An advantage to stating equations in single-dimensional natural units (Planck units) is the ease with which we see the arrangements of physical quantities that conserve universal constants. For example, since
and
we see that
Notice that the product of Compton wavelength and rest mass is conserved; that is, its value remains constant as values of wavelength and mass change indicating an inverse relationship.
Rearranging terms highlights the correlation
The inverse relationship between Compton wavelength and rest mass is so precise that we can produce the Planck constant using CODATA values of electron and muon Compton wavelengths and rest masses to 11 digits.
We can see the correlated, non-dimensional natural quantities that allow us to calculate Compton wavelength and rest mass from the Planck constant; these are the ratios of wavelength and mass to the Planck scale.
The following illustrations are a visually engaging way to help students appreciate the natural structure we don’t see when calculating in arbitrary units. These relationships are revealed in natural units.
Teachers are encouraged to download these images to incorporate into their instruction on natural units and dimensional analysis.
Electric Constant
Conservation
| Image | Name | File Type | File Type | |
| Electrostatic force | PNG | PSD | |
| Electric potential energy | PNG | PSD | |
| Electric potential | PNG | PSD | |
| | | | |
Correlation
| Image | Name | File Type | File Type | |
| Electrostatic force | PNG | PSD | |
| Electric potential energy | PNG | PSD | |
| Electric potential | PNG | PSD | |
| | | | |
Magnetic constant
Conservation
| Image | Name | File Type | File Type | |
| Magnetic force | PNG | PSD | |
| | | | |
Correlation
| Image | Name | File Type | File Type | |
| Magnetic force | PNG | PSD | |
| | | | |
Gravitational constant
Conservation
| Image | Name | File Type | File Type | |
| Schwarzschild Radius | PNG | PSD | |
| Escape Velocity | PNG | PSD | |
| Gravitational acceleration | PNG | PSD | |
| Gravitational potential energy | PNG | PSD | |
| Gravitational force | PNG | PSD | |
| | | | |
Correlation
| Image | Name | File Type | File Type | |
| Schwarzschild Radius | PNG | PSD | |
| Escape Velocity | PNG | PSD | |
| Gravitational acceleration | PNG | PSD | |
| Gravitational potential energy | PNG | PSD | |
| Gravitational force | PNG | PSD | |
| | | | |
Planck constant
Conservation
| Image | Name | File Type | File Type | |
| deBroglie wavelength | PNG | PSD | |
| | | | |
Correlation
| Image | Name | File Type | File Type | |
| deBroglie wavelength | PNG | PSD | |
Speed of light
Conservation
| Image | Name | File Type | File Type | |
| Photon wavelength and period | PNG | PSD | |
| | | | |
Correlation
| Image | Name | File Type | File Type | |
| Photon wavelength and period | PNG | PSD | |
| | | | |
Planck constant and speed of light
Conservation
| Image | Name | File Type | File Type | |
| Photon wavelength and energy | PNG | PSD | |
| Compton wavelength & rest mass | PNG | PSD | |
Correlation
| Image | Name | File Type | File Type | |
| Photon wavelength and energy | PNG | PSD | |
| Compton wavelength & rest mass | PNG | PSD | |
