Units and Constants

The default is to keep $\hbar$ explicit and to keep $c$ explicit unless a page declares a unit system in which one or both constants are set to one. Introductory, undergraduate, and core-formalism pages normally use dimension-complete formulas. Advanced bridge pages may use natural units when that choice clarifies structure.

A unit declaration applies only within its stated scope. Writing “natural units” is incomplete when it is unclear whether the author has set $\hbar=1$, $c=1$, $k_B=1$, or some combination.

Default Policy

Use these defaults unless a page states otherwise.

• Keep $\hbar$ explicit in commutators, phases, time evolution, and differential operators.
• Keep $c$ explicit in nonrelativistic pages and when comparing relativistic with nonrelativistic scales.
• Use SI-compatible dimensions for laboratory quantities unless an atomic, natural, or problem-specific system is declared.
• Write $e>0$ for the elementary positive charge; the electron has charge $q_e=-e$.
• Distinguish ordinary frequency $f$ from angular frequency $\omega=2\pi f$.
• Distinguish wave number $k$ from momentum $p=\hbar k$.
• Distinguish temperature $T$ from thermal energy $k_B T$.
• State the energy, length, and time units used in numerical tables or plots.

Constants Used Most Often

The following SI defining constants are exact:

$$\begin{aligned} c&=299\,792\,458\ {\rm m\,s^{-1}},\\ h&=6.626\,070\,15\times10^{-34}\ {\rm J\,s},\\ e&=1.602\,176\,634\times10^{-19}\ {\rm C},\\ k_B&=1.380\,649\times10^{-23}\ {\rm J\,K^{-1}}. \end{aligned}$$

The reduced Planck constant is

$$\hbar=\frac{h}{2\pi}.$$

Because the numerical factor $2\pi$ is exact, $\hbar$ is fixed exactly by the defined value of $h$, although decimal tables necessarily truncate it.

Particle masses, the fine-structure constant, the vacuum permittivity in the revised SI, and many other quantities are measured rather than defining constants. When numerical precision matters, state the constants release and uncertainty used by the calculation.

Dimensional Ledger

With mass $M$, length $L$, and time $T$ as base dimensions,

$$\begin{aligned} [E]&=ML^2T^{-2},\\ [p]&=MLT^{-1},\\ [\hbar]&=ML^2T^{-1},\\ [c]&=LT^{-1}. \end{aligned}$$

Several recurring combinations are therefore dimensionless:

$$\frac{Et}{\hbar}, \qquad \frac{px}{\hbar}, \qquad \frac{mc^2}{E}.$$

Quantum phases require dimensionless arguments. The operators

$$\begin{aligned} U(t) &=\exp\left(-\frac{iHt}{\hbar}\right),\\ T(a) &=\exp\left(-\frac{iaP}{\hbar}\right) \end{aligned}$$

make the action dimensions of $Ht$ and $aP$ explicit.

Dimensions are a necessary check, not a proof. They cannot determine a sign, a factor of $2\pi$, a dimensionless coupling, an ordering convention, or a boundary condition.

Planck’s Constant in Quantum Mechanics

Planck’s reduced constant connects generators to finite transformations and wave variables to mechanical variables:

$$\begin{aligned} [X,P]&=i\hbar I,\\ P&=-i\hbar\frac{d}{dx},\\ E&=\hbar\omega,\\ p&=\hbar k. \end{aligned}$$

Ordinary frequency obeys

$$E=hf, \qquad \omega=2\pi f, \qquad h=2\pi\hbar.$$

Mixing $f$ with $\omega$ is a common source of missing $2\pi$ factors. A phase written as $e^{-i\omega t}$ uses angular frequency; a phase written as $e^{-2\pi i f t}$ uses ordinary frequency.

The canonical position–momentum transform and its $\hbar$ factors are fixed in Fourier Transform Conventions.

The Speed of Light

Keep $c$ explicit in relativistic dispersion:

$$E^2=c^2\mathbf p^2+m^2c^4.$$

This formula makes three scales transparent:

$$\begin{aligned} E_{\rm rest}&=mc^2,\\ p_{\rm rel}&\sim mc,\\ \lambda_C&=\frac{\hbar}{mc}. \end{aligned}$$

Setting $c=1$ identifies mass, momentum, and energy dimensions. It does not mean that the physical speed of light has changed or that relativistic effects are absent. It means that time and length units, and mass and energy units, have been related using $c$.

Charge and Minimal Coupling

Use $e>0$ for the magnitude of the elementary charge. A particle’s signed charge is denoted $q$. Thus

$$q_{\rm electron}=-e, \qquad q_{\rm proton}=+e.$$

For a particle of charge $q$, the canonical minimal-coupling replacement is

$$\mathbf p\longrightarrow\mathbf p-q\mathbf A,$$

and a common nonrelativistic Hamiltonian convention is

$$H =\frac{1}{2m} \left(\mathbf p-q\mathbf A\right)^2 +q\Phi.$$

For an electron, this becomes

$$H_e =\frac{1}{2m_e} \left(\mathbf p+e\mathbf A\right)^2 -e\Phi.$$

The sign follows from $q=-e$; it should not be memorized as a separate electron-only rule. Pages using another convention for the covariant derivative, potentials, or charge must state it. The canonical physics is developed in Minimal Coupling.

SI-Compatible Units

Use SI-compatible units when:

• comparing with laboratory measurements;
• displaying dimensional estimates;
• calculating fields, voltages, currents, temperatures, or times;
• writing introductory examples where hidden constants would obscure the scale;
• interfacing with experimental or engineering data.

Energies may be quoted in joules or electronvolts. The electronvolt is defined by the exact elementary charge:

$$1\ {\rm eV} =1.602\,176\,634\times10^{-19}\ {\rm J}.$$

Using electronvolts does not imply natural units. Length may still be in meters, time in seconds, and $c$ and $\hbar$ explicit.

Natural Units

Relativistic and field-theory bridge pages may declare

$$\hbar=c=1.$$

Then energy, mass, momentum, inverse length, and inverse time can all be expressed through one energy unit:

$$\begin{aligned} [m]&=[E], & [p]&=[E],\\ [x]&=[E]^{-1}, & [t]&=[E]^{-1}. \end{aligned}$$

This notation suppresses conversion constants but does not erase dimensions. A statement such as “the mass is $1$” remains incomplete until the energy unit is specified.

If $k_B=1$ is also declared, temperature is measured in energy units and a thermal factor becomes

$$e^{-E/T}$$

rather than $e^{-E/(k_B T)}$.

Restoring constants

Use three checks together.

1. Determine the physical dimensions of the desired quantity.
2. Identify the available powers of $\hbar$ and $c$.
3. Compare with a known convention-complete limit or defining equation.

Examples:

$$\begin{aligned} E^2&=\mathbf p^2+m^2,\\ &\longrightarrow E^2=c^2\mathbf p^2+m^2c^4,\\ e^{-iEt} &\longrightarrow \exp\left(-\frac{iEt}{\hbar}\right),\\ \frac{1}{m} &\longrightarrow \frac{\hbar}{mc}. \end{aligned}$$

The final row applies when $1/m$ represents a length scale.

The last translation depends on knowing that the target is a length. The same $1/m$ in a time scale would restore as $\hbar/(mc^2)$.

Atomic Units

Atomic and quantum-chemistry calculations often declare Hartree atomic units:

$$\hbar=m_e=e=4\pi\epsilon_0=1.$$

The natural length and energy scales are then the Bohr radius and Hartree energy:

$$\begin{aligned} a_0 &=\frac{4\pi\epsilon_0\hbar^2}{m_e e^2},\\ E_h &=\frac{e^2}{4\pi\epsilon_0 a_0}. \end{aligned}$$

In this system, $a_0=1$ and $E_h=1$. The speed of light is not one; it is approximately the inverse fine-structure constant in atomic units. Declaring “atomic units” therefore does not license every natural-unit simplification.

Use Atomic Units and Scales for the atomic-physics conversion table and physical hierarchy.

Thermal Units

The canonical Boltzmann factor is

$$\exp\left(-\frac{E}{k_B T}\right).$$

When $k_B=1$, temperature and energy share units. When comparing with kelvin, restore $k_B$. Do not insert $k_B$ into a formula that already treats $T$ as an energy.

Similarly, inverse temperature may be written

$$\beta=\frac{1}{k_B T}$$

or $\beta=1/T$ after declaring $k_B=1$.

Numerical Reporting

A reproducible numerical result should state:

• the unit system;
• the numerical values and source of measured constants when precision matters;
• whether energies are angular frequencies, ordinary frequencies, joules, or electronvolts;
• the conversion used for plots or tables;
• the number of significant figures justified by inputs and approximation error.

Do not report more digits than the model, numerical method, and input constants support.

Common Mistakes

• Setting $\hbar=1$ or $c=1$ without declaring the scope.
• Mixing $\omega$ with $f$ and losing a factor of $2\pi$.
• Mixing momentum $p$ with wave number $k$ and losing a factor of $\hbar$.
• Treating the electron charge as $e$ rather than $-e$ after declaring $e>0$.
• Combining SI electromagnetic formulas with Gaussian-unit formulas.
• Restoring constants in an exponent but not in measures, fields, or couplings.
• Assuming every quantity becomes dimensionless in natural units.
• Treating atomic units as identical to $\hbar=c=1$.
• Converting electronvolts to joules while leaving a time expressed through an angular frequency without $\hbar$.
• Reporting precise constants without naming the constants release when the uncertainty matters.

Exercises

Exercise 1: Frequency conventions

A transition has ordinary frequency $f$. Write its energy and time-dependent phase using both $f$ and $\omega$.

Solution

The two frequency variables satisfy $\omega=2\pi f$. Therefore

$$\begin{aligned} E&=hf=\hbar\omega,\\ e^{-iEt/\hbar} &=e^{-i\omega t} =e^{-2\pi i f t}. \end{aligned}$$

All three phases agree. Writing $E=\hbar f$ would mix ordinary and angular frequency conventions.

Exercise 2: Restore the Schrödinger equation

In units with $\hbar=1$, the free Schrödinger equation is written

$$i\partial_t\psi =-\frac{1}{2m}\nabla^2\psi.$$

Restore $\hbar$.

Solution

The dimension-complete equation is

$$i\hbar\partial_t\psi =-\frac{\hbar^2}{2m}\nabla^2\psi.$$

The left side has dimensions of energy times $\psi$. On the right, $\hbar^2\nabla^2/m$ also has energy dimensions. Restoring $\hbar$ only on the left would fail this check.

Exercise 3: Electron in electromagnetic potentials

Starting from

$$H =\frac{1}{2m} \left(\mathbf p-q\mathbf A\right)^2 +q\Phi,$$

specialize to an electron using $e>0$.

Solution

For an electron $q=-e$, so

$$\begin{aligned} \mathbf p-q\mathbf A &=\mathbf p+e\mathbf A,\\ q\Phi&=-e\Phi. \end{aligned}$$

Hence

$$H_e =\frac{1}{2m_e} \left(\mathbf p+e\mathbf A\right)^2 -e\Phi.$$

The relative signs come from one signed-charge convention.

Exercise 4: Restore a Compton scale

In $\hbar=c=1$ units, a characteristic inverse mass is written $\ell=1/m$. Restore constants if $\ell$ is a length, and state the corresponding time scale.

Solution

The length is the reduced Compton wavelength,

$$\ell=\frac{\hbar}{mc}.$$

Dividing by $c$ gives the associated time scale,

$$\tau=\frac{\ell}{c} =\frac{\hbar}{mc^2}.$$

Dimensions distinguish the two restorations even though both appear as $1/m$ in natural units.

References

• Bureau International des Poids et Mesures, The International System of Units (SI Brochure), 9th ed., updated 2022.
• P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA Recommended Values of the Fundamental Physical Constants: 2022”, Journal of Physical and Chemical Reference Data 54, 033105 (2025).
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• I. N. Levine, Quantum Chemistry, 7th ed., Pearson, 2014.