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UNDERSTANDING THE PLANCK CONSTANT AND THE BEHAVIOUR OF PHOTON PARTICLES FROM A MECHANICAL PERSPECTIVE

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I am grateful that after nearly six years independent research, this paper has been accepted for publishing in Canadian Journal of Pure and Applied Sciences. Vol. 15, No. 3, Oct 2021. 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.
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
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
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  )
(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 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
. 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
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
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
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
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
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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... As the damping is extremely weak, the resonant frequency is approximately equal to the natural frequency 0 . Therefore, the energy dissipated by the viscous force over a cycle (a period time of ≪ ) from the ℎ cycle of the photon [16,19,20] shall be, ...
... where and are subsequently the angular frequency and the amplitude of the oscillating of the photon particle at the ℎ cycle, = 2 , = 2 . And, ℎ = 2 2 [19,20], which is the Planck constant for photons. A period time of one cycle ( ) is relatively a short time, hence, ...
... where (0) and (0) are the mass and energy of the matter particle at its state of the lowest mass and energy at = 0. Remembering is approximately 2.29 × 10 −18 −1 and 0 < 1 ≤ 1 , therefore during a relatively short period of time, for example days or years, the mass and energy change are extremely tiny. However, the tiny change of mass and energy accompanies an inward force, 20 the force leads to contracting, vibrating and spinning of the matter particle around its equilibrium position because the interacting with photons with linear and circular polarizations. ...
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The momentum of light in a transparent material has remained a subject of an extensive scientific controversy for more than a century. The controversy has culminated in the difficulty to establish an unambiguous expression for the photon momentum and in particular in formulating a consistent theory to choose between the Abraham momentum p=hk/n and the Minkowski momentum p=nhk, where n is the refractive index and k is the wavenumber in vacuum. For both momenta, there exists experimental and theoretical support. In this work, we prove that, as a direct consequence of the fundamental conservation laws of nature and the special theory of relativity, the energy and momentum of light propagating in a medium are carried by quasiparticles, coupled states of field and matter, which have a finite rest mass and the Minkowski form of momentum. The total momentum of the quasiparticle, the Minkowski momentum, is the only directly measurable momentum of light in a medium since, due to the coupling, the momenta of the field and the matter cannot be separately measured. The finite rest mass of the quasiparticle directly leads to a photon mass drag effect optomechanically displacing the medium along the photon flow. After photon transmission, part of the initial photon energy is dissipated to heat when the mass distribution of the medium returns to equilibrium. This energy is however negligible, of the relative order of 10^-27 depending on the material. We also establish a transparent connection between the one-photon model and the electrodynamics of continuous media accounting for the mass transfer effect. The coupled field-matter description of light agrees with available experimental data and changes our understanding of light in a fundamental way.
Book
This textbook provides a unified approach to acoustics and vibration suitable for use in advanced undergraduate and first-year graduate courses on vibration and fluids. The book includes thorough treatment of vibration of harmonic oscillators, coupled oscillators, isotropic elasticity, and waves in solids including the use of resonance techniques for determination of elastic moduli. Drawing on 35 years of experience teaching introductory graduate acoustics at the Naval Postgraduate School and Penn State, the author presents a hydrodynamic approach to the acoustics of sound in fluids that provides a uniform methodology for analysis of lumped-element systems and wave propagation that can incorporate attenuation mechanisms and complex media. This view provides a consistent and reliable approach that can be extended with confidence to more complex fluids and future applications. Understanding Acoustics opens with a mathematical introduction that includes graphing and statistical uncertainty, followed by five chapters on vibration and elastic waves that provide important results and highlight modern applications while introducing analytical techniques that are revisited in the study of waves in fluids covered in Part II. A unified approach to waves in fluids (i.e., liquids and gases) is based on a mastery of the hydrodynamic equations. Part III demonstrates extensions of this view to nonlinear acoustics. Engaging and practical, this book is a must-read for graduate students in acoustics and vibration as well as active researchers interested in a novel approach to the material. • Provides graduate-level treatment of acoustics and vibration suitable for use in courses and for self-guided study • Highlights fundamental physical principles that can provide independent tests of the validity of numerical solutions and computer simulations • Demonstrates use of approximation techniques that greatly simplify the mathematics without substantial decrease in accuracy • Includes end-of-chapter problems and "Talk like an Acoustician" boxes to highlight key terms introduced in the text
Article
A discussion is given on the interpretation and physical importance of the Minkowski momentum in macroscopic electrodynamics (essential for the Abraham-Minkowski problem). We focus on the following two facets: (1) Adopting a simple dielectric model where the refractive index $n$ is constant, we demonstrate by means of a mapping procedure how the electromagnetic field in a medium can be mapped into a corresponding field in vacuum. This mapping was presented many years ago [I. Brevik and B. Lautrup, Mat. Fys. Medd. Dan. Vid. Selsk {\bf 38}(1), 1 (1970)], but is apparently not well known. A characteristic property of this procedure is that it shows how natural the Minkowski energy-momentum tensor fits into the canonical formalism. Especially the spacelike character of the electromagnetic total four-momentum for a radiation field (implying negative electromagnetic energy in some inertial frames), so strikingly demonstrated in the Cherenkov effect, is worth attention. (2) Our second objective is to give a critical analysis of some recent experiments on electromagnetic momentum. Care must here be taken in the interpretations: it is easy to be misled and conclude that an experiment is important for the energy-momentum problem, while what is demonstrated experimentally is merely the action of the Abraham-Minkowski force acting in surface layers or inhomogeneous regions. The Abraham-Minkowski force is common for the two energy-momentum tensors and carries no information about field momentum. As a final item, we propose an experiment that might show the existence of the Abraham force at high frequencies. This would eventually be a welcome optical analogue to the classic low-frequency 1975 Lahoz-Walker experiment.