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