Units

This page is a unit-system lookup table. It states what is being set to one and what must be restored when translating formulas.

SI Base Units Used Here

Quantity	SI Unit	Symbol
Length	meter	$\mathrm m$
Mass	kilogram	$\mathrm{kg}$
Time	second	$\mathrm s$
Electric current	ampere	$\mathrm A$
Temperature	kelvin	$\mathrm K$
Amount of substance	mole	$\mathrm{mol}$
Luminous intensity	candela	$\mathrm{cd}$

Common Derived Units

Quantity	Unit	Definition
Energy	joule	$\mathrm J=\mathrm{kg\,m^2\,s^{-2}}$
Frequency	hertz	$\mathrm{Hz}=\mathrm{s^{-1}}$
Charge	coulomb	$\mathrm C=\mathrm{A\,s}$
Electric potential	volt	$\mathrm V=\mathrm{J\,C^{-1}}$
Magnetic field	tesla	$\mathrm T=\mathrm{kg\,s^{-2}\,A^{-1}}$
Angular momentum	joule second	$\mathrm{J\,s}$
Wave number	inverse meter	$\mathrm{m^{-1}}$

Quantum-Mechanics Unit Systems

Unit System	Set to One	Common Use	Main Risk
SI	no constants suppressed	laboratory values, metrology, engineering	formulas can be cluttered by $4\pi\epsilon_0$, $c$, and $\hbar$
Natural units	$\hbar=c=1$	relativistic and field-theory bridge work	dimensions collapse into powers of energy
Atomic units	$\hbar=m_e=e=4\pi\epsilon_0=1$	atomic and molecular calculations	charge and electromagnetic conventions differ from SI-looking formulas
Thermal units	$k_B=1$	statistical mechanics and many-body theory	temperature and energy share dimensions

Energy Conversions

Conversion	Value
Electron volt	$1\ \mathrm{eV}=1.602176634\times10^{-19}\ \mathrm J$
Hartree	$1\ E_h\approx27.211386245988\ \mathrm{eV}$
Rydberg energy	$1\ \mathrm{Ry}=E_h/2$
Temperature-energy	$1\ \mathrm{K}=k_B\ \mathrm K\approx8.617333262\times10^{-5}\ \mathrm{eV}$

Restoring Constants

Dimensionless Convention	Restoration Rule
$\hbar=1$	restore $\hbar$ by dimensional analysis in actions, commutators, angular momenta, and frequencies
$c=1$	restore $c$ when converting mass, energy, momentum, and time
$k_B=1$	restore $k_B$ when converting temperatures to energies
$4\pi\epsilon_0=1$	restore SI Coulomb factors carefully; do not mix with Gaussian conventions

Common Mistakes

• Adding quantities with incompatible dimensions after setting constants to one.
• Treating angular frequency $\omega$ and ordinary frequency $\nu$ as identical without the factor $2\pi$.
• Forgetting that $E=\hbar\omega$ but $E=h\nu$.
• Restoring $c$ but not $\hbar$ in relativistic formulas.

Canonical Links

• Units and Constants
• Fourier Transform Conventions
• Common Convention Translations
• Constants
• AMO Constants and Conversions gives the derivation-backed $\mathrm{eV}$, $\mathrm{Hz}$, $\mathrm{cm^{-1}}$, $\mathrm K$, magneton, and debye conversions used in atomic and molecular work.

References

• BIPM SI Brochure
• NIST CODATA Fundamental Physical Constants
• P. W. Atkins and R. S. Friedman, Molecular Quantum Mechanics, 5th ed., Oxford University Press, 2010.