Natural Units: Conversion Reference

Natural units simplify formulas by setting constants to one. They are useful only after the suppressed constants are named.

Common Choices

Convention	What It Does	Common Use
$\hbar=1$	action and angular momentum become dimensionless in units of $\hbar$	quantum dynamics and many-body theory
$c=1$	time and length share units; mass, momentum, and energy share units	relativity and QFT bridge pages
$k_B=1$	temperature is measured in energy units	statistical mechanics
$\hbar=c=1$	length and time are inverse energy scales	relativistic quantum theory

Restoration Examples

Natural-Unit Formula	SI-Restored Form
$E=\omega$	$E=\hbar\omega$
$p=k$	$p=\hbar k$
$E^2=p^2+m^2$	$E^2=p^2c^2+m^2c^4$
$\beta=1/T$	$\beta=1/(k_BT)$
$e^{-iEt}$	$e^{-iEt/\hbar}$

Dimensional Rule

When $\hbar=c=1$, a length has dimensions of inverse energy and a time has dimensions of inverse energy. This is powerful for estimates, but it can hide which experimental unit must be restored.

Common Mistakes

• Mixing a natural-unit formula with an SI Coulomb potential.
• Restoring $c$ but forgetting $\hbar$ in phases.
• Treating mass as energy without deciding whether it means $m$ or $mc^2$.
• Using $T$ for temperature in a $k_B=1$ formula and then inserting kelvin directly.

Canonical Links

• Hbar Conventions
• Units
• QFT Bridge Problem Map
• Klein-Gordon Equation

References

• M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Westview Press, 1995.
• S. Weinberg, The Quantum Theory of Fields, Volume I, Cambridge University Press, 1995.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.