Planck’s constant is the famous constant of proportionality discovered by Max Planck at the end of the 19^{th} century. The constant is instrumental in calculating physical quantities on tiny scales including the properties of elementary particles.

Calculating a particle’s wavelength, momentum, and energy is simple. Multiply Planck’s constant by the right inputs and out comes the answer. For well over a century, the constant has improved our understanding of the quantum universe. But is there *more* to Planck’s constant than simply producing the right answers? Does the quantity 6.626 x 10^{-34} *kgm ^{2}/s* have deeper meaning?

The answer lies in the elementary form of the constant. Planck’s constant is made up of Planck *units* which are natural quantities of length, mass, and time.

Planck’s constant has four units—two units of length, one unit of mass, and one unit of time. These units are arranged into useful ratios for calculating the mechanical properties of elementary particles and systems.

We can show the reduced Planck constant using only the elementary Planck units, or we can simplify the constant using 𝑐 for the speed of light.

### elementary form

### simple form

The defining characteristic of Planck’s constant is the ratio 𝑙_{𝑃}𝑚_{𝑃}/𝑡_{𝑃}—the Planck momentum. Multiplying the constant by 𝑐 produces the Planck energy. The remaining quantity 𝑙_{𝑃} in the numerator pairs up with a particle’s wavelength in the denominator to create a *dimensionless* operator.

In formulas with Planck’s constant, it is really the *Planck units* doing the heavy lifting. This is evident in the granular structure of these formulas.

## Momentum

The traditional formula for momentum uses Planck’s constant in the numerator and a particle’s wavelength in the denominator. Replacing Planck’s constant with natural units of length, mass, and time produces a formula

The natural form of the equation has two parts:

- The maximum potential of the unit dimensions we are solving for—in this case, momentum. The product of Planck mass and the speed of light creates the maximum momentum potential.
- A dimensionless proportionality operator in the ratio of Planck length to the particle’s wavelength. This dimensionless ratio reduces the maximum momentum potential in the right proportion.

The following illustration emphasizes the two distinct parts of each natural formula.

## Photon Energy

The formula for photon energy includes another quantity of 𝑐 in the numerator, creating the Planck energy potential on the right and the same ratio of Planck length to particle wavelength on the left.

We can think of the photon’s wavelength as *diluting* the Planck energy, with longer wavelengths generating less energy and shorter wavelengths producing more. A photon reaches the maximum energy limit when its wavelength is equal to the Planck length.

## Wavelength

Planck’s constant can also calculate a particle’s wavelength. For massive particles, the formula uses inputs of rest mass 𝑚_{0} and velocity 𝑣 in the denominator to find the particle’s de Broglie wavelength. Each input pairs up with its maximum potential to create a dimensionless operator.

Multiplying the Planck length by the inverse operators produces the particle’s wavelength. The formula also works for particles moving at the speed of light when velocity is 𝑐.

The second wavelength formula gives a particle’s Compton wavelength.

Replacing Planck’s constant with elementary Planck units reveals hidden information about the natural world that is explained by the New Foundation Model of physics. This model gives new meaning to abstract mathematical ideas and explains traditional formulas in terms of the physical unit dimensions length, mass, and time.

What do you think about the stochastic significance of the Planck’s constant?

See : S.Dumitru : arXiv:1205.3892v5 [quant-ph] 7 Sep 2020

Spiridon Dumitru

s.dumitru42@yahoo.com