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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 ×10−34,

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.

1

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=h∆tmfJ

∆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=h∆f=J

∆

∆

s=J

s(8)

2

2018 The OM Particle 3

Here, Pis power and h∆has the numerical value of

Planck’s constant (6.626 ×10−34) 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

3

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:

∆x∆p≥h(11)

∆E∆t≥h(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:

∆E∆t≥h∆∆t(13)

Here, ∆t=tmand interprets as time interval. The ∆t’s

cancel on each side and you end up with the following

relation:

∆E≥h∆(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 ×10−34 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:

∆x∆p≥h∆∆t(15)

Dividing both sides by ∆tgives the following:

∆x∆p

∆t≥h∆(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:

∆E≥h∆(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 ×10−34 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)

4

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 ×10−11K. 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 ×10−11Kcan 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 ×10−10 K,

just one order of magnitude higher than the lowest tem-

perature possible, if in fact h∆is 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, h∆is

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

5

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

ss−1

mm] = [1−1](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:

6

2018 The OM Particle 7

(wt −kz)∆s

ss−

∆m

mm= [∆s−∆m](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=h∆∆t

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 h∆and care constants, a new quantum

constant, corresponding to the quantum of momentum,

pq, can be written:

h∆

c=pq=2.2102189 ×10−42 [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,

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8 L. Gardi: On the Units of Planck’s Constant Revisited Vol. 1

(1m−1)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 ×10−22[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 ×10−19[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:

p∂x

∂t=∂Dx

∂x∂x

∂t(39)

The ∂Dxterms cancel and you end up with the time

changing electric ﬂux lines in the xdirection:

p∂x

∂t=∂Dx

∂t(40)

On the left, charge density times velocity is equal to

current density leading to the following:

p∂x

∂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.

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2. Gardi, Lori-Anne. "Calibrating the universe, and why we

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3. Gardi, Lori-Anne. "Planck?s Constant and the Law of

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4. Park, Jee Woo, Sebastian A. Will, and Martin W. Zwierlein.

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