Constants and Conversions

Atomic, molecular, and optical physics routinely describes one energy in joules, electronvolts, kelvin, hertz, angular frequency, inverse centimetres, and atomic units. These are not competing physical descriptions. They are different numerical coordinates on the same quantity, provided the conversion uses the correct constant and the author states whether a frequency is cyclic or angular.

This page is a versioned quick reference for those translations. Its central identity is

$$E =h\nu =\hbar\omega =hc\tilde{\nu} =k_{\mathrm B}T_E,$$

where $T_E$ is an energy-equivalent temperature, not automatically the thermodynamic temperature of a sample. Most factor-of-$2\pi$ and factor-of-100 errors in AMO calculations can be found by returning to this line and checking the units.

Canonical Scope

This page owns the AMO-facing conversion workflow:

• photon and transition energies in $\mathrm J$, $\mathrm{eV}$, $\mathrm{Hz}$, $\mathrm{rad\,s^{-1}}$, $\mathrm{cm^{-1}}$, and $\mathrm K$;
• the distinction among frequency, angular frequency, wavenumber, and wavevector;
• practical magnetic-moment and electric-dipole conversion scales;
• rules for reporting exact, derived-exact, and measured constants; and
• examples that expose the convention choices hidden by bare numbers.

It does not rederive Hartree atomic units. That material lives in Atomic Units and Scales and the Atomic Units reference entry. The global Constants table remains the canonical compact list of constants across all volumes, while Fundamental Constants explains how measured constants are inferred, adjusted, correlated, and updated.

Source Version and Status

The numerical values below use two different kinds of authority:

1. The International System of Units is taken from version 4.01 of the ninth edition of the BIPM SI Brochure, updated in June 2026.
2. Measured constants use the 2022 CODATA recommended values released by NIST in 2024 and published in full in 2025. NIST identifies this as the latest available CODATA set; the next regular adjustment is the 2026 adjustment.

The distinction matters. Since 20 May 2019, the numerical values of $h$, $c$, $e$, and $k_{\mathrm B}$ are fixed in the SI. The electron mass, fine-structure constant, Rydberg constant, Bohr magneton, and nuclear magneton remain measured or adjusted quantities. Their recommended values can change when CODATA incorporates new evidence.

Three status labels

This page uses the following labels.

Label	Meaning	Example
exact	the numerical value is fixed by the SI definition	$h$, $c$, $e$, $k_{\mathrm B}$
derived exact	the value follows exactly from defining constants, but its decimal expansion may not terminate	$\hbar=h/(2\pi)$, $1\ \mathrm{eV}$ in joules
measured	the value has a standard uncertainty and may change in a future adjustment	$m_e$, $\alpha$, $R_\infty$, $\mu_{\mathrm B}$

An ellipsis after a derived-exact decimal means that digits were truncated, not that the quantity has measurement uncertainty. Parentheses after a measured value have a different meaning: for example, $7.297\,352\,5643(11)\times10^{-3}$ gives a standard uncertainty of $0.000\,000\,0011\times10^{-3}$.

Quantity Dictionary

The same symbols are not used uniformly across every subfield. This page uses the following convention.

Quantity	Symbol	Definition	Preferred unit
energy	$E$	Hamiltonian eigenvalue difference or photon energy	$\mathrm J$ or $\mathrm{eV}$
cyclic frequency	$\nu$ or $f$	cycles per unit time	$\mathrm{Hz}$
angular frequency	$\omega$	$2\pi\nu$	$\mathrm{rad\,s^{-1}}$
vacuum wavelength	$\lambda_0$	$c/\nu$	$\mathrm m$ or $\mathrm{nm}$
vacuum wavenumber	$\tilde{\nu}$	$1/\lambda_0=\nu/c$	$\mathrm{m^{-1}}$ or $\mathrm{cm^{-1}}$
wavevector magnitude	$k$	$2\pi/\lambda_0$	$\mathrm{rad\,m^{-1}}$
energy-equivalent temperature	$T_E$	$E/k_{\mathrm B}$	$\mathrm K$

IUPAC recommends $\tilde{\nu}$ for vacuum wavenumber and notes that $\sigma$ may be used for wavenumber in a medium. In an AMO paper, the safest practice is to define the symbol once rather than rely on local custom.

Two pairs are especially easy to confuse:

$$\omega=2\pi\nu, \qquad k=2\pi\tilde{\nu}.$$

Thus $1\ \mathrm{Hz}$ and $1\ \mathrm{rad\,s^{-1}}$ do not represent the same numerical frequency, even though both can be reduced dimensionally to inverse seconds. Likewise, $\tilde{\nu}$ in $\mathrm{cm^{-1}}$ is not the same quantity as the wavevector $k$.

Defining Constants Used in AMO

The values in this table are exact in the present SI.

Constant	Symbol	SI value	AMO role
speed of light in vacuum	$c$	$299\,792\,458\ \mathrm{m\,s^{-1}}$	wavelength–frequency conversion
Planck constant	$h$	$6.626\,070\,15\times10^{-34}\ \mathrm{J\,s}$	energy per cycle frequency
reduced Planck constant	$\hbar$	$1.054\,571\,817\ldots\times10^{-34}\ \mathrm{J\,s}$	energy per angular frequency
elementary charge magnitude	$e$	$1.602\,176\,634\times10^{-19}\ \mathrm C$	electronvolt and electromagnetic coupling
Boltzmann constant	$k_{\mathrm B}$	$1.380\,649\times10^{-23}\ \mathrm{J\,K^{-1}}$	energy–temperature conversion

The symbol $e$ denotes a positive magnitude. The electron charge is $q_e=-e$. A Hamiltonian written with a particle charge $q$ must therefore use $q=-e$ for an electron; silently replacing $q$ by $e$ reverses electric and magnetic signs.

Because

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

$h$ belongs with cyclic frequency and $\hbar$ belongs with angular frequency:

$$E=h\nu=\hbar\omega.$$

Mixing $h$ with $\omega$ or $\hbar$ with $\nu$ inserts or removes a factor of $2\pi$.

Measured AMO Constants

The following rounded values come from the 2022 CODATA adjustment. The parenthetical digits are one-standard-deviation uncertainties in the final quoted digits.

Constant	Symbol	2022 CODATA value
electron mass	$m_e$	$9.109\,383\,7139(28)\times10^{-31}\ \mathrm{kg}$
fine-structure constant	$\alpha$	$7.297\,352\,5643(11)\times10^{-3}$
inverse fine-structure constant	$\alpha^{-1}$	$137.035\,999\,177(21)$
Rydberg constant	$R_\infty$	$10\,973\,731.568\,157(12)\ \mathrm{m^{-1}}$
Bohr magneton	$\mu_{\mathrm B}$	$9.274\,010\,0657(29)\times10^{-24}\ \mathrm{J\,T^{-1}}$
nuclear magneton	$\mu_{\mathrm N}$	$5.050\,783\,7393(16)\times10^{-27}\ \mathrm{J\,T^{-1}}$

These constants are related, not independent entries in a bag of numbers. For example,

$$\alpha =\frac{e^2}{4\pi\epsilon_0\hbar c}, \qquad \mu_{\mathrm B} =\frac{e\hbar}{2m_e}, \qquad \mu_{\mathrm N} =\frac{e\hbar}{2m_p}.$$

After the 2019 SI revision, $\epsilon_0$ and $\mu_0$ are measured quantities because they inherit the uncertainty of $\alpha$. In particular, $\mu_0=4\pi\times10^{-7}\ \mathrm{N\,A^{-2}}$ is no longer exact.

The Rydberg constant describes the infinite-nuclear-mass Coulomb scale:

$$R_\infty =\frac{\alpha^2m_ec}{2h}, \qquad hcR_\infty =13.605\,693\,122\,990(15)\ \mathrm{eV}.$$

It should not be inserted as though it were the exact line wavenumber of a finite-mass atom. Reduced-mass, recoil, relativistic, radiative, nuclear-size, and hyperfine corrections separate an observed transition from the simple $R_\infty$ formula.

Master Energy Conversion

For a vacuum photon or an energy splitting represented by an equivalent frequency,

$$\begin{aligned} E &= h\nu,\\ &= \hbar\omega,\\ &= \frac{hc}{\lambda_0},\\ &= hc\tilde{\nu},\\ &= k_{\mathrm B}T_E. \end{aligned}$$

Each equality identifies the conversion constant:

• divide by $h$ to convert energy to hertz;
• divide by $\hbar$ to convert energy to radians per second;
• divide by $hc$ to convert energy to inverse length;
• divide by $k_{\mathrm B}$ to convert energy to kelvin; and
• divide by $e$ to convert joules to electronvolts.

Dimensional analysis catches most mistakes:

$$\frac{\mathrm J}{\mathrm{J\,s}} =\mathrm{s^{-1}}, \qquad \frac{\mathrm J} {\mathrm{J\,K^{-1}}} =\mathrm K.$$

The first result is not enough to decide whether the number represents cycles per second or radians per second. That decision comes from whether the divisor was $h$ or $\hbar$.

Core Conversion Table

The displayed digits below are rounded. The underlying factors are derived from exact SI defining constants, so the ellipses represent nonterminating decimal expansions rather than experimental uncertainty.

Starting quantity	Joules	Hertz	Inverse centimetres	Kelvin
$1\ \mathrm{eV}$	$1.602\,176\,634\times10^{-19}$	$2.417\,989\,242\,085\times10^{14}$	$8\,065.543\,937\,349$	$11\,604.518\,121\,55$
$1\ \mathrm{cm^{-1}}$	$1.986\,445\,857\,149\times10^{-23}$	$2.997\,924\,58\times10^{10}$	$1$	$1.438\,776\,877\,504$
$1\ \mathrm{Hz}$	$6.626\,070\,15\times10^{-34}$	$1$	$3.335\,640\,951\,982\times10^{-11}$	$4.799\,243\,073\,366\times10^{-11}$
$1\ \mathrm K$	$1.380\,649\times10^{-23}$	$2.083\,661\,912\,333\times10^{10}$	$0.695\,034\,800\,486$	$1$

For electronvolts and kelvin,

$$1\ \mathrm K =8.617\,333\,262\,145\times10^{-5}\ \mathrm{eV},$$

and therefore

$$1\ \mathrm{meV} \approx11.6045\ \mathrm K.$$

For wavenumbers,

$$1\ \mathrm{cm^{-1}} =29.979\,2458\ \mathrm{GHz}.$$

This exact frequency factor follows from $c$ and the exact relation $1\ \mathrm{cm^{-1}}=100\ \mathrm{m^{-1}}$.

Electronvolts

The electronvolt is the energy acquired by a charge of magnitude $e$ through an electric potential difference of one volt:

$$1\ \mathrm{eV} =e\times1\ \mathrm V =1.602\,176\,634\times10^{-19}\ \mathrm J.$$

Its SI value is exact because $e$ is a defining constant. The electronvolt is not a volt, and an energy in electronvolts should not be divided by $e$ a second time.

Useful scale markers are:

Regime	Typical energy
radiofrequency and microwave hyperfine control	$\mathrm{neV}$ to $\mathrm{\mu eV}$
molecular rotation	$\mathrm{\mu eV}$ to $\mathrm{meV}$
molecular vibration	$\mathrm{meV}$ to tenths of an $\mathrm{eV}$
visible atomic and molecular transitions	roughly $1$–$4\ \mathrm{eV}$
core-electron and x-ray transitions	$\mathrm{keV}$ and above

These ranges orient a calculation; they are not classification boundaries.

Wavenumbers

Spectroscopists often report an energy as a vacuum wavenumber:

$$\tilde{\nu} =\frac{1}{\lambda_0} =\frac{\nu}{c} =\frac{E}{hc}.$$

The common unit $\mathrm{cm^{-1}}$ is an inverse length, not “per centimetre of path traveled” and not an angular wavevector. Because $1\ \mathrm{cm^{-1}}=100\ \mathrm{m^{-1}}$,

$$E =hc\left(100\tilde{\nu}_{[\mathrm{cm^{-1}}]}\right)$$

when $\tilde{\nu}_{[\mathrm{cm^{-1}}]}$ is the numerical value quoted in $\mathrm{cm^{-1}}$.

For a vacuum wavelength expressed in nanometres,

$$\tilde{\nu}\,[\mathrm{cm^{-1}}] =\frac{10^7} {\lambda_0\,[\mathrm{nm}]}.$$

A reported wavelength must say whether it is a vacuum wavelength or an air wavelength. Frequency is continuous across a stationary interface, while the wavelength in a medium is

$$\lambda_{\mathrm{med}} =\frac{\lambda_0}{n(\nu)}.$$

At low precision, “589 nm” may be enough context. At precision-spectroscopy accuracy, the refractive-index model, pressure, temperature, humidity, and composition of the medium can matter.

Frequency and Angular Frequency

NIST recommends explicit units because the numerical values of frequency and angular frequency differ by $2\pi$:

$$\omega=2\pi\nu.$$

Use hertz for cyclic frequency and radians per second for angular frequency:

$$\nu=\frac{\Delta E}{h}, \qquad \omega=\frac{\Delta E}{\hbar}.$$

This distinction propagates into time evolution. A stationary amplitude contains

$$e^{-iEt/\hbar} =e^{-i\omega t} =e^{-i2\pi\nu t}.$$

The same issue appears in detunings, Rabi frequencies, decay rates, and linewidths. A statement such as “$\Omega=10\ \mathrm{MHz}$” is incomplete unless the convention is established. Two unambiguous alternatives are

$$\frac{\Omega}{2\pi}=10\ \mathrm{MHz}$$

and

$$\Omega=10\ \mathrm{Mrad\,s^{-1}}.$$

They describe different angular frequencies by a factor of $2\pi$.

Rates and linewidths

A decay rate $\Gamma$ is often written in $\mathrm{s^{-1}}$, while a spectral linewidth may be quoted in $\mathrm{Hz}$ or $\mathrm{rad\,s^{-1}}$. Under the standard isolated, lifetime-limited two-level convention in which the excited population decays as $e^{-\Gamma t}$, the Lorentzian full width at half maximum is

$$\Delta\omega_{\mathrm{FWHM}}=\Gamma, \qquad \Delta\nu_{\mathrm{FWHM}}=\frac{\Gamma}{2\pi}.$$

This relation is convention-dependent once extra dephasing, power broadening, or a different definition of $\Gamma$ is introduced. A reliable report gives the decay law, the width definition, and the frequency unit rather than only a symbol.

Temperature as an Energy Unit

The mapping

$$T_E=\frac{E}{k_{\mathrm B}}$$

answers: “What temperature has thermal energy $k_{\mathrm B}T$ equal to this energy scale?” It does not assert that a single photon, level splitting, trap depth, or recoil energy is itself a thermodynamic temperature.

Examples include:

• a trap depth $U_0/k_{\mathrm B}=1\ \mathrm{mK}$;
• a recoil scale $E_r/k_{\mathrm B}=200\ \mathrm{nK}$;
• a hyperfine splitting expressed as an equivalent kelvin; and
• a vibrational quantum compared with $k_{\mathrm B}T$ to estimate thermal occupation.

For a two-level energy gap $\Delta E>0$ in thermal equilibrium,

$$\frac{p_e}{p_g} =e^{-\Delta E/(k_{\mathrm B}T)}.$$

Here the ratio depends on both the gap-equivalent temperature $T_E=\Delta E/k_{\mathrm B}$ and the actual ensemble temperature $T$. Conflating them erases the statistical-mechanical content.

Negative binding energies require care. Converting $E=-13.6\ \mathrm{eV}$ to a negative kelvin is algebraically possible, but a thermal comparison usually uses the positive ionization energy $|E|/k_{\mathrm B}$.

Magnetic Energy Scales

The Bohr and nuclear magnetons convert magnetic fields into electron-scale and nuclear-scale energies:

$$\mu_{\mathrm B} =\frac{e\hbar}{2m_e}, \qquad \mu_{\mathrm N} =\frac{e\hbar}{2m_p}.$$

Useful frequency forms are

$$\begin{aligned} \frac{\mu_{\mathrm B}}{h} &=13.996\,244\,9171(44)\ \mathrm{GHz\,T^{-1}},\\ \frac{\mu_{\mathrm N}}{h} &=7.622\,593\,2188(24)\ \mathrm{MHz\,T^{-1}}. \end{aligned}$$

Since $1\ \mathrm G=10^{-4}\ \mathrm T$,

$$\frac{\mu_{\mathrm B}}{h} =1.399\,624\,49171\ \mathrm{MHz\,G^{-1}}.$$

For a weak-field hyperfine Zeeman shift,

$$\Delta E =g_F\mu_{\mathrm B}B\,\Delta m_F,$$

so the shift in hertz is

$$\Delta\nu =g_F\frac{\mu_{\mathrm B}}{h}B\,\Delta m_F.$$

The sign depends on $g_F$, the magnetic-field direction, and the definition of the magnetic quantum number. The magneton supplies a scale, not the full magnetic moment. Nuclear moments also include species-dependent nuclear $g$ factors and may be quoted with signs.

Dipole Moments and Electric Fields

Molecular electric dipoles are commonly reported in debye. IUPAC defines the debye as a non-SI unit with

$$1\ \mathrm D \approx3.335\,64\times10^{-30}\ \mathrm{C\,m}.$$

The 2022 CODATA atomic unit of electric dipole moment is

$$ea_0 =8.478\,353\,6198(13)\times10^{-30}\ \mathrm{C\,m} \approx2.54175\ \mathrm D.$$

For a matrix element $d$ coupled to an electric field amplitude $\mathcal E$,

$$E_{\mathrm{int}}\sim d\mathcal E.$$

A useful laboratory scale is

$$\frac{(1\ \mathrm D)(1\ \mathrm{kV\,cm^{-1}})}{h} \approx503.412\ \mathrm{MHz}.$$

This is a scale estimate, not a promise of a linear Stark shift. A parity eigenstate has no first-order diagonal electric dipole unless degeneracy, state mixing, orientation, or another symmetry-breaking mechanism permits it. For an optical transition, the relevant $d$ is a transition matrix element and field conventions may distinguish peak, root-mean-square, and complex amplitudes.

Wavelength Shortcuts

For vacuum wavelength in nanometres,

$$\begin{aligned} \nu\,[\mathrm{THz}] &=\frac{299\,792.458} {\lambda_0\,[\mathrm{nm}]},\\ E\,[\mathrm{eV}] &=\frac{1\,239.841\,984\,332\ldots} {\lambda_0\,[\mathrm{nm}]},\\ \tilde{\nu}\,[\mathrm{cm^{-1}}] &=\frac{10^7} {\lambda_0\,[\mathrm{nm}]}. \end{aligned}$$

These are compact forms of $E=hc/\lambda_0$, not independent empirical rules. Keeping the wavelength unit in the denominator is safer than memorizing a bare value such as “1240.”

For spectroscopy, frequency is usually the cleaner invariant quantity. Wavelength becomes nonlinear under uncertainty propagation:

$$\lambda_0=\frac{c}{\nu}, \qquad \frac{\delta\lambda_0}{\lambda_0} \approx-\frac{\delta\nu}{\nu}$$

for small changes. The minus sign says that a higher frequency has a shorter wavelength.

Finite Nuclear Mass

The Rydberg constant $R_\infty$ is an infinite-nuclear-mass quantity. At the leading nonrelativistic level for a nucleus of mass $M$, replace $m_e$ by the reduced mass

$$\mu =\frac{m_eM}{m_e+M}.$$

The corresponding scale is

$$R_M =R_\infty\frac{\mu}{m_e} =\frac{R_\infty}{1+m_e/M}.$$

Expanding for $m_e/M\ll1$ gives

$$R_M \approx R_\infty \left(1-\frac{m_e}{M}\right).$$

This leading correction explains isotope-dependent shifts in simple hydrogenic reasoning, but precision predictions require recoil, nuclear-size, relativistic, radiative, and structure corrections. Do not use reduced-mass scaling as a substitute for a precision theory model.

Worked Examples

A 780 nm photon

Take a vacuum wavelength $\lambda_0=780\ \mathrm{nm}$. The cyclic frequency is

$$\nu =\frac{c}{\lambda_0} =3.84349\times10^{14}\ \mathrm{Hz} =384.349\ \mathrm{THz}.$$

The photon energy is

$$E =h\nu =2.54672\times10^{-19}\ \mathrm J =1.58954\ \mathrm{eV}.$$

The vacuum wavenumber and energy-equivalent temperature are

$$\tilde{\nu} =12\,820.5\ \mathrm{cm^{-1}}, \qquad \frac{E}{k_{\mathrm B}} =18\,445.9\ \mathrm K.$$

The last number does not mean a 780 nm laser beam has a temperature of $18\,446\ \mathrm K$. It states the single-photon energy in kelvin units.

The cesium defining transition

The SI fixes the unperturbed ground-state hyperfine transition frequency of cesium-133 to

$$\Delta\nu_{\mathrm{Cs}} =9\,192\,631\,770\ \mathrm{Hz}.$$

Its energy is

$$\Delta E =h\Delta\nu_{\mathrm{Cs}} =6.091\,102\,297\,113\,866\,55 \times10^{-24}\ \mathrm J.$$

Equivalent forms are

$$\begin{aligned} \Delta E &=38.0177\ \mathrm{\mu eV},\\ \frac{\Delta E}{k_{\mathrm B}} &=0.441177\ \mathrm K,\\ \frac{\Delta E}{hc} &=0.306633\ \mathrm{cm^{-1}}. \end{aligned}$$

The frequency is exact as part of the SI definition. The statement does not say that every realized cesium clock is exact; real clocks carry statistical and systematic uncertainty.

A Rabi-frequency convention

Suppose a resonant drive is reported as

$$\frac{\Omega}{2\pi}=10\ \mathrm{MHz}.$$

Then

$$\Omega =2\pi\times10^7\ \mathrm{rad\,s^{-1}},$$

and the resonant $\pi$-pulse duration is

$$t_\pi =\frac{\pi}{\Omega} =50\ \mathrm{ns}.$$

Reading “$10\ \mathrm{MHz}$” as $\Omega=10^7\ \mathrm{rad\,s^{-1}}$ would instead give $t_\pi\approx314\ \mathrm{ns}$. The physics did not change; the unstated convention did.

A Zeeman shift estimate

For $g_F=1/2$, $\Delta m_F=1$, and $B=100\ \mathrm{\mu T}$,

$$\Delta\nu =\frac12 \left(13.9962\ \mathrm{GHz\,T^{-1}}\right) \left(10^{-4}\ \mathrm T\right) \approx699.8\ \mathrm{kHz}.$$

This is a weak-field linear estimate. Near hyperfine avoided crossings or in the Paschen–Back regime, diagonalize the appropriate Hamiltonian rather than extrapolate the linear formula.

Conversion Workflow

For a reliable conversion:

1. Identify the physical quantity, not only its dimensions.
2. Write the source value with its unit.
3. Choose one bridge equation from the master identity.
4. Carry the unit algebra through the calculation.
5. State whether a frequency is cyclic or angular.
6. State vacuum or medium for wavelength and wavenumber.
7. Round only after the conversion.
8. Preserve the source uncertainty and correlations when they matter.

For example, converting a wavenumber $\tilde{\nu}=125\ \mathrm{cm^{-1}}$ to energy should begin with

$$E=hc\tilde{\nu},$$

and explicitly replace

$$125\ \mathrm{cm^{-1}} =12\,500\ \mathrm{m^{-1}}.$$

Writing the factor of 100 makes the unit conversion visible and prevents a two-order-of-magnitude error.

Reporting Precision and Uncertainty

Conversion does not create information. If a measured line is $384.230(5)\ \mathrm{THz}$, reporting a wavelength with fifteen digits is misleading even though a calculator can produce them.

For $y=y(x)$ and a small standard uncertainty $u(x)$,

$$u(y) \approx \left|\frac{dy}{dx}\right|u(x).$$

Thus for $\lambda=c/\nu$,

$$u(\lambda) \approx \frac{c}{\nu^2}u(\nu).$$

If several measured constants enter, use their covariance matrix:

$$u^2(y) = \sum_{i,j} \frac{\partial y}{\partial x_i} \frac{\partial y}{\partial x_j} \operatorname{cov}(x_i,x_j).$$

CODATA values are correlated because they come from a joint least-squares adjustment. Treating them as independent can overstate or understate a precision uncertainty. For routine AMO scale estimates, rounded values are appropriate. For precision metrology, retrieve the current recommended values, uncertainties, and correlation coefficients from the official database.

A minimum numerical report should identify:

• the constant set, such as “2022 CODATA”;
• the frequency convention;
• the wavelength medium;
• the quoted uncertainty type;
• the number of significant digits justified by the input; and
• any atomic-unit, Gaussian-unit, or SI convention used in the source model.

Common Mistakes

Treating hertz and radians per second as interchangeable

Their dimensions can both be reduced to inverse seconds, but their numerical values differ by $2\pi$. Keep $\mathrm{Hz}$ with $h$ and $\mathrm{rad\,s^{-1}}$ with $\hbar$.

Calling wavenumber a wavevector

$\tilde{\nu}=1/\lambda_0$, whereas $k=2\pi/\lambda_0$. A quoted $10\,000\ \mathrm{cm^{-1}}$ is normally a spectroscopic wavenumber.

Forgetting centimetres

The factor

$$1\ \mathrm{cm^{-1}}=100\ \mathrm{m^{-1}}$$

must appear before using SI $h$ and $c$.

Confusing the Rydberg constant and Rydberg energy

$R_\infty$ has units of inverse length. The Rydberg energy is $hcR_\infty$, and the Hartree energy is $2hcR_\infty$.

Treating an equivalent kelvin as a sample temperature

$E/k_{\mathrm B}$ is a scale comparison. Thermodynamic temperature requires an ensemble and an operational or statistical definition.

Using the wrong magneton

Electron magnetic scales are usually set by $\mu_{\mathrm B}$; nuclear magnetic scales are usually set by $\mu_{\mathrm N}$ multiplied by a nuclear $g$ factor. The two magnetons differ by roughly the proton-to-electron mass ratio.

Ignoring the sign of charge or magnetic moment

$e$ is a positive magnitude, but the electron charge and electron magnetic moment carry signs. State whether a formula uses signed moments or positive scales.

Treating every decimal as measured

$1\ \mathrm{eV}=1.602\,176\,634\times10^{-19}\ \mathrm J$ is exact in the current SI. A truncated decimal for $\hbar$ is also derived from exact $h$. By contrast, $m_e$ and $\alpha$ have measurement uncertainty.

Keeping obsolete exactness

After the 2019 SI revision, $\mu_0$ and $\epsilon_0$ are not exact. Old references that set $\mu_0=4\pi\times10^{-7}\ \mathrm{N\,A^{-2}}$ exactly use the pre-2019 SI.

Omitting vacuum or air

At precision accuracy, a wavelength without a medium convention is underspecified. Prefer frequency when comparing results across environments.

Exercises

Exercise 1: Derive the conversion spine

Starting from $E=h\nu$ and the definitions $\omega=2\pi\nu$, $\lambda_0=c/\nu$, and $\tilde{\nu}=1/\lambda_0$, derive

$$E=\hbar\omega=hc\tilde{\nu}.$$

Explain why $k=2\pi\tilde{\nu}$ does not introduce a contradiction.

Solution

Because $\hbar=h/(2\pi)$ and $\omega=2\pi\nu$,

$$\hbar\omega =\frac{h}{2\pi}(2\pi\nu) =h\nu =E.$$

The vacuum wavelength relation gives

$$\nu=\frac{c}{\lambda_0},$$

so

$$E =h\frac{c}{\lambda_0} =hc\tilde{\nu}.$$

The wavevector magnitude is $k=2\pi/\lambda_0=2\pi\tilde{\nu}$, so the same energy can also be written

$$E =\hbar ck.$$

The factors of $2\pi$ move together: $h$ pairs with $\tilde{\nu}$, while $\hbar$ pairs with $k$.

Exercise 2: Sodium D-line scale

Treat $589\ \mathrm{nm}$ as a vacuum wavelength. Convert it to terahertz, electronvolts, inverse centimetres, and energy-equivalent kelvin. Keep four significant figures.

Solution

The frequency is

$$\nu =\frac{299\,792.458} {589}\ \mathrm{THz} =508.9855\ \mathrm{THz}.$$

The photon energy is

$$E =\frac{1\,239.841984}{589}\ \mathrm{eV} =2.104995\ \mathrm{eV}.$$

The wavenumber is

$$\tilde{\nu} =\frac{10^7}{589}\ \mathrm{cm^{-1}} =16\,977.93\ \mathrm{cm^{-1}},$$

and

$$T_E =\frac{E}{k_{\mathrm B}} =24\,427.45\ \mathrm K.$$

To four significant figures:

$$509.0\ \mathrm{THz}, \quad 2.105\ \mathrm{eV}, \quad 1.698\times10^4\ \mathrm{cm^{-1}}, \quad 2.443\times10^4\ \mathrm K.$$

Exercise 3: One inverse centimetre

Derive the frequency corresponding to $1\ \mathrm{cm^{-1}}$ without using the conversion table.

Solution

Convert the inverse length first:

$$1\ \mathrm{cm^{-1}} =100\ \mathrm{m^{-1}}.$$

Since $\nu=c\tilde{\nu}$,

$$\nu = \left(299\,792\,458\ \mathrm{m\,s^{-1}}\right) \left(100\ \mathrm{m^{-1}}\right).$$

Therefore

$$\nu =2.997\,924\,58\times10^{10}\ \mathrm{Hz} =29.979\,2458\ \mathrm{GHz}.$$

The result is exact because $c$ and the centimetre-to-metre relation are exact.

Exercise 4: A π pulse

A laboratory note states “Rabi frequency $5\ \mathrm{MHz}$.” Compute the $\pi$-pulse time under each of these interpretations:

1. $\Omega/(2\pi)=5\ \mathrm{MHz}$;
2. $\Omega=5\times10^6\ \mathrm{rad\,s^{-1}}$.

What should the note have written?

Solution

For a resonant two-level system,

$$t_\pi=\frac{\pi}{\Omega}.$$

Under the first interpretation,

$$\Omega =2\pi(5\times10^6)\ \mathrm{rad\,s^{-1}},$$

so

$$t_\pi =\frac{1}{2(5\times10^6)} =100\ \mathrm{ns}.$$

Under the second interpretation,

$$t_\pi =\frac{\pi}{5\times10^6} =0.6283\ \mathrm{\mu s}.$$

The answers differ by $2\pi$. The note should have written either $\Omega/(2\pi)=5\ \mathrm{MHz}$ or $\Omega=5\ \mathrm{Mrad\,s^{-1}}$.

Exercise 5: Weak-field Zeeman scale

Estimate the cyclic-frequency shift for $g_F=1/2$, $\Delta m_F=1$, and $B=100\ \mathrm{\mu T}$. Use $\mu_{\mathrm B}/h=13.9962\ \mathrm{GHz\,T^{-1}}$.

Solution

The shift is

$$\Delta\nu =g_F\frac{\mu_{\mathrm B}}{h}B\Delta m_F.$$

Since $100\ \mathrm{\mu T}=10^{-4}\ \mathrm T$,

$$\begin{aligned} \Delta\nu &=\frac12 \left(13.9962\times10^9\ \mathrm{Hz\,T^{-1}}\right) \left(10^{-4}\ \mathrm T\right)\\ &=6.9981\times10^5\ \mathrm{Hz}. \end{aligned}$$

Thus

$$\Delta\nu\approx699.8\ \mathrm{kHz}.$$

Exercise 6: Dipole-field coupling

Find the frequency scale $d\mathcal E/h$ for $d=2.0\ \mathrm D$ and $\mathcal E=5.0\ \mathrm{kV\,cm^{-1}}$. State one reason this need not equal an observed first-order Stark shift.

Solution

Using

$$\frac{(1\ \mathrm D)(1\ \mathrm{kV\,cm^{-1}})}{h} \approx503.412\ \mathrm{MHz},$$

gives

$$\frac{d\mathcal E}{h} \approx (2.0)(5.0)(503.412\ \mathrm{MHz}) =5.034\ \mathrm{GHz}.$$

This is the bare interaction scale. A parity eigenstate can have zero first-order diagonal dipole moment; rotational averaging, matrix-element geometry, degeneracy, state mixing, and field polarization can all modify the observed shift.

Exercise 7: Exactness audit

Classify each statement as correct or incorrect.

1. $1\ \mathrm{eV}$ in joules has experimental uncertainty.
2. The displayed decimal for $\hbar$ may be truncated even though $\hbar$ is derived exactly from $h$.
3. $\mu_0=4\pi\times10^{-7}\ \mathrm{N\,A^{-2}}$ is exact in the current SI.
4. A CODATA parenthesis such as $(11)$ marks uncertainty in the final digits.

Solution

1. Incorrect. The elementary charge is fixed exactly, so $1\ \mathrm{eV}=1.602\,176\,634\times10^{-19}\ \mathrm J$ is exact.
2. Correct. $\hbar=h/(2\pi)$ is derived exact, but its decimal expansion is nonterminating.
3. Incorrect. Since the 2019 SI revision, $\mu_0$ inherits uncertainty through the measured fine-structure constant.
4. Correct. For a measured CODATA value, parenthetical digits give the one-standard-deviation uncertainty in the corresponding final digits.

Exercise 8: Repair an underspecified result

A report says:

The linewidth is 6.1 MHz, the transition wavelength is 780.24 nm, and the Rabi frequency is 10 MHz.

List at least four missing convention or uncertainty statements needed for a reproducible precision comparison.

Solution

A defensible report should state at least:

• whether the linewidth is a full width or half width;
• whether the linewidth is in cyclic frequency or angular frequency;
• the fitted line-shape model and whether power or inhomogeneous broadening is included;
• whether the wavelength is in vacuum or in a specified medium;
• the uncertainty and calibration basis of the wavelength or frequency;
• whether the Rabi number means $\Omega/(2\pi)=10\ \mathrm{MHz}$ or an angular frequency of $10\ \mathrm{Mrad\,s^{-1}}$;
• whether the field amplitude is peak, root-mean-square, or a complex amplitude; and
• which transition, polarization, magnetic sublevel, and detuning convention were used.

The original numbers may be useful estimates, but they are not yet a reproducible precision statement.

Cross-Links

• Reference and Data is the task-oriented gateway to AMO lookup pages and source-provenance rules.
• Spectroscopy Nomenclature distinguishes vacuum and medium wavelength, ordinary and angular frequency, spectroscopic wavenumber, and coordinate-density Jacobians.
• AMO Atomic Units gives the quantity-specific Hartree multipliers, Rydberg diagnostics, and dimensional restoration rules.
• Units and Constants gives the global reporting policy.
• Constants is the compact site-wide numerical table.
• Units compares SI, atomic, natural, and common quantum-mechanical units.
• Atomic Units and Scales derives Hartree units and their physical hierarchy.
• Fundamental Constants develops adjustment, correlations, and precision determinations.
• Line Shapes and Broadening defines spectral widths and broadening mechanisms.
• Rabi Oscillations fixes the drive-amplitude and angular-frequency conventions used in coherent control.
• Atomic Units is the canonical atomic-unit translator.
• Hbar Conventions explains how to restore $\hbar$ after a natural-unit calculation.

References

1. Bureau International des Poids et Mesures, The International System of Units (SI), 9th ed., version 4.01 (2026), doi:10.59161/AUEZ1291.
2. E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, “CODATA Recommended Values of the Fundamental Physical Constants: 2022,” Journal of Physical and Chemical Reference Data 54, 033105 (2025), doi:10.1063/5.0279860.
3. NIST Physical Measurement Laboratory, Fundamental Physical Constants, 2022 CODATA web version 9.0, updated May 2024.
4. NIST Physical Measurement Laboratory, Conversion Factors for Energy Equivalents, based on the 2022 CODATA adjustment.
5. National Institute of Standards and Technology, The International System of Units: NIST Special Publication 330, section 2, “SI units.”
6. International Union of Pure and Applied Chemistry, “Wavenumber,” Compendium of Chemical Terminology, 5th ed. (2025), doi:10.1351/goldbook.W06664.
7. International Union of Pure and Applied Chemistry, “Frequency,” Compendium of Chemical Terminology, 5th ed. (2025), doi:10.1351/goldbook.FT07383.
8. International Union of Pure and Applied Chemistry, “Debye,” Compendium of Chemical Terminology, 5th ed. (2025), doi:10.1351/goldbook.D01533.