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Planck's Constant and the Nature of Light

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Planck's constant, h, is a central character in light theory and quantum mechanics and is traditionally referred to as the quantum of action. The classical definition of action is "kinetic energy minus potential energy, multiplied by time", with units [J s]. The units of the energy equation, E = h f , are traditionally written as [J s 1/s] where units of frequency, f , are [1/s] (cycles per second or Hertz), and the units of Planck's constant, h, are [J s], or the units of action. The two [s] units of this energy equation cancel and the units of E = h f reduce to [J] as expected. Or do they? Using Modified Unit Analysis (MUA), it is shown that the units [J s 1/s] do not equal [J], leading to a flaw in the logic of the standard interpretation of light. This new line of thinking leads to a more classical definition of Planck's constant as the energy of one oscillation (one wavelength, one period) of an electromagnetic wave with frequency f. This new definition of Planck's constant leads to a much simpler interpretation of light which can be modeled using the same mathematical construct (e ^ iφ =cos(φ)+isin(φ)) as all other oscillating wave phenomenon.
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2018 The OM Particle 1
Planck’s Constant and the Nature of Light
Lori Gardi
lori.anne.gardi@gmail.com
Planck’s constant, h, is a central character in light theory and quantum mechanics and is traditionally
referred to as the quantum of action. The classical definition of action is "kinetic energy minus potential energy,
multiplied by time", with units [J s]. The units of the energy equation, E=h f , are traditionally written as [J s
1/s] where units of frequency, f, are [1/s] (cycles per second or Hertz), and the units of Planck’s constant, h,
are [J s], or the units of action. The two [s] units of this energy equation cancel and the units of E=h f reduce
to [J] as expected. Or do they? Using Modified Unit Analysis (MUA), it is shown that the units [J s 1/s] are
not exactly equal to [J], leading to a flaw in the logic of the standard interpretation of light. Using the MUA
approach, the numerical value of Planck’s constant is seen as the energy of one oscillation (one wavelength,
one period) of an electromagnetic wave with frequency f. This reinterpretation of Planck’s energy equation
leads to a much simpler interpretation of light which can be modeled using the same mathematical construct
(eiφ=cos(φ) + isin(φ)) as all other oscillating wave phenomenon.
Keywords: Planck’s constant, unit analysis, domain, wavelength, frequency, period, cycle, wavefunction, sine,
cosine, photon, action, energy, uncertainty principle, luminosity, capacitance, inductance, Maxwell
1. Introduction
Planck’s constant, h, named after the physicist Max
Planck, is an important fundamental quantity in quantum
physics. It links the amount of energy that a "photon"
carries to the frequency of its electromagnetic wave. The
inputs of Planck’s energy equation, E=h f , are action
[J s] and frequency [1/s] and the output, E, interprets
as the energy of one photon with units [J]. Photons are
literally defined as fundamental particles, but what does
this mean? Electrons are fundamental particles because
all electrons contain the same mass-energy. Protons are
also fundamental particles since all protons contain the
same mass-energy. Photons on the other hand, can take
on many energies, so how can they be fundamental? Here
are the questions we should be asking. What is a photon?
How much time does it take to generate a photon? What
is the energy of one oscillation of light? And finally, if
the frequency is variable and the wavelength is variable
and the photon energy is variable, then what does the
constant, h, correspond to?
The purpose of this article is to convince the reader,
beyond a reasonable doubt, that (the numerical value of)
Planck’s constant is the energy of one oscillation of light,
no matter what frequency we are investigating. Although
this appears to be a challenge to standard thinking, and it
is, once you finish reading this article, you will have no
choice but to conclude that the constant, 6.626 ×1034,
is the energy of one oscillation (i.e., one period or one
wavelength) of electromagnetic energy and interprets as
the quantum of energy, h.
The problem seems to have arisen from an improper
treatment of the units of frequency. Historically, and by
convention, the units of frequency were defined as [1/s]
or Hertz. I’m sure this seemed reasonable at the time,
but here’s the problem. The numerical value of [1] in
the unit section of an equation is ambiguous. One what?
Everyone knows that it corresponds to 1-cycle but it is
not obvious at first glance. The SI unit of time is 1-
second. The unit of time has its own label [s]. This is
because time itself is its own domain, separate from the
domains of space, mass and charge.
Although unit analysis generally goes by the name "di-
mensional analysis", the term "dimensional" is also am-
biguous (it has many meanings). Unit analysis is better
thought of as "domain analysis". In SI units, the unit of
the Domain of Time is 1-second, [s]; the unit of the Do-
main of Space is 1-meter, [m]; the unit of the Domain
of Mass is 1-kilogram, [kg]; the unit of the Domain of
Charge is 1-Coulomb, [C].
In Modified Unit Analysis or MUA, as outlined in a
previous paper by the author [2], the Domain of Cycles
(a.k.a. Domain of Oscillation) is added as a unique do-
main of the system (separate from all the other domains).
The unit of the Domain of Cycles is 1-cycle and and is
assigned the unique label []. When this new label is
applied to the unit of frequency, [/s], a different pic-
ture of Planck’s energy equation emerges. This simple
change, , to the language of unit analysis, changes more
than just the interpretation of Planck’s energy equation.
It changes everything, as you will see.
By convention, in this article, all units are placed
within square brackets to clearly distinguish the unit sec-
tion of an equation from the main body of the equation.
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2 L. Gardi: On the Units of Planck’s Constant Revisited Vol. 1
2. The Energy Equation for Light
The energy equation for the photon is traditionally and
by convention written as follows:
E=h f J s 1
s(1)
Here, Eis the energy of the photon, his non-reduced
Planck’s constant and fis the frequency of the photon
in Hz. Planck’s constant, hhas units [J s] and frequency
has units [1/s]. Using Modified Unit Analysis (MUA),
the unit [1] in the unit section is replaced with the unit of
the Domain of Cycles, , as follows:
E=h f J s
s(2)
When written this way, it is clear to see that the units
for this equation are not balanced. When the units of
frequency are written as [1/s], it only appears as if the
units balanced because the [1] in the unit section is being
treated both as the unit of the Domain of Cycles and the
unit of the real numbers. Technically though, [J x 1] is
not equal to [J] because the [1] in the unit section stands
for 1-cycle and thus [J x 1] means [J x 1-cycle] which is
definitely not equal to [J]. This is a problem. There are
several, equivalent ways to solve this problem. This first
is as follows:
E=h f Js
s(3)
In this equation, [s/] reads "seconds per cycle" and
are the units for the period, T, of a wave with frequency
f. In this case, Planck’s constant, h, has units [J s/] and
literally interprets as the energy "of" one period of an
electromagnetic wave. The new [] unit could also be
applied as follows:
E=htmfJ
s
s(4)
In this setup, the units of Planck’s constant are [J/]
and interpret as energy per cycle or quantum of energy,
h. In this equation, measure-time, tmis explicitly shown
in the body of the equation. Technically speaking, we
have decoupled the unit of time from the unit of action.
The equation on the left of the unit section is now in-
terpreted as the equation of an experiment. In this case,
the experiment has to do with counting cycles. In the
above equations, (3) and (4), the unit section reduces to
the units of energy, [J], as required. Also, in both equa-
tions, Planck’s constant is associated with one cycle, one
oscillation, one period and/or one wavelength of an elec-
tromagnetic wave. This seems to imply that each oscilla-
tion or wavelength of light carries the SAME energy, h,
regardless of frequency, contrary to standard thinking. To
further this line of thinking, the body of the equation is
expanded as follows:
E=h
1tm
n
tmJ
s
s(5)
In this setup, all of the terms in the unit section corre-
spond to a term in the body of the equation. This gives
us a complete look at the photon energy equation and as-
sociated units. Here, h/1 interprets as energy per cycle
and n/tminterprets as cycles per second. The anomaly
here is the time parameter, tm. There is no time param-
eter in the original photon energy equation. However, if
we set tmequal to exactly 1 (1-second) and assume that
we don’t have to write the "1’s" in the body of the equa-
tion (because 1 is the unit of the real numbers), then we
recover the original E=h f . Here, fis the number of
cycles counted in exactly one second (i.e., n/1=f).
Although the time parameter of an experiment, tm, can
generally be set to any duration of time, it looks like
the measure-time variable in the photon energy equation
was inadvertently hard coded to exactly 1-second and
subsequently hidden. This implies that the output of this
equation is the energy collected (transported, absorbed)
in an arbitrary 1-second time interval and not the energy
of an elementary particle, i.e., a photon, as previously
thought.
The logic presented in this section also suggests that
each oscillation (cycle, wavelength, period) of light
"carries" the same quantity of energy, h[J/]no matter
which frequency we observe. Although this is clearly
contrary to standard thinking, this logic suggests a new
way of interpreting the "photon" energy equation.
3. A Power Equation for Light
Richard Feynman once said, "There is always another
way to say the same thing, that doesn’t look at all like
the way it was said before". There is in fact another
way to demonstrate the flaw in the logic of the original
interpretation of the "photon" energy equation. Here, we
are going to ignore the equation and look at the unit
section directly, using the notation of MUA:
[J] = J
s
s(6)
Divide both sides by [s]:
J
s=J
s(7)
Here, we have completely gotten rid of the concept
action since the units [J s] no longer appear in the unit
section. Now, we are going to write the equation that
corresponds to these units as follows:
P=hf=J
s=J
s(8)
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2018 The OM Particle 3
Here, Pis power and hhas the numerical value of
Planck’s constant (6.626 ×1034) but with the units of
[J/] or energy per cycle. In short, the energy equation
has been re-written in terms of power thus removing
the necessity for the concepts of action and the photon.
Written this way, it is clear to see that Planck’s constant
is the energy of one cycle or period of electromagnetic
energy and can be considered as a quantum of energy.
This equation (8) may be a more natural way of thinking
about light since the power spectrum of light generally
applies to the frequency domain. (The power spectrum
of light is the distribution of the energies of a complex
waveform among its different frequency components.)
Here again, h, interprets as energy per cycle. This, I
argue is the actual fundamental quantity of light. The
photon, historically described as fundamental, is merely
an arbitrary accumulation of light quanta, h, over an
arbitrary 1-second time interval (i.e., photon energy is
VARIABLE and therefore not fundamental).
We still need to reconcile how all oscillations (wave-
lengths) of light can embody the same energy, no matter
what the frequency.
4. The Luminous Intensity of Light
In this section, it is argued that each cycle (wave-
length) of light contains the same energy (no matter what
frequency) and that it is the intensity of light that changes
with wavelength. The units for intensity are as follows:
J
m2s(9)
From these units, it is clear to see that there are several
ways to increase the intensity of a luminous signal. One
way is to increase the energy [J] delivered to the same
area in the same interval of time. There are two other
ways to increase the intensity of the signal. One is to
keep the energy constant but reduce the surface area with
which the energy is being delivered and the other is to
decrease the time interval with which the energy is being
delivered. In a previous paper by the author, "Planck’s
Constant and the Law of Capacitance"[3], it was shown
that both the area and the time interval decrease with a
decrease in wavelength. A depiction of this is seen in
Figure 1.
The top wave depicts a lower frequency (longer wave-
length) signal. The bottom wave depicts a higher fre-
quency (smaller wavelength) signal. To the right is the
depiction of a detector of some kind. Each dot in this fig-
ure represents one quantum of energy. As you can see,
for each frequency, each wavelength "contains" the same
unit of energy. However, in the bottom signal, the ’same’
energy is delivered to a smaller area of the detector in a
shorter period of time. From this, it is clear to see how
all wavelengths can carry the same energy and yet de-
liver different energy intensities to a detector (or solar
Figure 1. Three different wavelengths/frequencies of light.
panel). Longer wavelengths take longer to deliver the
same energy to a larger area of the detector (i.e., intensity
is lower). Also, from this schematic, it is clear so see that
more energy units are delivered to the detector in unit
time by the higher frequency signal than the lower fre-
quency signal. This explains why higher frequency light
"contains" more energy than lower frequency light.
5. Discussion 1
The purpose of the previous sections was to explore a
previously discovered but not well known flaw in the unit
section of of Planck’s energy equation. According to the
research of Dr. Juliana Mortenson, the problem stemmed
from Planck’s adoption of Wien’s method for determin-
ing energy density which inadvertently removed the time
parameter from his equations[1]. This explains why there
is no time parameter in the equation E=h f . There also
appears to be a flaw in the definition of frequency itself.
1960, the term "cycles per second" or [cps] was used for
the unit of frequency. In 1960, the term was officially
replaced by the Hertz with units of inverse time. Thus
[1/s] replaced [cps] or cycles per second. In this defini-
tion, [1] appears to be the unit of the domain of cycles.
The assumption here was that, since cycles are count-
able, the proper unit for the domain of cycles was the
unit of countable numbers which is the numerical value
of [1]. The flaw in this logic can be easily seen when one
realizes that seconds are also countable (one steamboat,
two steamboat etc, three steamboat, etc.). The same can
be said of meters and kilograms. As well, as seconds,
meters and kilograms are divisible, so too are cycles. If
this weren’t true, then we would not be able to reference
a half cycle, a degree or a radian for that matter. Why
should the domain of cycles be treated any differently
than the domains of time, space, mass and charge? In
Modified Unit Analysis, the Domain of Cycles was ex-
plicitly introduced as its own domain, separate from the
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4 L. Gardi: On the Units of Planck’s Constant Revisited Vol. 1
domains of time, space, mass and charge, and the label,
[1], is replaced by the label, [], as the unit of this do-
main.
Another flaw comes from the fact that 1 is also the
unit of the real numbers. The numerical value of [1] in
the unit section has different meaning than the numerical
value of 1 in the body of the equation. It cannot be used
in the same manner. For example, E x 1 = E but the units
[J x 1] does not evaluate to [J] since the [1] in the unit
section corresponds to 1-cycle, not the real number 1.
Technically, [J x 1] evaluates to [J x 1-cycle] = [J ]
which is technically not equal to [J].
Finally, looking at equation 3, it is obvious that the
[s] from the historical action constant is meant to cancel
with the [s] from the frequency term, just as it is obvious
that the two [] units should cancel. Historically, the [s]
from the action constant was allowed to cancel with one
of the [s] terms from the units of [J] leaving hwith the
units of angular momentum:
E=h f "kg m
s
m
s
/s
/1
s=kg m2
s
1
s#(10)
In MUA, this is not recommended since the [s] from
the action constant is meant to pair with the [s] from
the frequency term. Although it may appear now that the
frequency term is unit-less, it is in fact not unit-less. Even
though it looks like they should cancel, for completeness,
they still need to be written. In MUA, cancellation of
units is highly discouraged as it destroys information. As
such, in MUA, the units [s/×/s] need not cancel.
The historical interpretation of Planck’s energy equa-
tion has lead to some strange concepts such as the uncer-
tainty principle, the action constant, the photon and the
quantization of angular momentum. MUA offers a more
complete notation of unit analysis that, when applied to
Planck’s energy equation, leads to a different interpreta-
tion of light as you will see. All equations that contain
Planck’s constant, hor ¯
hneed to be reexamined from the
perspective of MUA. In the next section, MUA is applied
to the Heisenberg Uncertainty Principle leading to a sur-
prisingly certain conclusion.
6. The Heisenberg Uncertainty Principle
The Heisenberg uncertainty principle asserts that there
is a fundamental limit to the precision with which certain
pairs of physical properties of a particle can be known.
The two relations in question are as follows:
xph(11)
Eth(12)
The standard interpretation of the first relation is that
the position and momentum of a particle cannot be si-
multaneously measured with arbitrary high precision.
There is a similar interpretation for the product of the
uncertainties of energy and time as seen in the second
relation. In equation (4) of this article, the time parame-
ter from the unit section was decoupled from the unit of
Planck’s constant and a measure-time variable, tm, was
added to the body of the equation. Using the logic of
this equation (4) a similar approach is taken with the two
above relations. Let’s do the second one first:
Etht(13)
Here, t=tmand interprets as time interval. The t’s
cancel on each side and you end up with the following
relation:
Eh(14)
The interpretation of this relation is quite simple.
The smallest change in energy that can possibly be de-
tected and measured is the quantum of energy or h=
6.626 ×1034 Joules. Next we will address the position-
momentum relation in (11). Again, using the logic from
equation (4), the position-momentum relation is written
as:
xpht(15)
Dividing both sides by tgives the following:
xp
th(16)
The units of distance, x, are [m], phas the units of
momentum, [kg m/s], and the units of time, tare [s].
Thus, the units of the left side of this relation are [kg m/s
m/s] or the units of energy. This reduces to the previous
relation:
Eh(17)
In short, by decoupling the extra [s] unit from Planck’s
constant, as was done in equation (4), a clear, understand-
able and very certain relationship presents itself. There is
nothing special nor is there anything uncertain about this
relationship as it tells clearly that one cannot measure
a change in energy smaller than 6.626 ×1034 Joules.
If this is true, then there must be a smallest detectable,
measurable temperature as well.
7. Boltzmann’s Constant and Temperature
If the minimal detectable energy is the quantum of en-
ergy or h, then this must correspond to a minimum de-
tectable temperature. The units for Boltzmann’s constant
are:
KbJ
K(18)
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2018 The OM Particle 5
In order to convert energy to temperature, one must
divide energy, E, by Boltzmann’s constant:
T=E
Kb
(19)
When we apply the quantum of energy, h, to this
equation, we get a value of 4.799243 ×1011K. Accord-
ing to relation (14), there exists a quantum of energy be-
low which no energy can be detected. In other words, no
temperature below 4.799243 ×1011Kcan ever be cre-
ated nor detected. The coldest temperature ever "created"
was by a group at MIT who created and measured a tem-
perature of around 500 picokelvin [4] or 5 ×1010 K,
just one order of magnitude higher than the lowest tem-
perature possible, if in fact his the smallest detectable
energy.
In the previous sections, MUA is used to fix the
flaw in the units of the "photon" energy equation and
remove the uncertainty of the uncertainty principle. The
photon energy equation was re-written in terms of power
thus removing the concept of action altogether. Most
importantly, the units for luminosity were used to show
"how" all wavelengths of light can contain the same
energy unit, h, no matter what the frequency, contrary
to standard thinking. Using Modified Unit Analysis, as
summarized below, the true nature of light can finally be
revealed.
8. Summary of Modified Unit Analysis
Using the logic outlined in the previous sections, Mod-
ified Unit Analysis, MUA, is specified as follows:
1. The Domain of Cycles, separate from the domains
of time, space, mass and charge, is added to unit
analysis.
2. The unit of the Domain of Cycles has the label
[]. This applies to both cyclic frequency and flow
frequency and generalizes to the unit of smallest
change.
3. The unit must be explicitly written in the unit
section.
4. The unit of frequency, f, is [/s].
5. The unit of period, T, [s/].
6. The unit of wavelength, λ, is [m/].
7. The unit of wavenumber, k, is [/m],
9. The Electronic Nature of Light
There is another power equation that is very similar to
equation (8):
P=V I [J
C
C
s] = [J
s](20)
This is the equation for power in an electronic circuit.
Here, Voltage, V, has units [J/C] and current, I, has units
[C/s]. Notice how the units for this equation look very
similar to the units for equation (8) only the unit "cycle"
[] is replaced by the unit "coulomb" [C]. An analogy
between these two equations will now be discussed.
One coulomb is defined as the charge transported (in a
circuit) by a current of 1 ampere during a 1 second time
interval. In other words, the coulomb is defined in terms
of flow rate or frequency. As light propagates through
space, charge propagates through a circuit. The rate with
which wavefronts of light pass by a point in space de-
termines the electromagnetic frequency. The rate with
which charge passes by a point in the circuit determines
the flow frequency of the current. In this manner, fre-
quency of light and current flow rate within a circuit are
conceptually analogous (self-similar). In these two equa-
tions (8 and 20), cycle is analogous to Coulomb, his
analogous to voltage (V) and frequency, f, is analogous
to current (I). The implication here is that light "flowing"
through the medium of space can be modelled in terms
of current flowing through a circuit of some kind.
In a previous paper by the author, "Planck’s Constant
and the Law of Capacitance"[3], the exact value of his
derived from first principles using the law of capacitance.
It is well known that an oscillating circuit can be built
using capacitors and inductors. Since electromagnetic
radiation is an oscillating phenomenon, then it could be
modelled using capacitors and inductors as follows:
Figure 2. Electromagnetic wave as an electronic circuit.
The capacitors in parallel ensure that the voltage re-
mains constant. The inductors in series ensure that the
current remains constant. The arrangement of this circuit
would ensure constant power during propagation (trans-
mission). This is in fact a schematic diagram of a power-
line circuit. The similarities between equation (8) and
equation (20) suggests an isomorphism between electro-
magnetic "waves" and current flowing through a circuit.
Although this section may seem out of context with
the subject matter of this article, its main purpose was to
show the similarity between the power equation for an
electronic circuit, equation (20), and the power equation
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6 L. Gardi: On the Units of Planck’s Constant Revisited Vol. 1
derived earlier, equation (8), and to give a context with
which the electromagnetic nature of light can be visual-
ized.
10. The Wave ΨNature of Light
Technically, in wave theory, all waves are continu-
ously sloshing between potential energy and kinetic en-
ergy. In fact, this is true of all oscillators, from pendu-
lums to springs to water waves to sound waves. This is
also true in the electronic oscillator made up of capaci-
tors and inductors. As capacitors store potential energy,
inductors store kinetic energy. In an oscillating circuit,
voltage (potential) and current (kinetic) are 90 degrees
out of phase with each other, that is, when the current
through the inductor is maximal, the voltage over the ca-
pacitor is minimal, and vice versa. Sine waves and cosine
waves are also 90 degrees out of phase with each other
which is why these mathematical entities are used to
model electronic circuits. If electromagnetic wave propa-
gation is analogous to (isomorphic to) the propagation of
power through power lines, as depicted in Figure 2, then
it makes logical sense to model electromagnetic waves
using sine and cosine waves as follows:
Figure 3. Euler’s formula and depiction of the wavefunction.
Historically, and by convention using Maxwell’s equa-
tions, electromagnetic wave propagation is modelled us-
ing two cosine waves (via the plane wave derivation)
such that the electro (potential) component and magneto
(kinetic) components are in phase with each other as de-
picted in the following diagram:
Figure 4. Maxwell’s Plane Wave
This is the standard diagram used throughout the
physics community to depict the propagation of light
through the vacuum of space. However, in ALL other
oscillators and wave phenomenon propagating in a
medium, the kinetic component and potential compo-
nents are 90 degrees out of phase as shown in Figure 3.
Why should light waves behave so much differently than
all other oscillating wave phenomenon? Short answer;
they don’t. Figure 3 is the correct mathematical model
for light propagation and Figure 4 is just "plane" wrong.
So, what happened? It turns out that Maxwell et al may
have made a mistake too. See Appendix B.
11. Maxwell’s Plane Wave Equation
In this section, we demonstrate how MUA affects
the interpretation of the plane wave equation, given as
follows:
Ey
z2=εµ 2Ey
t2(21)
Here:
Ey=A1cos(wt kz) + A2cos(wt +kz)(22)
and:
k=w
c(23)
The symbols εand µcorrespond to the permittivity and
permeability of the medium and kis the wavenumber
with units [1/m]. Applying standard unit analysis to the
term (wt - kz) gives as follows:
(wt kz) [1
ss1
mm] = [11](24)
In other words, this expression is unitless. Clearly, one
can see how the numerical value of [1] in the unit section
of an equation can lead to ambiguity. In MUA, the units
are written as follows:
(wt kz)
ss
mm(25)
Here, the [1] is replaced by the unit of the domain of cy-
cles, []. As stated earlier, canceling terms in Modified
Unit Analysis is highly discouraged as it destroys infor-
mation. For example, in (25), if we cancel the [s] and
the [m] terms, we would end up with [] which is
still ambiguous as it give the false impression that these
two terms can operate on each other and give a mean-
ingful result. Looking more closely, you will notice that
the term on the left has frequency in the Domain of Time
and the term on the right has frequency in the Domain
of Space. To distinguish these two terms, a subscript is
added as follows:
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2018 The OM Particle 7
(wt kz)s
ss
m
mm= [sm](26)
Now, we can cancel the [s] and [m] terms in the unit
section as the information about what domain the
term belongs to is now recovered. Here’s the problem.
The sterm on the left and the mterm on the right
belong to different domains, the Domain of Time and
the Domain of Space respectively. These two domains
cannot operate on each other directly. They can only
be compared through a ratio: [m/s] for example. The
domains of time and space are not orthogonal and so
they do not form a vector space. In other words, the
two terms, wt and kx, cannot be added (nor can they
be subtracted). How does one add things that have no
common basis. Aside from the fact that they both contain
frequency terms (in different domains), wt and kz have
absolutely no basis with which to be compared (i.e., they
have nothing in common).
In MUA, frequency terms in the Domain so Space can-
not be added to (or subtracted from) frequency terms in
the Domain of Time, and vice versa, thus, the expres-
sion (wt - kz) makes no logical or mathematical sense.
Therefore, any mathematical derivation that uses this
expression, including Schroedinger’s equation, is erro-
neous (i.e., inherently flawed).
12. Discussion 2
Paul Dirac was once asked, "How do you find new
laws of physics" and he replied, "I play with equa-
tions." The author herein, "plays" with unit analysis.
Unit analysis is about interpretation and playing with
unit analysis, via Modified Unit Analysis, has led to
some interesting interpretations that differ from standard
thinking. The interpretations of the past has led to some
strange concepts such as the uncertainty principle, the
quantum of action and the photon. What if this was
all because we got unit analysis wrong when we were
first developing quantum mechanics? What if all the
strangeness goes away if we just make this small change,
, to the language of unit analysis? What if the correct
model for light is the same as the model for all other
oscillating (wave) phenomenon? This would certainly
simplify a lot of things.
13. Conclusion
It is the contention of the author that a mistake was
made by convention more than 100 years ago when
the units of frequency were given as [1/s] (since the
[1] in the unit section is ambiguous). In Modified Unit
Analysis (MUA), the Domain of Cycles is added as
a domain of the system, separate from the domains
of time, space, mass and charge. The unit of cycle is
assigned a unique label, . The units of frequency in
the domain of time are written as [/s]; the units of
frequency in the domain of space (the wavenumber) are
written as [/m]; the units of wavelength are written
as [m/]; the units of period are written as [s/]. If
both temporal and spatial frequency terms appear in
the same expression, a subscript is added to the delta
term, sand mto disambiguate the delta terms. This
provides a more complete and balanced unit analysis
when used correctly. In particular, it leads to a different,
and hopefully, more understandable concept of light as a
wave, just like any other "wave" propagating through a
medium.
13.1. Appendix A: The Momentum Equation
In this section, a new constant of nature, quantum of
momentum, is derived. This is not new, and has been
done before [7], but the logic outlined herein uses the lan-
guage and notation of MUA. Traditionally the relation-
ship between Planck’s constant and momentum is written
as follows:
p=h
λ(27)
Using the logic of equation (4), the following is writ-
ten:
p=ht
λ(28)
Dividing both sides by tgives the following force
equation:
p
t=h
λ(29)
This equation has the units of force. Looking at this
equation, it is clear to see that force increases with
shorter wavelength (higher frequency) light as expected.
Using the relationship between wavelength and fre-
quency, λ=c/fthe following is written:
p
t=h
cf(30)
Since both hand care constants, a new quantum
constant, corresponding to the quantum of momentum,
pq, can be written:
h
c=pq=2.2102189 ×1042 [kg m
s] = [P](31)
Interestingly, this exact value appears in the NIST
standard[5] but with units [kg] and is referred to as the
"inverse meter-kilogram relationship". Even using the
standard units of Planck’s constant, [J s], the units of h/c
do not evaluate to [kg] but instead, evaluates to [kg m].
In the NIST Reference on Constants, Units, and Uncer-
tainty, the inverse meter-kilogram relationship is written,
7
8 L. Gardi: On the Units of Planck’s Constant Revisited Vol. 1
(1m1)h/c. Notice how they are mixing units with con-
stants in this relationship? In MUA, this is strictly for-
bidden. The units for h/c are either [kg m] in standard
unit analysis, or [kg m/s] in MUA. Using this new con-
stant (of nature), the corresponding force equation in the
frequency domain, can now be written as follows:
F=pqfPs
s=P
s(32)
Since there currently is no standard label for the unit
for momentum, MUA uses [P] to represent the unit
of momentum [kg m/s]. This equation (and associated
units) looks similar to equation (8) but is a force equation
instead of a power equation and the constant, pq, inter-
prets as the quantum of momentum. This new constant
can be used in the following manner. Multiplying the
quantum of momentum, pqby the Compton frequency
of the electron, fe=1.2356 ×1020 Hz, gives the follow-
ing:
Fe=pq×fe=2.7309 ×1022[P
s](33)
NIST refers to this exact value as the natural unit
of momentum. The above equation, however, inter-
prets as a force. A similar technique can be applied to
the proton using the proton Compton frequency ( fp=
c/protonComptonWavelength = 2.2687×1023 Hz) as fol-
lows,
Fp=pq×fp=5.01439 ×1019[P
s](34)
As you can see, MUA opens up the door to a new con-
stant of nature, quantum of momentum, which has phys-
ical meaning. Interestingly, the values fe,fp,pqand Fe
all appear in the NIST standard but, for some reason, Fp
does not, even though equation (33) has similar interpre-
tation to equation (34). Further investigation as to why is
suggested.
13.2. Appendix B: Maxwell’s Displacement Current
Maxwell’s equations are generally thought to encoded
everything we know about electricity and magnetism.
That said, a group at IBM recently discovered a new, sub-
tle, effect which they are calling the "camelback effect"
that is not encoded into Maxwell’s equations (A New Ef-
fect in Electromagnetism Discovered: 150 years later). In
other words, Maxwell’s equations are either incomplete
or inherently flawed. Robert Distinti, an electrical engi-
neer of 30+ years, has discovered a flaw in the at least one
of Maxwell’s equations. Distinti gets credit for discover-
ing this problem that has illuded the physics community
for the last 150 years. If Maxwell’s equations are not just
incomplete, but inherently flawed, then the derivation of
the plane wave equation may also be flawed along with
the plane wave model of light as depicted in Figure 4 of
this article. The equation in question is as follows:
H=J+D
t(35)
Thought experiment: A circular length of wire is imag-
ined as a capacitor and the inductor in series. An imagi-
nary barrier is place between the imaginary plates of the
imaginary capacitor in the loop. The number of flux lines
that cut the imaginary boundary is referred to as the dis-
placement current. This is represented by the term on the
right of the above equation, D/t. The flaw in the logic
of this equation will now be outlined.
The divergence of the electric field lines gives you the
charge density within a finite region:
p=·D(36)
Putting into point form gives:
p=Dx
x+Dy
y+Dz
z(37)
For now, we will look at the xcomponent only:
p=Dx
x(38)
Multiply both sides by x/t:
px
t=Dx
xx
t(39)
The Dxterms cancel and you end up with the time
changing electric flux lines in the xdirection:
px
t=Dx
t(40)
On the left, charge density times velocity is equal to
current density leading to the following:
px
t=Jx=Dx
t(41)
Putting back into point form gives the following:
p
t=J=D
t(42)
This result leads to the following conclusion:
H=J+D
t=2×J(43)
This herein lies the problem. How can this possibly be
true? If this were true, then why didn’t Maxwell write
H=2J? According the the logic presented in this
section, the following appears to be true:
8
2018 The OM Particle 9
H=J=D
t(44)
In other words, displacement current and current den-
sity are one in the same. If equation (35) is wrong, then
this calls into question Maxwell’s plane wave derivation
along with the plane wave interpretation of light as de-
picted in Figure 4. Further investigation into this problem
is highly recommended.
As a side note, it should be mentioned that Maxwell’s
equations are rarely used by electrical engineers. The
standard wavefunction as depicted in Figure 3 is the
equation that is used in practice. The Radar Handbook
[6], for example barely mentions Maxwell’s equations.
13.3. Acknowledgments
Many thanks to Robert Distinti for his fearless efforts
in "slaying the dragon" of modern physics. Your genius,
diligence and persistent search for truth was the inspi-
ration that I needed to complete this article. Also, many
thanks for your enlightening YouTube video, "The Great
Displacement Current Caper" which was loosely tran-
scribed into the Appendix section of this article.
REFERENCES
1. Brooks, Juliana HJ. "Hidden variables: the elementary
quantum of light." In The Nature of Light: What are
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for Optics and Photonics, 2009.
2. Gardi, Lori-Anne. "Calibrating the universe, and why we
need to do it." Physics Essays 29, no. 3 (2016): 327-336.
3. Gardi, Lori-Anne. "Planck?s Constant and the Law of
Capacitance."
4. Park, Jee Woo, Sebastian A. Will, and Martin W. Zwierlein.
"Ultracold Dipolar Gas of Fermionic [superscript 23] Na
[superscript 40] K Molecules in Their Absolute Ground
State." (2015).
5. National Institute of Standards and Tech-
nology, Gaithersburg MD, 20899, From:
http://physics.nist.gov/constants, All values (ascii),
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(retrieved March 15, 2016). NOTE: this is an ascii table
only and does not have page numbers but is easily
searchable.
6. Skolnik, Merrill Ivan. "Radar handbook." (1970).
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mass, and momentum." In The Nature of Light: What are
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for Optics and Photonics, 2011.
9
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