The Structure and Meaning of Planck’s Constant

Planck’s constant plays an important role in quantum mechanics but has little physical meaning apart from its contribution to the formulas. Evaluating each of the constant’s unit dimensions gives a more granular account of the physical phenomena described by the equations. These formulas show how classical and quantum mechanical equations are related, and they give consistent descriptions of phenomena like momentum and energy in classical and quantum systems.


Introduction
Planck's constant plays a principal role in quantum physics, appearing regularly in formulas describing the natural world on small scales. The initial motivation for the constant was to explain the spectral radiance of black bodies, but its importance has grown over more than a century as the constant has been linked with a wide variety of quantum phenomena.  These enigmatic ratios offer little or no ontological meaning for the concepts of length, mass, and time; yet the natural units are ingrained in formulas for calculating numerous physical phenomena.
An equivalent way of employing the universal constants is by expressing them in each unit dimension of length, mass, and time as shown in table 2. This approach has the considerable advantage of introducing more granular elements into the equations. That each of the universal constants may be neatly reduced to natural units of the correct magnitude and unit dimensions suggests that perhaps Planck should have presented the natural units as fundamental and the universal constants as their compound derivatives. It is well-established that the speed of light is the ratio of Planck length to Planck time; and it is the concepts of length and time that give physical meaning to velocity. It is similarly the case that the physical meaning of Planck's constant is found in each of the unit dimensions it contributes to the formulas. These contributions are discussed in section 3.
The question of whether to describe the physical universe in terms of Planck's constant and the gravitational constant, or in terms of the natural units they embody cannot be resolved mathematically.
The formulas are equivalent and either approach produces the same result.
The case for deriving a better description of the physical world from natural units is made in this paper; first by demonstrating a mathematically equivalent form of the equations in natural units that contains more granular information; and second by showing that these formulas produce a consistent physical description of the universe that isn't apparent in the standard formulas.   Instead, the natural units reveal a consistent pattern in the formulas by which physical attributes of a particle or system are proportional to their Planck scale limits. For example, the natural unit formula for momentum is written by replacing with l P m P c:

Natural Unit Formulas
The important structural insight is that Planck's constant supplies natural units of length and momentum. The formula input of wavelength in the denominator and the Planck length in the numerator produce a dimensionless ratio that quantifies a given particle's momentum proportional to the Planck momentum.
Taken to its limit, we see that a photon with a Planck length wavelength has Planck momentum.
The natural unit formula suggests that it is not the composite value and dimensions of Planck's constant that define momentum. It is the ratio between Planck length and the particle's wavelength that quantifies the correct momentum.  for calculating physical phenomena like momentum and energy at all scales. This is why we obtain meaningful results by setting certain constants like , G, and c to one. However, the natural unit formulas offer a deeper level of insight and control over this natural structure. In the natural unit formulas, the role of each unit dimension is clarified whereas the composite constants are more difficult to manipulate.
Most importantly, the natural unit formulas show that certain particle properties are correlated as shown in section 4.
This natural pattern is not unique to photon momentum but repeats in formulas containing the universal constants [5]. In addition, natural formulas consistently describe mechanical, gravitational, The natural unit formula for photon energy is similar to the momentum formula, but because the photon's velocity is the speed of light, no additional inputs are required to quantify the photon's velocity.
Photon energy is proportional to wavelength In the photon energy formula, Planck's constant and the speed of light produce Planck energy and Planck length. The photon's wavelength in the denominator makes the Planck length dimensionless and quantifies the photon's energy as a ratio of the Planck energy.

The structure of matter and electromagnetic radiation
Classical and quantum physics are considered separate domains in which different rules and formulas apply. While it is generally believed that classical physics is the large-scale manifestation of quantum phenomena, the relationships between them-and the transition from one to the other-are not welldefined.
Natural formulas give a common language for describing both of these domains, showing explicitly how classical and quantum mechanical formulas are related. Writing the equations in natural units gives a single, coherent description of physical concepts like momentum, which today is characterized separately by two different formulas.
Humpherys has given a detailed description of this structure [2] which is briefly summarized here as it is essential for explaining the physical meaning of Planck's constant.
We can show that classical physics and quantum mechanics describe a common underlying structure by comparing the classical and quantum energy formulas. The classical kinetic energy formula can be stated equivalently for a particle of matter such as an electron by expressing momentum and velocity as ratios of the Planck scale Note that the natural formula is also valid for a system of n particles where the body's rest mass gives the correct extensive quantity of mass and velocity is an intensive property that works at any scale.
We can compare this structure with the quantum mechanical energy formula (2) in which the momentum and velocity components of the formula are each specified Similarities are already evident in the structures of the classical and quantum formulas. But we can go further by resolving the apparent discord between the classical and quantum mechanical momentum formulas. Momentum is understood to pertain to both matter and radiation, but the different language of classical and quantum formulas conceals two important relationships. These correlations between the wavelength, momentum, and velocity of elementary particles are easily derived from the Compton and de Broglie wavelength formulas [2] and they are described in appendix B for reference. These two relationships are summarized in table 4 The symbol β represents a ratio between 0 and 1 where 1 is the Planck scale. The subscript λ designates the ratio of Planck length to wavelength and the subscript v designates the ratio of velocity to the speed of light.
The essential insight is that rest mass and velocity are bulk properties of matter that give the same ratio of the Planck scale as particle wavelength. Rest mass quantifies a particle's Compton wavelength and velocity quantifies the ratio between its Compton wavelength and de Broglie wavelength. The Compton wavelength can be characterized more applicably here as the non-relativistic limit of the de Broglie wavelength as the particle's velocity approaches the speed of light.
Replacing classical momentum with the equivalent quantum form gives the following structural comparison between classical and quantum mechanical energy formulas Table 5 demonstrates that the classical kinetic energy formula gives the same result as the natural unit formula, and the quantum formula is equal to the natural unit formula according to 2.  The importance of wavelength and velocity ratios must be emphasized since they are embedded in both the classical and quantum formulas. Furthermore, the mechanical properties of elementary particles of matter and radiation can be quantified entirely from these two ratios as shown in table 6. Table 6: Classical and quantum formulas quantify the mechanical properties of matter and radiation using ratios of particle wavelength and velocity.
Physical property Symbol Ratios Unit potential Natural formula

Two degrees of freedom
The ratios β λ and β v quantify two degrees of freedom in the mechanical properties of elementary particles. Both matter and radiation have the same degree of freedom in their momenta, characterized physically by a particle's wavelength and not by velocity as the classical formula implies. Only the quantum mechanical description of momentum can be applied consistently to both matter and radiation.
The ratio l P / correctly quantifies the momentum of both, even if m 0 v is the only practical method of measuring momentum in classical systems.
The second degree of freedom is a particle's velocity. Particles with rest mass have variability in their velocity while particles without rest mass do not. The correlation between a matter particle's wavelength For matter and radiation, kinetic energy is characterized by two components: 1. An energy potential generated by a particle's wavelength, proportional to l P .
The product of these two degrees of freedom accurately quantifies the kinetic energy of matter and radiation 1 as shown in equations 6 and 7, and table 5.
1 a 1/2 coefficient is also required for matter.

Inertial mass
The conflicting descriptions of momentum given by classical and quantum mechanical momentum formulas can be resolved by treating momentum as a function of wavelength instead of rest mass and velocity. Removing unit dimensions of velocity from momentum gives a quantity of inertial mass m that is inversely proportional to a particle's wavelength Inertial mass should not be confused with rest mass. The natural formulas describe the effect of rest mass as reducing a massive particle's energy potential from the Planck scale down to the Compton wavelength scale, where the Compton wavelength represents the minimum limit of the particle's wavelength at the velocity limit, c. The particle's inertial mass is less than its rest mass by the and can easily be calculated by dividing any quantity of momentum by c.
The relationship between inertial mass and rest mass is also described by the energy-momentum relation. Replacing momentum with inertial mass in the formula gives a vector sum of the two masses

The physical structure of kinetic energy
The structure of matter and radiation described by natural unit formulas is shown in figure 3. The figure compares the kinetic energies of a photon and an electron; the wavelength degree of freedom is shown in orange and the velocity degree of freedom in blue. Each of the particles has the same wavelength while the photon's velocity is c and the electron's velocity is v.
As the illustration shows, the inertial mass of the photon and electron are equal since the particles have the same wavelength. Multiplying inertial mass by velocity gives the kinetic energy of each particle; but note that each result must be multiplied by a superfluous quantity of c to obtain the familiar unit dimensions of energy, and matter requires a one-half coefficient. This state can only be applied to matter.
The second state characterizes momentum as a kind of mass density or energy potential that is inversely proportional to a particle's wavelength. This state can be applied equally to matter and radiation.
A worthy standard for encoding physical quantities as mathematical symbols and operations is the proper alignment between the symbols and operations in the formula and the physical attributes and dynamics they represent. This standard demands greater contextual awareness of a formula's physical meaning in a language besides mathematics. Without this accountability, formulas become their own ontologies no matter how they are expressed.
For example, the physical characterization of kinetic energy as a function of particle wavelength and velocity explains why the kinetic energy of matter is proportional to velocity squared. Since a change in velocity is accompanied by a proportional change in wavelength according to 33, only one instance of velocity characterizes kinetic energy. The second instance of v is the proportional relationship between velocity and wavelength. Figure 4 illustrates how the wavelength and velocity description of kinetic energy produces the correct quantitative result while also giving a meaningful and consistent physical description of kinetic energy.

Figure 4:
The two-part structure of energy explains why the kinetic energy of matter is proportional to velocity squared. Energy is spread equally across a matter particle's wavelength and velocity degrees of freedom, correlating these two physical properties.

E = m 0 c 2
The simple two-part structure of energy described by natural unit formulas and illustrated in figure 3 gives a physical explanation of mass-energy equivalence. To see how, consider three equivalent energy formulas expressed in classical, quantum mechanical, and natural terms The physical description of 14 is that a matter particle's latent rest mass has a kinetic energy potential equal to the radiant energy of a photon with the particle's Compton wavelength. Equation 13 shows how this rest mass component combines with the particle's inertial mass component, quantified by its wavelength, to formulate the total energy.
The structure illustrated in figures 3 and 4 also gives a meaningful physical interpretation of the ratio v 2 c 2 (15) which appears in the Lorentz transformation governing the behavior of relativistic interactions. Since one of the v terms is proportional to changes in wavelength, the combined ratio v 2 /c 2 quantifies the combined difference between a particle's wavelength and its minimum wavelength limit; and between its velocity and the speed of light. Each of the conserved pairs in 16, 17, and 18 may be considered a symmetry in which the product of the pair is invariant to changes in either term. Furthermore, any pair is equivalent to the Planck unit values of the pair. Although they appear as three different symmetries, they are all reflections of a single symmetry between particle wavelength and inertial mass. Figure 5 illustrates why this is the case. Table 8: Certain pairs of particle properties are conserved including wavelength-inertial mass, wavelengthmomentum, and time-energy. All three relationships can be explained in terms of a single length-mass symmetry. Velocities of the charged leptons were chosen arbitrarily and any velocity gives similar results. The photon's wavelength was also chosen arbitrarily and any wavelength gives similar results. § × 2 for matter and 1 for radiation † × 0.5 for matter and 1 for radiation

Quantum of action
The combination m is the physical representation of all three symmetries. A given wavelength has an inversely proportional quantity of inertial mass, and this is true for matter and radiation.
The pairs p and T E are conserved because each of them multiplies m by constants. Wavelengthmomentum multiplies the symmetrical pair m by the speed of light, and Time-energy multiplies that by the particle's oscillation period in the numerator and denominator.
The subsequent application of the Schrodinger equation to the atomic model incorporated the same idea.
Bohr had observed discrete electron orbitals and assigned integer numbers to them. It is easy to see from 19 how one could presume that Planck's constant is a discrete unit of angular momentum that is responsible for creating the orbital intervals. But while n quantifies the correct electron wavelengths in each orbital, the meaning of Planck's constant in the formula is more subtle. It may be that something else is responsible for quantizing the orbitals and Planck's constant is simply used to calculate electron properties.
Restating equation 19 as a ratio between radius and wavelength gives the de Broglie wavelength formula which is not generally limited to atomic orbitals. Bohr published his paper more than a decade before In 27 the variable components of the inequality are position and velocity. This much is clearly understood, but the physical structure generally is not. The two variables are related by 8 and 33 which show that changes in matter particle wavelength are proportional to velocity. The ratio m P /m 0 multiplied by the Planck length quantifies the fixed length of the particle's Compton wavelength-the minimum limit of its physical wavelength. The ratio c/v re-quantifies this length to the full de Broglie wavelength at the given velocity v, according to 33. Equation 27 can be stated explicitly in terms of wavelength by substituting the intensive ratio c/v with the same ratio / C (note that this removes unit dimensions of velocity from the formulas) The formulas support the conclusion that the Heisenberg uncertainty relation is a statement about the incompatibility of simultaneously characterizing matter as both a particle and a wave. However, the uncertainty is not over the full particle wavelength but only half of it. A possible explanation is that only the orthogonal directions of a particle's phase are positively distinguishable by any single measurement.
An alternative form of the uncertainty relation can be explained in the same terms as shown in section 5. The time-energy relation multiplies the

B Structural relationships of elementary particles
Several important structural relationships of matter and radiation are represented in table 7. These relationships are explained by Humpherys [2] and summarized below.

B.1 Correlation between Compton wavelength and rest mass
Rest mass is inversely proportional to Compton wavelength [10], [11], [12]. The product of rest mass and Compton wavelength is a constant, equal to the product of Planck length and Planck mass C m 0 = l P m P = 3.518 × 10 −43 kgm This relationship can also be expressed as an equivalence between dimensionless ratios of wavelength and mass.

B.2 Correlation between wavelength and velocity
Kinetic energy is spread equally across wavelength and velocity degrees of freedom in a matter particle.
This equipartition of energy gives a relationship between the particle's wavelength and velocity The following table demonstrates the relationship between wavelength and velocity. Velocities in the table are arbitrary and any velocity will give similar results. The following table demonstrates that momentum is a function of wavelength for matter and radiation, regardless of whether a particle has rest mass or whether its velocity varies. The electron velocity shown was chosen arbitrarily and the muon and tau velocities were calculated to give the same wavelength as the electron. The photon wavelength was selected to match the lepton wavelengths.