1. Introduction

The previous paper of author [9] preliminarily reached a conclusion that the physical meaning of Planck’s constant is integral of one cycle EM wave energy per frequency ν. It is believable that the conclusion is reasonable correct. However, considering some derivation and calculation seems not very perfect and may not be very rigorous, this paper would like to provide complement derivation and calculation together with previous results to build a complete perfect theory and conclusion.

To corroborate and re-clarify accurately why physical meaning of Planck’s constant h is integral of one cycle EM wave energy per frequency ν, some key points are worth noting.

1. The quanta energy E = hν of EM wave being produced or being absorbed by electrons is the measurable smallest energy unity and the only direct or indirect detection tool is electron (inside the sensor.)

2. The energy of EM wave produced by motion of electron or point source are not consist of quanta energies before detected; only the energy absorbed or detected by an electron is quanta energy hν. Critically, because when motion of an electron generates EM wave, multi-electrons located at various directions and distances can sense or receive the quanta wave energy hν separately and independently.

3. The decisive and critical measure to calculate energy of a periodical signal or wave is to set the domain (boundary) or (lower - upper) limit of the integral to one wavelength 𝜆 or one period T.

1.1. Analyzing and Discussing Some Crucial Points in Precedent Derivations of Physical Meaning of Planck’s Constant

1.1.1. Radiation of EM Wave from a Point Source (an Electron) before Detected is not Quanta Energy, and is not nhν

It is said that the EM wave emitted from a point source is spherical wave. Radius of a spherical wave varies in a sinusoidal function. The wave propagates outward from the point source at the wave propagation speed.

The energy in a spherical wave propagates equally in all directions; no one direction is preferred over any other. When the source emits power, at any distance r from the source must be distributed over a spherical surface of area 4πr². Hence, the wave intensity at a distance r from the point source is For example, the intensities at distances (for examples) r₁ and r₅ are and the ratio of intensities on these two spherical surfaces is

When one electron emits EM wave, various electrons (inside receiving antennas or sensor) at any direction and at any distance away from the point source can receive the signal of EM wave, although the intensities at different distances (different radius) and different angles are various.

According to axioms of spherical wave, the emitted power density, at any distance r away from the point source must be distributed over a spherical surface of area 4πr ², however, in practices, electrons (sensors) either being located at light year away or located meters away can receive the EM wave as well.

Every electron inside a sensor which is randomly located, will generate a closed tiny circle subtend a solid angular centered at the point source of the spherical wave. They have same patch of areas with different distance (r) from the point source, different areas subtend different solid angles, thus densities of energy or (densities of photons) distribution they experienced must be different, and as farer as lower. As shown in Fig.1.1.

Figure 1.1

The radiation from a transmitting antenna can be received by various receiving antennas located in different directions and distances. As shown in Fig.1.2.

Figure 1.2. Movements of electron inside a transmitting antenna produces EM wave, electrons insider various receiving antenna can detect quanta energy independently and separately

If EM wave consist of quanta energies or photons, how can their energy densities at cosmic space light years away keep as same high as meters away?

Max Planck comes up with a formula for the spectral energy density of blackbody radiation , the formula does not manifest physical meaning of Planck’s constant h, theoretically and practically; one cannot quantitatively control the radiation energy of EM wave of a point source blocked by a closed surface to scale of single quanta, therefore one cannot detect EM energy of a monochromatic frequency emitted from a point source blocked by a closed surface to the scale of exact one quanta.

These above mentioned phenomena verify that the EM wave energy before received by electrons is not ready quanta energy (photon). Before detected, EM wave energy is not existed in the form of quanta but globally distributed field of EM wave. Therefore, the existing of quanta energy is not depend upon the theoretical energy density of the EM wave but upon the energy absorbed by an electron or say an electron can absorb only quanta EM wave energy.

Quanta energy reflects the response characteristics of electron (inside the sensor or detector) to EM wave, not the intrinsic property of EM wave itself.

If one can measure or receive the EM wave by other sensor or particle (except electron) and verifies it being also quanta energy (hν), then one can say that the EM wave originally is consist of quanta energies, or the EM wave energy before absorbed or detected by electron is consist of quanta energies.

1.1.2. We can Conclude the Decisive Points as Following

So far as we can only make sure is that the EM wave energy packet absorbed or detected by electron is quanta energy.

Both EM waves and light waves are produced by oscillating or back and forth movements, i.e., acceleration – deceleration motions of electrons and technologies we can rely on to measure the EM wave or light waves and X-rays directly or indirectly is no alternative choice but electrons.

The EM radiation emits from a point source can be received by multi electrons located at various angles and distance.

Thus, photoelectric effect measured quanta energy reflects only the response characteristic of electron corresponding to light wave or EM wave, not the original intrinsic properties of EM wave.

Therefore, when one calculates the quanta energy, one should consider the quantity of energy around a single detecting electron, in other words, it is most reasonable to consider integral of EM energy packet over one wave cycle which is absorbed by an electron.

1.1.3. The Determination of Boundary or Limits of Upper and Lower of Integral of EM Wave Energy

It is worth noting that, as shown in Fig.1.3, the tracks of one EM wave cycle left no energy behind the current cycle, only the current instant cycle has energy, when calculating the integral of EM wave energy respect to time.

As shown in Fig.1.4, the cycles following the current cycle are either the other wave cycles (other quanta energy) or tracks of same cycle with empty energy, when calculating the integral of EM wave energy respect to distance.

Therefore when one calculate the integral of a periodic signal, it is meaningless to set the domain or limit of integration as non-integer periods, and it is even more ignorant and absurd to set upper limit of the integral to infinite or more than one wavelength 𝜆 or more than one period T.

Since the signal is repeated periodically, the boundary of the periodic signal must be bounded by the wavelength. The total energy of one wave cycle is equal to the integral of energy over one wave cycle. Therefore, it is properly right and reasonable to set lower-upper limit of integral to one wavelength 𝜆 or one period T.

Figure 1.3

Figure 1.4

1.1.4. Discussions of Some Precedent Approaches and Viewpoints in Deriving Physical Meaning of Planck’s Constant h

Some researchers have published their approaches (e.g. Brucholz [7], Chang [8], Kwiat [9] and Liu [10]) in their articles and essays, trying to derive physical meaning of Planck’s constant. It is helpful to have some evaluation, research, analysis and discussion on these approaches by the way.

1.1.4.1. Bruchholz [7]

Bruchholz asserts in an essay that a photon must have a geometric boundary, the integration of its energy density over a bounded volume must have

The field strengths respectively flow densities become

Bruchholz emphasized, this volume integral would be impossible without the boundary, because the linear solution, being alone, is not physically meaningful for the infinite extension.

Discussion

The assertion of photon having boundary sounds right.

However, it is inefficient to set up the domain or range of integration as a volume integral rather than a line integral.

And Bruchholz did not try to determine the “boundary” or limit of integral in the essay.

1.1.4.2. Chang [8]

In the article, Chang treats the photon as a wave packet which is made up of an oscillating electro-magnetic field as a wave packet of electro-magnetic radiation and directly calculating the total energy and momentum contained within the wave packet.

Then examine whether E is proportional to the oscillating frequency v. If yes, the proportional constant will be identified as the Planck’s constant ℎ.

The integration of the energy density described in equation below over the entire volume of the wave packet: U

Discussion

The train of thought of treating the photon as a wave packet and calculating the total energy within the wave packet deems correct.

However, the hν or quanta energy is connected or proportional to a monochromatic frequency ν which should be treated as constant, thus it may not be necessary to analyze frequency distribution.

It is inefficient to set up the domain or limit of integration as a volume integral rather than a line integral and setting the upper limit of integral to infinite rather than one wavelength may not be proper.

1.1.4.3. Kwiat [9]

In the approach of the paper, coupled string like real wave functions is assumed, therefore Planck constant h is interpreted as result of exchange interactions between two coupled strings.

Discussion

Schrödinger equation is a wave equation which is not derived from any classical mechanics model. Kwiat assumes seems equivalent to trying set up a classical mechanical wave model similar to spring wave mechanical model with tension or sound wave mechanical model with bulk modulus.

However, the assumption seems absence of concrete mechanical ground.

The “string theory” seems an idea out of expediency, temporary substitute of medium ether, may not be the result of comprehensive considerations.

The problems are:

Why only two strings interacts each other rather than multi strings are interacting roundly?

Any so called string, its two ends must be fixed then can the interaction between two strings be coupled, thus, how are the ends of string fixed?

By this way, it may be better to try to find the “bulk modulus” of the continuous uniform distributed medium ether (or modern ether) rather than consider “tension” of discrete “strings”.

(Modern) ether theory could comprehensively contain string theory. To seek physical meaning of Planck’s constant consider more on the quanta energy original emitting and receiving objects (electrons producing EM wave and response to EM wave when detecting EM wave) may be more efficient.

1.1.4.4. Liu [10]

The wave energy expressions in the approach are inspired by expression of Poynting vector.

According to the principle of wave energy flow density vector to find the flow rate density with respect to time t

Similar to the Poynting vector, the derivative of energy flow density respect to time represents flow rate density through a unit surface area perpendicular to the direction of wave propagation, is expressed as

Discussion

The result and conclusion should be not wrong, however, the coming up expressions and equations are needed to be checked whether are compatible with the conventional electromagnetic wave expression and theory. For this reason, this paper aims to improve the derivation and calculation more rigorously and accurately to conform to the Maxwell’s equation and formula of EM wave.

1.2. More Accurate and Rigorous Calculation of the Integral of EM Wave Energy over One Wavelength Based on Maxwell Equations

1.2.1. The E Wave and the B Wave should have a Phase Difference

Examining prevailing description of EM wave formulas, the amplitudes of E wave field and B wave field are reaching the crests and troughs without phase difference. Thus the instantaneous sum of the E energy and M energy is not a constant.

At the nodes, the sum of total energy is zero, where is the energy of the wave stored to restore the amplitudes of the oscillator of EM wave? This obviously violates the energy conservation law. As shown in Fig.2.1.a. Therefore, regarding the law of energy conservation, the EM wave should be depicted as shown as Fig.2.1.b, where point of the crest of E wave falls in the node of B wave, whilst the point of crest of B wave falls in the node of E wave. I.e. the E wave and the B wave should have a phase difference Hence, any instantaneous value of the sum of potential energies is a constant.

Figure 2.1

However, the mainstream of physics community may not agree with the above measure the moment. For the sake of a complementary consideration, we take the two assertions separately into account when we calculating the EM wave energy.

1.2.2. General Measure

In EM wave, both the electric potential energy density and magnetic potential energy density are proportional to and transforming each other.

The key point is that if EM wave is described by the general solution in the x axis direction,

The monochromatic (constant frequency) harmonic waves in the x direction, namely

The could be considered as equivalent velocity in y direction or velocity of amplitude, then the equivalent kinetic energy density of wave must be proportional to second power of i.e.,

The equivalent kinetic energy must be equal to potential energy, therefore

1.2.3. Deriving the Formula of Calculation of the EM Wave Energy over One Cycle considering the E and M without Phase Lag

According to Fig.2.1a depicted (the case 1), considering the wave propagates along the x-axis, then the energy density of an electro-magnetic field is known to be

(2.1)

Where and are the dielectric permittivity and magnetic permeability of the vacuum, E and B are electric field and magnetic induction, respectively. According to the Maxwell’s theory, E and B can be derived from the scalar potential Φ and the vector potential A:

(2.2)

Consider the x-axis direction propagation, (assume being the direction of an electron of the detecting sensor), since then

The energy packet density is then described in following equation U,

(2.3)

We can calculate the integral of the total energy over one wavelength by integrating (2.3).

If we take a snapshot of the wave at time then the energy density in a wavelength 𝜆 is the integral of the expression respect with x over one wavelength 𝜆:

(2.4)

Substituting into (2.4) obtains,

(2.5)

Substituting into (2.5) we have the expression of energy in one wavelength as

(2.6)

1.2.4. Deriving the Formula of Calculation of the EM Wave Energy over One Cycle Considering the E and M has 90° or Lag

According to Fig.2.1b depicted, (case2 the energy conservation law is obeyed), the E and M has phase lag, the energy density of an electro-magnetic field should be

(2.7)

The integral of energy over one cycle is

(2.8)

Substituting the equations and into (2.8), we obtain the expression of energy over one wavelength is

(2.9)

Comparing Eq.(2.9) with Eq.(2.6), the result is same.

1.2.5. Estimation or a Hypothesis of the Amplitude of EM Wave

Regarding both of transmitting and receiving of the EM wave are using by electrons; the maxim amplitude of EM wave could be equal to or proportional to the radius (R_e=D_e /2) of electron. (There were some preliminary discussions of this hypothesis in [6].)

An explanatory illustrational diagram is shown as Fig.2.2.

Figure 2.2. Amplitude A0 is equal to or proportional to radius of electron (De /2)

1.3. Rigorous Logical Deductions And New Discovery of a General Law about Wave Energy Re-Clarify and Corroborate the Planck’s Constant Being the Integral of One cycle EM Wave Energy Per Frequency ν

1.3.1. There is a Novel Uncovered General Law of One Cycle Wave Energy

It is worth noting that there is a previously un-clarified or covered general principle or law about wave energy that the integral of energy over one cycle of any wave is proportional to a constant h_x and its frequency f.

Normally, the linear distributed potential energy density U of a sinusoidal wave is proportional to second power of ω y (), where ω is angular frequency, y is the amplitude.

If potential energy density U (or power) of wave is proportional to , then at least mathematically, the integral of wave energy over one cycle or one wavelength 𝜆 is definitely proportional to a constant and frequency f or ν.

This novel uncovered general law is not exception for electromagnetic waves and not be noticed by researchers previously.

1.3.2. Reviewing the Integral of Energy over One Cycle of Sound Wave with Respect to Frequency

“To evaluate the kinetic energy of this volume of air, we need to know its speed. We have

” [1]

We know that the angular frequency is and velocity is Insert the equations and velocity into the energy expression of one wavelength obtains

Let ; the density of medium, A and v remain unchanged, the coefficient is a constant, therefore the total energy in one wavelength of sound wave is , i.e., proportional to a constant and frequency.

1.3.3. Reviewing the Integral of Energy over One Cycle of String Wave with Respect to Frequency

“To obtain the total potential energy in one wavelength, we integrate this expression over all the string segments in one wavelength:

The total energy in one wavelength of the wave is the sum of the potential and kinetic energies:

” [1]

We know that the angular frequency is and the velocity is . Insert the equations and into the one wavelength obtains

Express the as , the magnitudes of 𝜇, v and A remain unchanged, the coefficient is constant, then the total energy in one wavelength of string wave is or , i.e., proportional to a constant and frequency.

1.3.4. The Novel Uncovered General Law of One Cycle Wave Energy is Even Actually Applicable to Simple harmonic Motion e.g. the Block-Spring System

As we know (as shown in Fig. 3.1), the displacement of a block-spring system is

The maxim displacement x_max actually is the amplitude A, and .

The equivalent wavelength 𝜆 actually is equal to 2x_max (2A), and the average velocity is

As we well know prevailingly, the total energy of block-spring system is

(3.1)

Substituting , and ω = 2πf into (3.1) obtains

Since the coefficient is a constant, therefore, the total energy of a block-spring system is also proportional to a constant h_bs and frequency f.

Figure 3.1

1.3.5. Reviewing and Analyzing the Integral of Energy over One Cycle of EM Wave with Respect to Frequency ν

Review the equation (2.9) derived in Section (2.3), we have found the total EM wave energy over one wavelength:

(3.9)

Where c as being light speed is constant; A₀ as being max amplitudes is constant; coefficient 𝜺 as being permittivity is constant. Then the coefficient is constant.

Notice that the quanta energy (hν) is Planck’ constant h multiplying by the wave frequency , the integral of EM wave energy over one wavelength is also a constant multiply by the wave frequency ν. Therefore the Planck’s constant must be equal to the coefficient .

1.4. Analysis Manifests that the Planck’s Constant h is Decisively Proportional to “Bulk Modulus and Density” (ε₀ Combine with μ₀) of Vacuum or Modern ether

1.4.1. Relation of h_SoundWave and Bulk Modulus of Sound Wave Medium Air

The energy expression of one wavelength of sound wave is

(4.1)

Where v is speed , substituting into (4.1), obtains

(4.2)

(4.3)

1.4.2. Relation of h_String and Bulk Modulus (Tension T) of String Wave Medium

The total energy in one wavelength of string wave is

(4.4)

Where v is speed: , T is tension of string, substituting into (4.4), obtains

(4.5)

(4.6)

1.4.3. Relation of h and Bulk Modulus and Density of Vacuum or Ether

The total EM wave energy over one wavelength is:

(4.7)

Where c is speed substituting into (4.7), obtains

(4.8)

Therefore Planck’s constant h is

(4.9)

1.4.4. Further Derivation and Analysis

We know that for sound wave, for string wave and for EM wave.

The and is density and bulk modulus of medium of sound wave in below expression,

(4.10)

The and is linear density and tension of string wave in below expression,

(4.11)

The and are the dielectric permittivity and magnetic permeability of the vacuum in below expression,

(4.12)

Make a comparison:

We see that, the combine with is playing corresponding roles similar to the linear density 𝜇 of string wave and density 𝜌 of medium of sound wave, tension T of string wave and bulk modulus B of medium of sound wave.

One could say that the combine with are density and bulk modulus of vacuum or free space if one insists that ether does not exist.

(Factually ether is existed, because if light or EM wave “medium is wave itself”, then the changes of amplitude of wave is following up with and synchronized with the wave propagation speed c.

This synchronous actions of relationship result in amplitude of wave not being actually manifested changes at a particular stationary position x of a receiving antenna.

Then the amplitude will be only function of x, the expression will be .

Thus any static sensor or antenna fixed on a particular position x, there is no AC signal can be perceived. Therefore photon is not existed, and medium ether must be existed! There are more details in reference [5] and [6].)

So as a matter of fact, alternatively, we can frankly say that the combine with are density and bulk modulus of (modern) ether if we objectively recognize that modern ether actually and factually exists.

Whatever, Planck’s constant h is mainly proportional to and determined density and bulk modulus of “modern ether” or “vacuum” or “free space”. (There are some preliminary discussions of the deduction in [5].)

1.5. Try to Calculate the Value of Planck’s Constant According to above Derived Physical Meaning and Deduced Conclusive Formula

In section 4.4, based on the result of integral of EM wave energy over one wavelength, we further derived the equation (4.12) for calculation of Planck’s constant:

(4.12)

As we well know the values of these parameters in the equation:

𝜺₀ = 8.854 2 × 10^-12 C²/N/m² and 𝜇₀ = 4π × 10^-7 Tm/A.

And we have a hypothesis in Section 2.5 of this paper that the maxim amplitude of EM wave could be equal to or proportional to the radius of electron R_e.

We try to use a most realistic approach to estimate the radius of electron R_e, to take the ratio of proton/electron mass, thus, divides the proton’s radius by the cube root of this number. This ratio would set the electron’s radius at approximately 12 times smaller than a proton: at R_e ≌ 9.1x10^-17 m.

Thus, substituting all available values of 𝜺₀, 𝜇₀ and R_e into (4.12), obtains

Taking the new theoretical calculated value of Planck’s constant to compare with the value from they are of the same order of magnitude.

Considering the radius of electron is an approximate value, the derivations of this paper are scientifically correct.

2. Conclusions

Based on the previous rigorous analysis, deductions and calculations in this article, we can generally and clearly conclude that:

1. The integral of energy over one cycle of any waves is proportional to a constant h_x determined by the wave parameters and is proportional to the wave frequency f or ν, i.e.,

However, for different waves the value of the constant h_x is distinctively different and for EM wave, the constant h_x had been called Planck’s constant h beforehand.

2. The Planck’s constant h actually and factually is the integral of energy over one wavelength of EM wave, i.e., per frequency ν.

3. Accordingly, the Planck’s constant

4. The maxim amplitude could be equal to or proportional to radius of electron R_e.

Planck’s constant h is mainly proportional to and determined density and bulk modulus of “modern ether”.