Reference and Data

Use this chapter when you need a conversion, a Hamiltonian, a selection rule, or the meaning of a spectroscopic quantity. The reference pages give compact answers with their assumptions and link to fuller explanations. A table is most useful when you can tell which physical quantity it describes: an angular frequency is not an ordinary frequency, and a symmetry-allowed transition need not produce a visible line.

Find the quantity you need

Question	Reference page
How do I convert energy, frequency, wavelength, and temperature-equivalent energy?	Constants and conversions, atomic units
Which degrees of freedom and interactions should my model retain?	AMO model index
What terms enter a Hamiltonian, and when do they apply?	Atomic Hamiltonians, molecular Hamiltonians
What do level labels and transition rules mean?	Term symbols, selection rules
How do I relate a matrix element to a transition strength or decay rate?	Oscillator strengths, Einstein coefficients
Which line width or spectral coordinate am I using?	Line shapes, spectroscopy nomenclature
How are laser, cavity, and drive parameters defined?	Laser nomenclature
Where should I start reading about an experiment or a subject?	AMO experiment index, bibliography and reading guide

The broader Reference collection connects these topics to operators, mathematical identities, models, and conventions used throughout quantum mechanics.

Units: one energy, several useful scales

For a photon in vacuum,

$$E=h\nu=\hbar\omega=hc\tilde\nu, \qquad \omega=2\pi\nu,\qquad \tilde\nu=\frac{1}{\lambda_{\mathrm{vac}}}.$$

Here $\nu$ is ordinary frequency in hertz, $\omega$ is angular frequency, and $\tilde\nu$ is spectroscopic wavenumber. If the wavelength is in metres, the inverse wavelength is in $\mathrm{m^{-1}}$; divide its numerical value by $100$ to express the same wavenumber in $\mathrm{cm^{-1}}$. The vacuum wavevector magnitude is $k=2\pi\tilde\nu$.

The energy-equivalent temperature $T_E=E/k_{\mathrm B}$ is another way to state a scale. It is not automatically the thermodynamic temperature of an ensemble. Likewise, setting $\hbar=m_e=e=4\pi\epsilon_0=1$ in Hartree atomic units changes the numerical representation of quantities, not their physical dimensions. Atomic units and scales derives the associated length, energy, time, and field units.

For a spectral line in a medium, use its frequency and the appropriate phase refractive index to relate in-medium and vacuum wavelengths. An air wavelength cannot be substituted for $\lambda_{\mathrm{vac}}$ without that conversion. The BIPM SI Brochure defines the units, while a named CODATA adjustment supplies measured constants and their correlated uncertainties. The 2022 adjustment is described by Mohr et al. (2025).

Line widths: population, coherence, and frequency

For an isolated radiative two-level transition with a stable ground state, no pure dephasing, and no driving field, let

$$P_e(t)=P_e(0)e^{-\Gamma t},\qquad \tau=\Gamma^{-1}.$$

The optical coherence decays at $\Gamma/2$. The corresponding Lorentzian natural line has angular-frequency HWHM $\Gamma/2$, angular-frequency FWHM $\Gamma$, and ordinary-frequency FWHM

$$\Delta\nu_{\mathrm{FWHM}}=\frac{\Gamma}{2\pi} =\frac{1}{2\pi\tau}.$$

These different widths describe the same ideal line. Pure dephasing increases coherence decay without the same increase in population decay. Decay of both levels, strong driving, Doppler broadening, or instrument response also changes the simple correspondence. Line shapes and broadening develops these distinctions; Bernath and Demtröder provide textbook treatments.

Allowed transitions and observable strengths

A selection rule gives a condition under which a matrix element $\mathcal M_{fi}=\langle f|\hat O|i\rangle$ must vanish. Passing that condition does not determine the matrix element’s size. Radial cancellation, state mixing, initial populations, polarization, and the detector can all make an allowed transition weak or absent.

Before converting an oscillator strength, Einstein coefficient, or line strength, check whether it is summed over final magnetic substates and averaged over initial ones. Also match polarization and ordinary-frequency versus angular-frequency spectral density. Degeneracy factors often account for an apparent disagreement between two formulas.

Level labels require the same care. In a weak-field coupling regime, $F$ may be a useful hyperfine quantum number; in a stronger field it can become only an approximate label. Term symbols distinguishes exact symmetry labels from labels for a dominant component.

From transition lines to a measured spectrum

For an incoherent sum of lines in a linear detection regime, a useful model is

$$S_{\mathrm{true}}(x)=\sum_j A_j\phi_j(x-x_j),$$

where $x_j$ is a line position, $A_j$ its integrated weight, and each $\phi_j$ is normalized in the spectral coordinate $x$. Different transitions can have different widths. A general linear instrument response gives

$$S_{\mathrm{obs}}(x)=\int K(x,x')S_{\mathrm{true}}(x')\,dx'+B(x).$$

Only when $K(x,x')=R(x-x')$ does the integral become a convolution. Only when all physical profiles are the same can the true spectrum itself be written as one stick spectrum convolved with a common profile. Coherent interference, saturation, or a nonlinear detector may require a different forward model.

In transmission experiments the instrument averages transmitted intensity. Taking a logarithm after this average generally differs from convolving the absorbance itself. With transmittance $T=I/I_0$,

$$\mathrm{OD}=-\ln T,\qquad A_{10}=-\log_{10}T, \qquad \mathrm{OD}=(\ln10)A_{10}.$$

The symbol OD is not universal: some laboratory conventions use it for a base-ten quantity. Define it by an equation. The spectroscopy nomenclature page connects these definitions to cross sections and integrated intensities.

Finding and comparing numerical data

The NIST Atomic Spectra Database provides evaluated atomic levels, wavelengths, and transition probabilities. Read each entry’s uncertainty, accuracy flags, and source references; a value may come from measurement, calculation, or an evaluation combining several results. The NIST Chemistry WebBook collects molecular spectra and thermochemical data. The IUPAC Gold Book defines terminology.

For a number you will use in a calculation, keep the source version or access date, isotope and charge state, level labels, units, uncertainty, and any air/vacuum convention. Include the original reference attached to a database entry. A hyperfine-resolved wavelength in an external field and a calculated fine-structure centroid may both be correct while referring to different quantities.

A measured or fitted number has uncertainty even when many digits are printed. For a small standard uncertainty $u_x$, a smooth conversion $y=f(x)$ gives $u_y\simeq|f'(x)|u_x$. In particular, for a positive vacuum wavelength and $\nu=c/\lambda$,

$$u_\nu\simeq\frac{c}{\lambda^2}u_\lambda, \qquad \frac{u_\nu}{\nu}\simeq\frac{u_\lambda}{\lambda}.$$

The absolute uncertainty changes its units and magnitude. Multiple uncertain inputs require their covariance; large relative uncertainties may require nonlinear propagation. Precision metrology explains these methods.

Exercises

1. One energy, four numbers

A transition is reported as a cyclic frequency $\nu$. State the equations needed to express it as angular frequency, photon energy, and spectroscopic wavenumber.

Solution

Use

$$\omega=2\pi\nu, \qquad E=h\nu, \qquad \tilde\nu=\frac{\nu}{c}.$$

The last relation gives inverse metres; divide by $100$ to express the result in $\mathrm{cm^{-1}}$. The symbol $\tilde\nu$ is spectroscopic wavenumber, not the wavevector magnitude $k=2\pi\tilde\nu$.

2. Allowed but absent

An E1 transition passes the angular-momentum and parity rules but is absent from a measured spectrum. List four checks before calling the selection rule wrong.

Solution

Check the reduced and radial matrix elements or configuration mixing; initial population; polarization and geometry; branching and competing decay; line overlap or broadening; detector sensitivity; and whether the assigned quantum numbers remain valid in the applied fields. Selection rules are normally necessary conditions, not guaranteed line-strength predictions.

3. Two linewidths called gamma

One source defines $\Gamma$ through population decay, while another calls $\gamma$ the optical-coherence HWHM in angular frequency. Can the symbols be equated directly?

Solution

No. Derive the coherence equation in each convention. For an isolated radiative two-level system without pure dephasing, the coherence decays at $\Gamma/2$, but extra dephasing changes that relation. Also verify whether the reported width is HWHM or FWHM and whether it uses $\omega$ or $\nu$.

References

• P. F. Bernath, Spectra of Atoms and Molecules, 5th ed., Oxford University Press, 2025, doi:10.1093/oso/9780197754498.001.0001.
• Bureau International des Poids et Mesures, The International System of Units (SI), 9th ed., 2026 update, doi:10.59161/AUEZ1291.
• W. Demtröder, Laser Spectroscopy 1: Basic Principles, 5th ed., Springer, 2014.
• International Union of Pure and Applied Chemistry, Compendium of Chemical Terminology, the Gold Book, 5th ed., online version 5.0.0, 2025.
• A. Kramida, Yu. Ralchenko, J. Reader, and the NIST ASD Team, NIST Atomic Spectra Database, version 5.12, National Institute of Standards and Technology, 2024, doi:10.18434/T4W30F.
• P. J. Linstrom and W. G. Mallard, eds., NIST Chemistry WebBook, NIST Standard Reference Database 69, National Institute of Standards and Technology.
• 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.