The Structure of Natural Formulas

The following table demonstrates the structure of natural formulas in which composite constants are replaced by fundamental Planck units:

Planck’s constant

Gravitational constant

physical quantity

historical formula

natural formula

Compton wavelength

$\color{darkgray} \lambdabar = \displaystyle\frac{\hbar}{mc}}$

Natural formula for Compton wavelength

de Broglie wavelength

$\color{darkgray} \lambdabar = \displaystyle\frac{\hbar}{mv}}$

Natural formula for de Broglie wavelength

momentum

$\color{darkgray} p = \displaystyle\frac{\hbar}{\lambdabar} = mv}$

Natural formula for momentum

photon energy

$\color{darkgray} E = \displaystyle\frac{\hbar c}{\lambdabar}}$

Natural formula for photon energy

photon energy

$\color{darkgray} E = \displaystyle\frac{\hbar c}{\lambdabar}}$

electron energy

$\color{darkgray} E = \displaystyle\frac{1}{2}mv^2}$

gravitational acceleration

$\color{darkgray} g = -\displaystyle\frac{GM}{r^2}}$

Natural formula for gravitational acceleration

gravitational force

$\color{darkgray} F = \displaystyle\frac{GMm}{r^2}}$

Natural formula for gravitational force

gravitational energy

$\color{darkgray} U = -\displaystyle\frac{GMm}{r}}$

Natural formula for gravitational energy

orbital velocity

$\color{darkgray} v_o = \sqrt{ \displaystyle\frac{GM}{r}}}$

Natural formula for orbital velocity

escape velocity

$\color{darkgray} v_e = \sqrt{ \displaystyle\frac{2GM}{r}}}$

Natural formula for escape velocity