natural time unit
is one full cycle. We can then normalize velocities by defining the orbital velocity v as v = /1, which
amounts to normalizing the length of the radius vector: a = 1. The magnitude of the velocity vector is
then expressed in radians (per unit time): 2 radians, to be precise, and v = = 2 rad.
This, in turn, allows us to choose a force unit such that this force unit times times T (the cycle time)
equals Planck’s quantum of action: TF = h. We can write this in reduced form by dividing and h by 2
so as to get the reduced form of Planck’s constant and write it as a product of energy (force times a
distance) and time:
ħ = FaT = (Fa)T = ET
Multiplying with 2 once more, gives us the second de Broglie relation, which gives us Planck’s constant
written in terms of a momentum (a force times a time interval) and a length:
h = FT2a = (FT)) = p
We can normalize the force unit too by equating ħ = 1 and then choosing the force unit such that F = 1.
Of course, we can think of force and momentum as vectors, which turns the equation into a vector
h = ET = p
Alternatively, we can also think of the radius as a radius vector and write
ħ = FaT = (Fa)T = (Fa)T = ET
Now we can define a mass unit using Newton’s force law F = dp/dt (the relativistically correct
expression) or F = mac (non-relativistic). Note that we have a centripetal acceleration vector a
hereand, yes, we added the subscript c so as to distinguish the centripetal acceleration vector ac from
the radius vector a.
And on and on it goes. We can now introduce derivatives and complex notation and introduce the
wavefunction = ae iθ = ae it = ae i(E/ħ)t for the electron, substituting a for a = ħ/mc = ħc/E and
ħ/m for ħc2/E, and, therefore, write Schrödinger’s equation as:
See our paper on the Zitterbewegung hypothesis and the scattering matrix.
The radius and force vector have opposite direction, so the vector dot product Fa reduces to Fa =Facosφ
= Fa1 = Fa
You should not confuse the c from centripetal with the c of lightspeed!
 
Now, this all looks very formidable, but it works out surprisingly well. Take the left side first:
 
Now the right side, but so there we have time as a variable and we want to take the (second-order)
derivative with respect to positionso how does that work, then? We can write x as x = ct or as x =
ct, perhaps
, and, therefore, substitute t for x/c, perhaps? Let us what we get:
Bingo! All is OK. We did not prove Schrödinger’s equation, but we did show it is dimensionally
consistent! 
Of course, now that you’ve got this, the real work starts: you must think about variabilizing the radius
and consider a particle that is not at rest. And you need to understand all about potentials, learn about
four-vectors, etcetera. For that, we refer you to our other K-12 level papers. 
If you do not want to go that far, you can continue thinking about the (possible) physical dimension of
the wavefunction. If it is an energy density, then the dimension (m) of its coefficient (the radius or
amplitude of the oscillation a) combines with the F/m2 or E/m3 of the complex exponential eiθ (it is just
a cyclical functionnothing to be mystified about: just a combination of two sinusoidally varying
orthogonal vectors), and so the whole aei(E/ħ)t expression then gets a F/m or an E/m2 dimension. That
makes a lot of sense when we think of the interpretation of the wavefunction in terms of probabilities of
actually finding the particle (or the pointlike charge?) at the x position x at time t!
Indeed, if we take the absolute square of the wavefunction
, and normalize that value by dividing it by
the (squared) energy density of the whole volume, we should get a probability: some pure (scalar)
number which varies between 0 and 1.
Of course, you may wonder: why this squaring business? The logic here is just the same as that which
we apply in statistics: plus and minus signs would cancel each other and, hence, to calculate a mean, we
should take a root mean square (aka as a quadratic mean) approach so as to get a meaningful result.
Is the above absolute truth? It surely is not. There are alternative ways of looking at the wavefunction:
we may think of it as modeling a field vector, for example. We then can analyze energies using the
Poynting vector or other ways of modeling (field) energy, which (also) involves the squaring of field
magnitudes. We say a few words on that in the Annex to this paper, so you may want to read on. 
We must define a convention here for the plus/minus sign of the velocity vector, associating one or the other
with a clock- and counterclockwise rotation, respectively. It is a fine matter (because we must take the minus sign
of the i3 = i factor into account), but we will not worry about it.
The correct term is: the absolute value of the square, but we prefer the shorthand term ‘absolute square’.
What is relative and absolute?
After all of this, you may wonder: what is real, and what is not? That is a philosophical question to which
there are (almost) as many answers as (great) philosophers. What we know is that we have a complete
and consistent description or representation of what we think of as reality. Now, that description
describes some things which are relative (relative in the sense as used in special or general relativity
theory) and some things which are not: the constants of Nature (think of the elementary charge,
lightspeed, and Planck’s quantum of action here). In addition, we have reduced all possible physical
dimensions to 7 base units (see the 2019 revision of the international system of units), and we also
devised a consistent set of concepts and operators (vectors, rotations, derivatives, integrals, etcetera).
So that allows to write all laws of physics as some combination of these constants and measurements.
As an example, we may combine Einstein’s mass-energy equivalence relation (E = mc2) with the Planck-
Einstein relation (E = hf) to get a combined or synthetic relation:
The m/f = h/c2 equation tells us that the ratio of the mass and the frequency (as given by the Planck-
Einstein relation) of any (elementary) particle must be equal to the ratio of Planck’s quantum of action
and the squared lightspeed. Do we understand this relation? Yes and no: to understand it, we must
analyze this equation in terms of the two fundamental relations which give us this equation. Hence, the
language of math, combined with the laws of physics, give us a representation which makes sense. Think
of it of some kind of storycall it the Book(let) of Nature or its mode d’emploi, if you want.
Space and time do remain somewhat special: these two concepts (categories of the mind, as Immanuel
Kant referred to them) link all of the physical concepts to our thinking about it, so we can think of the
wavefunction (and the wave equation) as some kind of link function (another concept from statistics,
which you may (or not) want to study further). Now, we could make this story this paper much
longer, but we do not want to do that, so we will just put in a diagram which reflects what we wrote
above and that will be it. 
The nuclear oscillation and quaternion math
In this paper, we talked a lot about the Zitterbewegung model of an electron, which is a model which
allows us to think of the elementary wavefunction as representing a radius or position vector. We write:
ψ = r = a·e±iθ = a·[cos(±θ) + i · sin(±θ)]
It is just an application of Parson’s ring current or magneton model of an electron. Note we
use boldface to denote vectors, and that we think of the sine and cosine here as vectors too! You should
note that the sine and cosine are the same function: they differ only because of a 90-degree phase shift:
cosθ = sin(θ + π/2). Alternatively, we can use the imaginary unit (i) as a rotation operator and use the
vector notation to write: sinθ = i·cosθ.
We also showed how and why this all works like a charm: when we take the derivative with respect to
time, we get the (orbital or tangential) velocity (dr/dt = v), and the second-order derivative gives us the
(centripetal) acceleration vector (d2r/dt2 = a). The plus/minus sign of the argument of the wavefunction
gives us the direction of spin, and we may, perhaps, add a plus/minus sign to the wavefunction as a
whole to model matter and antimatter, respectively (the latter assertion remains very speculative
One orbital cycle packs Planck’s quantum of (physical) action, which we can write either as the product
of the energy (E) and the cycle time (T), or the momentum (p) of the charge times the distance travelled,
which is the circumference of the loop λ in the inertial frame of reference (we can always add a classical
linear velocity component when considering an electron in motion, and we may want to write Planck’s
quantum of action as an angular momentum vector (h or ħ) to explain what the Uncertainty Principle is
all about (statistical uncertainty, nothing ontological), but let us keep things simple as for now):
h = E·T = p·λ
It is important to distinguish between the electron and the charge, which we think of being pointlike:
the electron is charge in motion. Charge is just charge: it explains everything, and its nature is, therefore,
quite mysterious: is it really a pointlike thing, or is there some fractal structure? Of these things, we
know very little, but the small anomaly in the magnetic moment of an electron suggests its structure
might be fractal. Think of the fine-structure constant here, as the factor which distinguishes the classical,
Compton and Bohr radii of the electron: we associate the classical electron radius with the radius of the
poinlike charge, but perhaps we can drill down further.
We also showed how the physical dimensions work out in Schrödinger’s wave equation. Let us jot it
down to appreciate what it might model, and appreciate once again why complex numbers come in
handy: 
This is, of course, Schrödinger’s equation in free space, which means there are no other charges around
and we, therefore, have no potential energy terms here. The rather enigmatic concept of the effective
This Annex is a copy of one of our blog posts on the same topic (math and physics), and there may be, therefore,
some repetition with the main body of the paper.
mass (which is half the total mass of the electron) is just the relativistic mass of the pointlike charge as it
whizzes around at lightspeed, so that is the motion which Schrödinger referred to as
its Zitterbewegung (Dirac confused it with some motion of the electron itself, further compounding
what we think of as de Broglie’s mistaken interpretation of the matter-wave as a linear oscillation: think
of it as an orbital oscillation). The 1/2 factor is there in Schrödinger’s wave equation for electron
orbitals, but he replaced the effective mass rather subtly (or not-so-subtly, I should say) by the total
mass of the electron because the wave equation models the orbitals of an electron pair (two electrons
with opposite spin). So, we might say he was lucky: the two mistakes together (not accounting for spin,
and adding the effective mass of two electrons to get a mass factor) make things come out all right.
However, we will not say more about Schrödinger’s equation for the time being (we will come back to
it): just note the imaginary unit, which does operate like a rotation operator here. Schrödinger’s wave
equation, therefore, must model (planar) orbitals. Of course, the plane of the orbital itself may be
rotating itself, and most probably is because that is what gives us those wonderful shapes of electron
orbitals (subshells). Also note the physical dimension of ħ/m: it is a factor which is expressed in m2/s, but
when you combine that with the 1/m2 dimension of the 2 operator, then you get the 1/s dimension on
both sides of Schrödinger’s equation. [The 2 operator is just the generalization of the d2r/dx2 but in
three dimensions, so x becomes a vector: x, and we apply the operator to the three spatial coordinates
and get another vector, which is why we call 2 a vector operator. Let us move on because we cannot
explain each and every detail here, of course!]
We need to talk forces and fields now. This ring current model assumes an electromagnetic field which
keeps the pointlike charge in its orbit. This centripetal force must be equal to the Lorentz force (F),
which we can write in terms of the electric and magnetic field vectors E and B (fields are just forces per
unit charge, so the two concepts are very intimately related):
F = q·(E + v×B) = q·(E + c×iE/c) = q·(E + 1×iE) = q·(E + j·E) = (1+ j)·q·E
We use a different imaginary unit here (j instead of i) because the plane in which the magnetic field
vector B is going round and round is orthogonal to the plane in which E is going round and round, so let
us call these planes the xy and xz-planes, respectively. Of course, you will ask: why is the B-plane not
the yz-plane? We might be mistaken, but the magnetic field vector lags the electric field vector, so it
is either of the two, and so now you can check for yourself of what we wrote above is actually correct.
Also note that we write 1 as a vector (1) or a complex number: 1 = 1 + i·0. [It is also possible to write
this: 1 = 1 + i·0 or 1 = 1 + i·0. As long as we think of these things as vectors something with a
magnitude and a direction it is OK.]
You may be lost in math already, so we should visualize this. Unfortunately, that is not easy. You may
to google for animations of circularly polarized electromagnetic waves, but these usually show the
electric field vector only, and animations which show both E and B are usually linearly polarized waves.
Let me reproduce the simplest of images: imagine the electric field vector E going round and round.
Now imagine the field vector B being orthogonal to it, but also going round and round (because
its phase follows the phase of E). So, yes, it must be going around in the xz or yz-plane (as mentioned
above, we let you figure out how the various right-hand rules work together here).
You should now appreciate that the E and B vectors taken together will also form a plane. This
plane is not static: it is not the xy-, yz or xz-plane, nor is it some static combination of two of these. No!
We cannot describe it with reference to our classical Cartesian axes because it changes all the time as a
result of the rotation of both the E and B vectors. So how we can describe that plane mathematically?
The Irish mathematician William Rowan Hamilton who is also known for many other mathematical
concepts found a great way to do just that, and we will use his notation. We could say the plane
formed by the E and B vectors is the EB plane but, in line with Hamilton’s quaternion algebra, we will
refer to it as the k-plane. How is it related to what we referred to as the i and j-planes, or the xy
and xz-plane as we used to say? At this point, we should introduce Hamilton’s notation:
he did write i and j in boldface (we do not like that, but you may want to think of it as just a minor
change in notation because we are using these imaginary units in a new mathematical space: the
quaternion number space), and he referred to them as basic quaternions in what you should think of as
an extension of the complex number system. More specifically, he wrote this on a now rather famous
bridge in Dublin:
i2 = -1
j2 = -1
k2 = -1
i·j = k
j·i= k
The first three rules are the ones you know from complex number math: two successive rotations by 90
degrees will bring you from 1 to -1. The order of multiplication in the other two rules ( i·j = k and j·i = k )
gives us not only the k-plane but also the spin direction. All other rules in regard to quaternions (we can
write, for example, this: i ·j·k = -1), and the other products you will find in the Wikipedia article on
quaternions) can be derived from these, but we will not go into them here.
Now, you will say, we do not really need that k, do we? Just distinguishing between i and j should do,
right? The answer to that question is: yes, but only when you are dealing with electromagnetic
oscillations! But it is a resounding no when you are trying to model nuclear oscillations! That is, in fact,
exactly why we need this quaternion math in quantum physics!
Let us think about this nuclear oscillation. Particle physics experiments especially high-energy physics
experiments effectively provide evidence for the presence of a nuclear force. To explain the proton
radius, one can effectively think of a nuclear oscillation as an orbital oscillation in three rather than just
two dimensions. The oscillation is, therefore, driven by two (perpendicular) forces rather than just one,
with the frequency of each of the oscillators being equal to ω = E/2ħ = mc2/2ħ.
Each of the two perpendicular oscillations would, therefore, pack one half-unit of ħ only. The ω =
E/2ħ formula also incorporates the energy equipartition theorem, according to which each of the two
oscillations should pack half of the total energy of the nuclear particle (so that is the proton, in this
case). This spherical view of a proton fits nicely with packing models for nucleons and yields the
experimentally measured radius of a proton:
 󰇨 
Of course, you can immediately see that the 4 factor is the same factor 4 as the one appearing in the
formula for the surface area of a sphere (A = 4πr2), as opposed to that for the surface of a disc (A = πr2).
And now you should be able to appreciate that we should probably represent a proton by
a combination of two wavefunctions. Something like this:
What about a wave equation for nuclear oscillations? Do we need one? We sure do. Perhaps we do not
need one to model a neutron as some nuclear dance of a negative and a positive charge. Indeed, think
of a combination of a proton and what we will refer to as a deep electron here, just to distinguish it from
an electron in Schrödinger’s atomic electron orbitals. But we might need it when we are modeling
something more complicated, such as the different energy states of, say, a deuteron nucleus, which
combines a proton and a neutron and, therefore, two positive charges and one deep electron.
According to some, the deep electron may also appear in other energy states and may, therefore, give
rise to a different kind of hydrogen (they are referred to as hydrinos). What do I think of those? I think
these things do not exist and, if they do, they cannot be stable. These researchers need to produce a
wave equation for them in order to be credible and, in light of what we wrote about the complications
in regard to the various rotational planes, that wave equation will probably have all of Hamilton’s basic
quaternions in it. [But so, as mentioned above, I am waiting for them to come up with something that
makes sense and matches what we can actually observe in Nature: those hydrinos should have a
specific spectrum, and we do not such see such spectrum from, say, the Sun, where there is so much
going on so, if hydrinos exist, the Sun should produce them, right? So, yes, I am rather skeptical here: I
do think we know everything now and physics, as a science, is sort of complete and, therefore, dead as a
science: all that is left now is engineering!]
But, yes, quaternion algebra is a very necessary part of our toolkit. It completes our description of
Let us now go back to our electron wavefunction and do some more calculations. We already said that a
field is a force per unit charge, so if we have the force, we can calculate the field strength. Now, we refer
to previous papers
for those force calculations. Here, we will just present the result for the electron:
If we think in terms of some force holding the pointlike charge in its orbit, then we calculate this force
for the electron as being equal to about 0.106 N. This is the formula (the ½ factor has something to do
with effective mass (half of the mass of the electron is kinetic and the other half is field energy), but we
will not bother you with that right now
That is a huge force at the sub-atomic scale: it is equivalent to a force that gives a mass of about 106
gram (1 g = 103 kg) an acceleration of 1 m/s per second! However, if you think it might be too huge to
make sense, think again: this is an electromagnetic force, and the nuclear force inside a muon-electron
and the proton is much stronger. Anyway, let us calculate the field strength now:
To help you to appreciate how humongous this value is, you should note that the most powerful man-
made accelerators may only reach field strengths of the order of 109 N/C (1 GV/m). So, does this make
any sense? We think it does, but you should, of course, always think for yourself. 
Oh what about that asymmetric potential for the nuclear force and energy conservation? Think about
it: the energy conservation law should hold because, for the nuclear force also, we will have the
equivalent of an electric and a magnetic field component, and so the associated energy sloshes back and
forth between them.
The only weird thing is that the inverse-square law does not seem to hold for the nuclear potential, but
that is perhaps because the nuclear force is so humongous (think of the massive proton versus the
volatile electron here) that it might, effectively, curve spacetime. It is not a very elegant solution to the
(field) energy conservation problem, but we do not see any other one, unfortunately. Perhaps you will
manage to show that gravity is, somehow, some residual forcea sort of Van der Waals force: that
what is left from the fundamental forces at the atomic or molecular scale. We should warn you,
however: many have tried their hand at this (or have trained their brain on it, I should say, but no one
has ever managed to do that. 
See, for example, our lecture on quantum behavior. The idea is this: energy is force over a distance, so we get
the force from using the Planck-Einstein and mass-energy equivalence relations, and substituting the distance by
the circumference of the loop.
The formula itself is derived from the definition of energy as a force over a distance: E = F (if the force is not
constant, then you need to calculate an integral). So, we can write F = E/ = (mc2)/(h/mc) = m2c3/h. This calculation
may be off with a factor 2 or a factor ½, so you should just think of it as an order of magnitude.
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