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Canadian Journal of Pure and Applied Sciences

Vol. 15, No. 3, pp. 0000-0000, Oct 2021

Online ISSN: 1920-3853; Print ISSN: 1715-9997

Available online at www.cjpas.net

UNDERSTANDING THE PLANCK CONSTANT AND THE BEHAVIOUR OF

PHOTON PARTICLES FROM A MECHANICAL PERSPECTIVE

Wenzhong David Zhang

Address: Hembury Avenue, Manchester, M19 1FH, UK

ABSTRACT

The Planck constant is derived from the analysing of the energy and frequency relationship of a simple harmonic oscillator

model for photon particles from a mechanical perspective. The correlation between the Planck constant, the inertial mass,

the angular frequency, and the corresponding radius of the harmonic oscillation of the photon particle is derived. The

photon particles have equal mechanical angular momentum. This insight is applied to the propagation of photon particles

in the free space and other transparent media based on the model of the cycloid motion of harmonic oscillators. The

interdependence of the properties of photon particles and their entanglements with their surrounding spaces are elicited.

Kepler’s second and third laws of motion are deduced for the specified photon particle in the free space and other

transparent media. For any particle in periodic motion with the rotational symmetry, its inertial mass, its time period, and

its space displacement are entangled together by its conserved mechanical angular momentum. The centre of the periodic

motion with the rotational symmetry is proposed as a universally applicable reference frame to simplify and unify the laws

of physics. The generalized Planck constant, generalized Planck-Einstein relation, and generalized de Broglie relation are

proposed for applications in both microcosms and macrocosms.

Keywords: Planck constant, mechanical angular momentum, cycloid motion, generalized Planck constant, generalized

Planck-Einstein relation, generalized de Broglie relation.

INTRODUCTION

The Planck constant is the pillar of modern quantum

physics. The Planck constant has profound ramifications in

three important areas: our technology, our understanding

of reality, and our understanding of life (Stein, JD. 2011.

Planck’s constant: The number that rules technology,

reality, and life.

http://www.pbs.org/wgbh/nova/blogs/physics/2011/10/pla

ncks-constant). The Planck constant entered physics in

the beginning of the 20th century as the result of Max

Planck’s attempts to provide a theoretical explanation for

the empirically discovered laws of thermal blackbody

radiation (Planck, 1901; Planck, 1900). He found that the

experimental observations of thermal blackbody radiation

spectrum can be speculated in perfect agreement, if one

adopted the concept that matter was a collection of discrete

harmonic oscillators (Oldershaw, 2013) emitting

electromagnetic radiation in packages (energy quanta) that

obeyed an energy (E) / frequency (f) law of the following

form:

(1)

One of the earliest applications of the Planck constant was

by Einstein to explain some aspects of Wien’s law for

blackbody radiation and to account for the photoelectric

effect (Einstein, 1905). Einstein introduced the idea of a

photon particle with energy related to its frequency as E =

hf called the Planck-Einstein relation. The establishment of

the Planck-Einstein relation is generally regarded as the

starting point of quantum physics. The Planck constant has

become one of the most essential universal constants.

However, the physical origin and nature of the Planck

constant have not been fully understood. Investigating the

physical origin and nature of the Planck constant is not only

important for advancing our understanding on the

foundation of modern quantum physics for microcosms, it

is also relevant to the current study of cosmology (Chang,

2017). Stellar Planck constant has been proposed to

describe the characteristic quantities of angular momentum

and action inherent in the objects of the stellar level of

matter (Fedosin, 2015; Fedosin, 2014). Basically, every

aspect of nature is associated with quantum phenomena

involving the Planck constant. Understanding the physical

origin and nature of the Planck constant may open the door

to unify the physical laws of the microcosm and the

macrocosm, the classical and the quantum.

The Planck constant is linked with periodic motion with

rotational symmetry. In order to get a better insight into the

physical origin and the nature of the Planck constant, an

approach is discovered to derive the Planck constant

through the analysing of the energy and frequency

relationship of a simple harmonic oscillator model for a

_____________________________________________________________________

Corresponding author e-mail: WenzhongZhang2008@gmail.com

Canadian Journal of Pure and Applied Sciences

2

photon particle from a mechanical perspective. The gained

insight into the physical origin and the nature of the Planck

constant is both applicable and powerful in elucidating the

property of photon particles in the free space (vacuum) and

other transparent media. All photon particles have an equal

mechanical angular momentum ħ. The interdependence of

the properties of photon particles and their entanglements

with their surrounding spaces are elicited. The inertial mass

of any particle in periodic motion with rotational symmetry

should not be equal to zero. Any particle in periodic motion

with rotational symmetry, its inertial mass, its time period,

and its space displacement are entangled together with its

conserved mechanical angular momentum.

The exploration of the physical origin and the nature of

the Planck constant from a mechanical perspective

The exploration begins from analysing the energy and

frequency relationship of a simple harmonic oscillator with

rotational symmetry as shown schematically in Figure 1.

For harmonic oscillation to occur, the system must possess

the following two quantities: elasticity and inertia. The

simplest example of a harmonic oscillating system is a

mass connected to a rigid foundation by way of a spring.

At this stage, we assume that the spring is undamped and

massless under ideal approximation. However, a nonzero

spring mass or damped spring can be easily accommodated

under some simplifying assumptions and can be developed

further for the more accurate understanding of reality

(Garret, 2017; Zhang, 2021a; Zhang, 2021b). The elasticity

of the spring k (spring constant) provides the elastic

restoring force that enables the system to return to its

equilibrium when the system is displaced from its

equilibrium position. The inertia of the mass m (inertial

mass) provides the overshoot that warrants the system to

pass the equilibrium position. The natural angular

frequency of the oscillation ω is related to the elastic and

inertia properties of the system.

Fig. 1. The corresponding harmonic circular motion of the

simple harmonic oscillator.

By applying Newton's second law F = ma and Hooke’s law

F = – kx, the equation of motion for the simple mass-spring

harmonic oscillating system is obtained as follows:

where ω is the natural angular frequency of the oscillation,

i.e.

(2)

The solution to the equation of motion takes the following

form:

where A is the amplitude of the oscillation or the

corresponding radius of the harmonic circular motion and

is the phase constant. Both the constants A and are

determined by the initial condition at a chosen time of t =

0 under the ideal approximation that the spring is

undamped and massless. Figure 1 shows schematically the

corresponding harmonic circular motion of the simple

harmonic oscillator. The period T (the time to complete one

corresponding circle), the frequency f, and the angular

frequency ω of the oscillating are defined as follows:

and

(3)

As the system oscillates, the total energy E of the system

remains constant and time-independent, and depends only

on the elasticity parameter k and the maximum

displacement A (or the mass m and the maximum

magnitude of velocity Vm = ωA), i.e.

(4)

where V is the magnitude of velocity (speed) of the point

of the mass m. For the corresponding harmonic circular

motion as shown schematically in Figure 1, one can write

down that

(5)

Differentiation of the total energy

in

equation (4) with respect to the angular frequency ω

generates the following expression:

(6)

Wenzhong David Zhang

3

Hence,

(7)

Comparing equation (7) with equation (1), the Planck

constant for the photon particle at the reference frame of

constant linear light velocity can be derived as follows:

(or ℏ )

(8)

where is the radius of the harmonic circular motion, is

the elasticity parameter of the space, is the equivalent

inertial mass of the photon particle and ℏ is the reduced

Planck constant or Dirac’s constant,

.

Incorporating equation (2), (5), and (8), especially for the

harmonic circular motion of the photon particle at the

reference frame of constant linear light velocity, it is

derived that

(9)

where V is the average of the magnitude of velocity (speed)

of the photon that represents the constant speed c in the free

space (vacuum) and the constant V in other transparent

homogeneous media. While is the mechanical

angular momentum of the harmonic circular motion of

photon particles. Hence, the reduced Planck constant is

the mechanical angular momentum of the harmonic

circular motion of photon particles.

It is worth to point out that the mechanical angular

momentum of photon particles from a mechanical

perspective is different to the optical angular momentum of

the corresponding photon waves from an electromagnetic

perspective (Bliokh and Nori, 2015), although they have

interdependence. The conservation of mechanical angular

momentum is fundamentally associated with periodic

motions of the rotational symmetry and can be calculated

using Noether’s theorem (Bliokh and Nori, 2015). For the

photon particles, equations (1), (8), and (9) disclose the

correlation and interdependence of and

ℏ. Physical science is all about the correlation of physical

quantities. If we take (ℏ) and the velocity of photon V =

c in the free space (vacuum) as some universal constants,

then the m (equivalent inertial mass), the (time period)

and the A (space displacement) of the photon particles are

entangled together with the constant mechanical angular

momentum .

From equation (9), it also can be seen that the reduced

Planck constant is a constant determined by the

interacting of the photon particle and its surrounding space.

If a simple dissipative element as a viscous damper was

added to the simple harmonic oscillator model to take into

account the weak friction force of the free space, it is

derived theoretically that the energy and the inertial mass

of the photon particle can decrease slowly, its

corresponding radius of harmonic circular motion

increases slowly (red-shift), its mechanical angular

momentum is kept as a constant according to equation (9)

(Zhang, 2021a; Zhang, 2021b). The harmonic oscillator

model of photon particles also paves the way for a possible

explanation that the energies and the inertial mass of

photon oscillators can be increased under driving forces

and works, meanwhile, their corresponding radii are

decreased (blue shift) according to equation (9). A driving

and dampening harmonic oscillator model may be further

developed to explain the production of high energy

photons, electron-positron pairs, and other elementary

particles.

Conservation of mechanical angular momentum in

macrocosms is well-known as Kepler’s second law or

Kepler’s equal area law. In microcosms, it is well-known

as the quantization of angular momentum, which was

initially proposed as one of Bohr’s key hypotheses in the

Bohr model of hydrogen. The mechanical angular

momentum is a conservative quantity in both microcosms

and macrocosms for periodic motions with rotational

symmetry, although the real trajectories of particles in

microcosms are quite different with the trajectories of

particles in macrocosms. The trajectories of particles in

microcosms are blurred to clouds in approximately round

shapes by transient energy fluctuations. It is unveiled that

the quantization of mechanical angular momentum in

quantum mechanics and Kepler’s second law in astronomy

arise from the same fundamental law of the conservation

of mechanical angular momentum for periodic motions

with rotational symmetry. For a particle in periodic motion

with rotational symmetry, its mechanical angular

momentum is a constant and can never be equal to zero.

Therefore, its energy forms into quantum (discrete energy)

in the frequency domain according to equation (7). The

Planck-Einstein relation can be generalized for any particle

in the periodic motion with rotational symmetry as follows:

(10)

(11)

where E is the energy of the particle in periodic motion

with the rotational symmetry, is the angular frequency

of the particle, HA is the conserved mechanical angular

momentum. HA may be named as the generalized Planck

constant. The particle can be a photon particle, an

elementary particle, an atom, a planet, a star, or a galaxy.

It shall be emphasized that from equation (7) to equations

(1) and (10), a zero-energy point is assigned while the

frequency of the periodic motion is equal to zero. If the

frequency is equal to zero, the motion shall be a motion

along an absolute straight line without spinning or

oscillation. This kind of motion may not exist in the

Universe. The typical motions at elementary particle scale,

atomic scale, stellar scale, and galactic scale are periodic

motions with the rotational symmetry. Hence, the

conservation of mechanical angular momentum is

Canadian Journal of Pure and Applied Sciences

4

fundamental in both microcosms and macrocosms. All

particles in periodic motion with the rotational symmetry

in both microcosms and macrocosms, their inertial masses,

their time periods, and their space displacements are

entangled together through their conserved mechanical

angular momentums. Einstein’s general relativity is a kind

of beautiful and abstractive mathematical description of the

physical world famously depicted as: Matter tells space-

time how to curve; space-time tells matter how to move

(Wheeler and Ford, 1998, 2000) that indicates the

entanglement of matter, space, and time. However, it does

not reveal the fundamental nature of the physical world,

why and how? Now it is uncovered that the fundamental

nature of the physical world is the entanglement of the

inertial mass, the time period, and the space displacement

of the particle in periodic motion with the rotational

symmetry, which arises from the fundamental law of the

conservation of mechanical angular momentum for

motions with rotational symmetry. Within the three

parameters of inertial mass, time period, and space

displacement of a particle in periodic motion with

rotational symmetry, only two parameters are independent.

Although the Planck constant is derived from analysing the

energy and frequency relationship of the simple harmonic

oscillator, it is applicable and powerful in elucidating the

property and propagation of photon particles in both the

free space (vacuum) and other transparent media, which

will be illustrated in the following sections. It will be

shown that Kepler’s third law (the Period law) can be

derived for a specified photon particle based on the new

insight into the physical nature and the physical origin of

the Planck constant.

The property and propagation of photon particles in

the free space (vacuum)

It is proved experimentally that the velocity of photons in

the free space is the constant speed c over a wide range of

frequencies, the relation between energy E and momentum

p of a propagating single frequency photon particle is

(Nelson and Kinder, 2017). The propagation of the

single frequency photon particle in the free space can be

treated as the cycloid motion of a harmonic oscillator from

a mechanical perspective shown schematically in Figure 2.

This model of the cycloid motion of the harmonic oscillator

for the single frequency photon particle illustrates vividly

the Wave Particle Durability of a photon and it can greatly

assist in the derivation of correlations between the

properties of photons. In the period (one cycle,

= 2

),

the single frequency photon particle rolls the wavelength

, which equals to one circumference as

shown in Figures 1 and 2, where

(12)

In the free space, it is experimentally approved that

(13)

From equations (2), (3), (8), (12), and (13), it can be

derived that

(14)

(15)

Incorporating equations (1), (2), (3), and (14), it is obtained

that

(16)

Fig. 2. The schematic diagram showing the cycloid motion

model for a photon particle.

The derived equality is the Einstein mass-energy

equation. However, the mass of the single frequency

photon particle in equation (16) shall be the inertial mass

of the single frequency photon particle from a mechanical

perspective. In this article, for the mass of the single

frequency photon particle, it is called either equivalent

mass or inertial mass to distinguish it with the rest mass in

Einstein’s special relativity. The single frequency photon

particle will be simply called the photon particle below.

Based on equation (16), the momentum of the photon

particle can be deduced as follows:

(17)

where

is the de Broglie relation. From equations

(3), (16), and (17), for the photon particle within one period

of or one wavelength of

, it can be derived that

(18)

(19)

Fascinatingly, for the specified photon particle with the

fixed frequency or wavelength, Kepler’s third law (the

Period law) can be derived from equations (2), (3), and (15)

as follows:

(20)

Wenzhong David Zhang

5

where is Kepler’s constant for the specified photon

particle in periodic motion in the free space. It is defined

by

(21)

Equations (20) and (21) disclose the correlation of

and for the photon particles in periodic

motions in the free space. Kepler’s laws of motion are

applicable in both microcosms and macrocosms. If we

substitute equation (14) into equation (20), it can be

derived that

(22)

How could equations (20) and (22) be both true? The

answer lies in the conserved mechanical angular

momentum of the cycloid motion of photon particles in the

free space, for the specified photon

particle. It may be easier to be understood from another

point of view, equation (22) is true for any photon particle

in any inertial reference frame at some constant speed in

the free space. Equation (20) is only true for an imagined

observer at the centre of the harmonic motion of the photon

particle. This is an imagined inertial reference frame at

constant linear light velocity moving together with the

rotation centre of the photon particle. Because of the

cycloid motion of photon particles, if we accept

= 2πA,

the measured parameters T, c,

and the calculated

parameter A in any inertial reference frame agree with each

other. Similar to Kkep, the inertial mass m of the photon

particle shall be the inertial mass in the imagined reference

frame at constant linear light velocity moving together with

the rotation centre of the specified photon particle.

According to equation (14), the photon particle has the

mass equivalent to the inertial mass of the photon particle

in the imaged reference frame at constant linear light

velocity from a mechanical perspective, which can be

calculated as m = h/(c

) = 2.21021906 × 10-42 (1/

). From

the imagined reference frame at constant linear light

velocity moving together with a stream of photon particles

to a laboratory reference frame on earth or in a satellite, the

inertial masses of photon particles cannot be transferred to

a zero inertial mass because the zero inertial mass cannot

be transferred back to a range of inertial masses. Hence, the

inertial masses of photon particles in a laboratory reference

frame on the Earth or inside a satellite cannot be equal to

zero. From a theoretical perspective, a finite inertial mass

for the photon particle will lead to Proca’s Lagrangian and

Yukawa potential (Proca, 1936; Tu et al., 2005; Caccavano

and Leung, 2013; Nyambuya, 2014), which is perfectly

compatible with the general principles of elementary

particle physics, which may lead to unified physics laws.

Maxwell’s Equations and Coulomb’s Law with great

successes in science and engineering are based on the ideal

assumptions of massless photon and the frictionless free

space, i.e. a vacuum. They are ideal approximations of

physical realities, which are brilliant enough for the scale

of size of the Earth-Moon system.

However, on a cosmic scale such as the Milky Way Galaxy

and above, modifications are needed (Zhang, 2021a;

Zhang, 2021b). Einstein’s theories of relativity sprang out

of Maxwell’s equations. Hence, the ideal approximations

of the massless photon and the frictionless free space were

inherited. Modern cosmology theories originated from

Einstein’s relativities have met difficulties and challenges

(Zhang, 2021a; Zhang, 2021b; Potter, 2009; Traunmüller,

2017). It is well-known that it is extremely difficult to

reconcile Einstein’s relativities with quantum mechanics.

The quantitative understanding of the inertial masses and

the entanglement of inertial masses, time periods, and

space displacements of particles in periodic motion with

rotational symmetry from a universally applicable

reference frame may help to resolve the difficulties and

challenges. The reference frame of the centre of the

periodic motion with rotational symmetry can be a

universally applicable reference frame, which may

simplify and unify the laws of physics.

Let us now focus back on the photon particles travelling in

the transparent media with refractive indices. Interestingly,

the inertial mass of the photon particle in a transparent

medium can have a sizable increase in comparison with the

one in the free space (vacuum) that will be elucidated in the

following section.

The property and propagation of photon particles in

transparent media with refractive indices

In the early 20th century, two possible candidates for the

momentum of a photon propagating through a transparent

medium emerged. One was proposed by Hermann

Minkowski (1908) that the photon’s momentum was given

by

. A year later, Max Abraham (1909) came

up with a different expression for the photon’s momentum:

. This controversy is known as the Abraham-

Minkowski dilemma (Partanen et al., 2017). These

rivalling momenta differ by , which is a sizeable factor

in most media: in water, n is approximately 1.33 and in

glass n is ~ 1.46 (Leonhardt, 2006). Over the years,

physicists have found supporting evidences, from both

first-principles arguments and experiments, for each

expression in different contexts, some of them have even

proposed ways that the apparent paradox might be resolved

(Partanen et al., 2017; Leonhardt, 2006; Cho, 2010;

Barnett, 2010; Sheppard and Kemp, 2016; Kemp, 2011;

Brevik, 1979; Brevik, 2017; Testa, 2013; Crenshaw, 2013).

Through computer simulations, Partanen et al (2017)

claimed that they proved the transfer of mass with the light

pulse representing the photon mass drag effect. The photon

mass drag effect gives an essential contribution to the total

momentum of the light pulse, which becomes equal to

Minkowski’s momentum. It is claimed that the Minkowski

Canadian Journal of Pure and Applied Sciences

6

momentum is the total momentum of the system, the

Abraham momentum is the portion of the momentum

carried by the light field, and the difference between the

two is carried by the atomic mass density wave. The

claimed experiments have only been taken in computer

simulations so far. Due to the coupling of the field and

matter, Abraham’s momentum is unable to be measured

directly, only the total momentum of the light pulse

(Minkowski’s momentum) can be directly measured.

The behaviour of photon particles in the transparent media

with refractive indices can be elucidated in a simple and

clear way based on the new insight into the physical origin

and nature of the Planck constant. Subtly, some materials

are transparent at least to some kinds of light. For example,

water and glass are transparent to visible light, air is

transparent to a wide range of frequencies of light. When a

stream of photon particles travels from the free space

(vacuum) to a transparent medium and travels through it

with negligible energy absorption, their energies and

frequenciescan be viewed as no change. However, the

elastic interactions with electrons inside the medium do

have the net effect of slowing the photons down (Nelson

and Kinder, 2017). The velocity of the photon in the

transparent medium is

, where is the velocity of

photons in the free space, is the medium’s refractive

index, a dimensionless number larger than 1. More

precisely, the index depends on the frequency , so the

relation can be better written as

(Nelson and Kinder,

2017). As now we focus theoretically on monochromatic

light with single frequency photon particles, instead of

will be used for simplicity.

Let’s recall the model of cycloid motion for photon

particles, as schematically shown in Figure 2. In the period

, the photon particle rolls the wavelength

in the

medium with the refractive index . The wavelength

equals to one circumference , where is the

amplitude of the harmonic motion in the transparent

medium with the refractive index , the radius of the

corresponding harmonic circular motion . Hence,

(23)

(24)

(25)

where is the elasticity of the medium, is the inertial

mass of the photon particle in the transparent medium. The

Planck constant is still a constant from the free space to the

transparent medium as the mechanical angular momentum

of the photon is conservative based on Noether’s theorem,

i.e.

(26)

(27)

From equations (23)-(26), it can be derived that

(28)

(29)

Incorporating equations (1), (24) and (28), it is obtained

that

(30)

Based on equation (30), the momentum of the photon

particle can be deduced as follows:

(31)

where

is the de Broglie relation. From

equations (30) and (31) for the photon particle within one

period of or one wavelength of

of its periodic motion,

it is derived as follows:

(32)

(33)

Equations (32) and (33) may be generalized for all particles

in periodic motion with the rotational symmetry, i.e.

(34)

(35)

where is the mechanical angular momentum of the

particle in periodic motion with rotational symmetry.

and are subsequently the energy, momentum,

wavelength, inertial mass, and the velocity of the particle

in periodic motion with rotational symmetry. Equation (35)

may be named generalized de Broglie relation. Equation

(34) is essentially another form of equation (10).

It is interesting to notice that the inertial mass of the single

frequency photon particle in the medium () increased in

comparison with the one in the free space (m):

(36)

where

is the wavelength of the photon in the free space

(vacuum). Partanen et al (2017) showed that with the light

pulse in a medium, the mass transfer equals to

with an assumption that the inertial mass of the photon

particle in the free space equals to zero. Our calculations

show that the inertial mass of the photon particle in the free

space is

and the inertial mass of the photon particle in

the transparent medium equals to

. The increase of

inertial mass from the free space to the transparent medium

is

. The results from equations (23) to (36)

imply that there are interdependences of the property of

Wenzhong David Zhang

7

photon particles, and the property of photon particles is

also entangled with their surrounding spaces in the

transparent media. Their surrounding spaces can be

different in the processes of emitting, propagation, and the

measurement of the photon particles.

It is fascinating to notice from equations (31) that the de

Broglie relation is kept the same form in the transparent

medium as in the free space. According to equations (24)

and (36), the inertial mass of the photon particle in the

transparent medium is , the photon particle

gains the inertial mass while it slows down. The square of

the velocity of the photon particle in the transparent

medium is

, hence, , the energy

is conservative. It can be seen also that from the free space

(vacuum) to the other transparent medium, the frequency

of the photon is kept the same and the Planck constant is

no change, thus the laws of conservation of both energy

and angular momentum are obeyed. The conservation of

energy arises from the negligible energy dissipation (under

the ideal approximation) in both the free space and the

transparent medium. The conservation of optical angular

momentum is associated with the rotational symmetry and

can be calculated using Noether’s theorem. Amazingly,

Kepler’s third law can be elicited from equations (23), (24),

(25), and (29) as follows:

(37)

where is Kepler’s constant for the specified photon

particle with the inertial mass in the transparent

medium, namely

(38)

Equations (37) and (38) disclose the correlation or

entanglement of and for the photon

particles in the transparent medium. The entanglement of

the photon particles with their surrounding spaces is

elicited. According to the derived equation (31), the photon

particle’s momentum calculated from the cycloid motion

model agrees with Minkowski's prediction

.

The Snell law of refraction may be re-analysed and

clarified as a piece of supportive experimental evidence.

The refraction of light in transparent media obeys Snell’s

law as shown schematically in Figure 3. The angles of

incidence (

) and refraction (

) at an interface are inversely

proportional to the respective indices of refraction and

, which are characteristics for each medium:

sin(θ) = sin(

)

(39)

The bending of light rays is regarded as the evidence for

the existence of a force acting at the interface between the

two media. The resultant force acting on the light is

managed in the direction normal to the interface (Buenker

and Muiño, 2004). Because of Newton's second law, the

component of the momentum of the photons, which is

parallel to the interface, shall be constant (Figure 3). Thus,

one can write that

sin(θ) = sin(

)

(40)

Fig. 3. The schematic diagram showing the refraction of

light at an interface between two transparent media.

Comparing equation (40) with equation (39), it can be

deduced that the total momentum of the photon particle is

always proportional to the refractive index of the given

medium. Assume that medium 1 represents the free space,

i.e. a vacuum. Hence, we have n1 = 1. Since the light is bent

toward the normal in transparent medium 2 (

, we are led to the conclusion that the momentum of the

photon particle is greater in transparent medium 2 in

comparison with the one in the free space (medium 1), i.e.

(41)

Hence, Snell’s law of refraction supports the derived

photon particle's momentum

that agrees with

Minkowski's prediction. The Snell law of refraction can be

viewed as a strong supportive experimental evidence of the

cycloid motion model of photon particles (Padyala, 2019).

It is not by chance that the mechanical model of the cycloid

motion of harmonic oscillator is able to illustrate vividly

the wave-particle durability of photons and it is powerful

in assisting the discovering of the interdependence and

entanglement of the properties of photons in both the free

space and other transparent media. The mechanical model

must be linked with the physical reality of photon waves of

the oscillating electric and magnetic fields from an

electromagnetic perspective. Further research is needed to

reveal the links in details.

CONCLUSIONS

The Planck constant is the pillar of modern physics.

However, the physical origin and nature of the Planck

constant have not been fully understood. In this article, the

Planck constant was derived through analysing the energy

and frequency relationship of the simple harmonic

Canadian Journal of Pure and Applied Sciences

8

oscillator model for the photon particle from a mechanical

perspective. It was derived that and

. All single photon particles

travelling in both the free space and other homogeneous

transparent media have equal mechanical angular

momentum ħ because they are in motion with identical

rotational symmetry. These discoveries are applied to the

propagation of photon particles in the free space by

viewing the propagation of the photon particle as the

cycloid motion of the harmonic oscillator. Amazingly, the

Einstein mass-energy equation and the de Broglie relation

are deduced subsequently. It is also elicited that the single

frequency photon particle has an inertial mass in the free

space such as

. It is further unveiled

that the quantization of mechanical angular momentum in

quantum mechanics and Kepler’s second law (the equal

area law in astronomy) arises from the same fundamental

law of the conservation of mechanical angular momentum.

Kepler’s third law of motion (the Period law) is derived for

the specified photon particle in both the free space and

other homogeneous transparent media. From a widely

applicable point of view, all particles in periodic motions

with the rotational symmetry, their inertial masses, their

time periods, and their space displacements are entangled

together by their conserved mechanical angular

momentums. These discoveries indicate the opening of the

door to unify the physical laws of the microcosm and the

macrocosm, the classical and the quantum. The reference

frame of the centre of the periodic motion with the

rotational symmetry is proposed as a universally applicable

reference frame to simply and unify the laws of physics.

Applying the discoveries to the propagation of photon

particles in the transparent medium with a refractive index,

it is deduced that the de Broglie relation is kept the same

form as in the free space. However, the equation for the

energy of the single frequency photon particle transformed

from to , while its energy and

mechanical angular momentum are both conservative. The

Abraham-Minkowski dilemma was briefly reviewed. The

derived momentum of the single frequency photon particle

based on the new method agrees with Minkowski's

prediction as most of experimental observations do. Snell’s

law of refraction was re-analysed and clarified as another

piece of supportive experimental evidence to Minkowski's

prediction. Snell’s law of refraction is an important

supportive evidence of the cycloid motion model of photon

particles.

The new insight into the physical nature and origin of the

generalized Planck constant, the generalized Planck-

Einstein relation, and the generalized de Broglie relation

can be equally applicable to atomic-scale systems, stellar

scale systems, and galactic scale systems. Further research

is needed to reveal the links between the cycloid motion of

photon particles and the oscillating electric and magnetic

fields of their corresponding electromagnetic waves.

ACKNOWLEDGEMENTS

The author gratefully acknowledges the encouragements

and supports from my family, friends, and colleagues-

researchers to these theoretical investigations.

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