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A model in which each excitation changes volume by a fixed increment $v$. If the all-ground configuration has volume $V_0$, then $V=V_0+v⟨N_{\mathrm{ex}}⟩$. Energies and statistical weights must be specified under the relevant mechanical constraint; volume endpoints alone do not define arbitrary intermediate configurations.

Assume nondegenerate hydrogen and helium in chemical equilibrium with a black-body radiation field at temperature $T$. Neglect doubly ionized helium, excited-state populations and the small ion/atom mass difference in translational factors. DefineThe Saha ionization equation and ground-state statistical weights givewhere $I_H$ and $I_{He}$ are the first ionization energies. The factor four comes from the ground-state degeneracy of neutral helium: the electronic spin singlet state has weight one, whereas the singly ionized helium ground state and free electron each have weight two. For hydrogen, the corresponding degeneracy ratio is one when nuclear spin is counted consistently.^[ref]

Both equations share the same free-electron abundance. Combining them with the free-electron fraction during hydrogen--helium recombination closes the algebraic system. As cosmological recombination falls out of chemical equilibrium, a kinetic evolution replaces these instantaneous Saha relations.

The effective area converting incident photon flux into a stimulated emission rate for a specified transition, frequency, polarization and sublevel distribution. For statistically populated energy levels of statistical weights $g_1,g_2$, the Einstein coefficients imply $g_1{\sigma }_{12}=g_2{\sigma }_{21}$. The related absorption cross section therefore differs when the statistical weights differ.

A momentum distribution describes the statistical weight of particle momenta, with its normalization stated. A one-particle distribution $n(p)$ can be normalized to one as a probability density or to the particle number; these conventions differ by a factor.

For a pure state with momentum-space wavefunction $\tilde{\psi }(p)$, the probability density is $|\tilde{\psi }(p){|}^2$ under the chosen Fourier normalization. In many-body systems, the one-body density matrix determines the single-particle momentum occupation. A narrow momentum peak can indicate coherence, but finite size, interactions and the measurement protocol also affect the observed profile.

Write $k=e^2/(4\pi {ϵ}_0)$ and retain the nuclear Coulomb potential. In spherical coordinates, the total potential energy iswhen $\beta =-\alpha /2$; $P_2(u)=(3u^2-1)/2$ is a Legendre polynomial. Treat the added term using quadratic anisotropic perturbation of hydrogen and degenerate perturbation theory in the $n=2$ hydrogen energy eigenbasis. It preserves parity and $m$, so it cannot mix $2s$ with $2p$, or distinct $2p$ magnetic sublevels.

For a hydrogen atom, the hydrogen radial second moment givesThe angular expectation values of ${\cos }^2\theta$ are $1/3$ for an $s$ orbital, $3/5$ for $p_0$, and $1/5$ for $p_{\pm 1}$, obtained by integrating the squared spherical harmonics. Therefore the five lowest spatial quantum states, counting statistical weight, have first-order energy levelsEach spatial quantum state has two electron spin states when spin-orbit coupling and the magnetic field are omitted. Thus these are five spatial states but only three distinct first-order energies; five distinct energies would be an incorrect reading. The accidental $2s,2p_{\pm 1}$ equality can be lifted at higher order. Quantitatively the added shifts must also be small compared with neighboring principal-shell gaps; $\alpha \ll k/a_0^3$ is an asymptotic condition, not a guarantee for arbitrary large $n$.

For general $\beta$, the $1s$ expectation value isIt does not cancel every order. At $\beta =-2\alpha$, $V_{\mathrm{ext}}=-2\alpha r^2{P}_2(\cos \theta )$ couples $1s$ to $d$ orbitals. For example its angular matrix element with $Y_{20}$ is $1/\sqrt{5}$, and the $3d$ radial matrix element is nonzero. The second-order nondegenerate perturbation theory sum has negative denominators, so this is a concrete nonzero higher-order contribution.

There is also a global limitation: for either negative $\beta$ above, $V(0,0,z)=-k/|z|+\beta z^2\to -\infty$. A localized wavepacket translated far along $z$ has fixed kinetic energy but arbitrarily negative potential energy, proving there is no true lowest energy level. The results describe local perturbative resonance poles, or a potential completed to a confining form far from the atomic nucleus. No globally exact ground-state cancellation exists for the potential as printed.

Specify the radiation normalization first. Let $u_{\nu }$ be radiation energy density per ordinary frequency, with SI units $\mathrm{J}{\mathrm{m}}^{-3}\mathrm{H}{\mathrm{z}}^{-1}$. Then $A_{21}$ is the spontaneous emission rate per upper-state chemical atom, while $B_{12}{u}_{\nu }$ and $B_{21}{u}_{\nu }$ are the induced absorption and stimulated emission rates, evaluated at a sharp transition frequency. For a finite spectral line profile, replace $u_{\nu }$ by its profile-weighted average.

The Einstein coefficients obeyHere $g_i$ are the statistical weights of the two energy levels, and the field is unpolarized and isotropic. The units and physical dimensions areThe apparently time-independent dimension of $B$ follows because a per-hertz density includes an extra unit of time. If instead $B$ multiplies the mean intensity $J_{\nu }$, use $B^{(J)}=(4\pi /c)B$ and $A/B^{(J)}=2h{\nu }_0^3/c^2$. These distinct normalizations must not be mixed in the radiative transfer equation.

For a closed absorbing two-level system, write $N=N_1+N_2$, let $A=1/\tau$, and use ${\sigma }_a={\sigma }_{12}$ for the absorption cross section. This differs from the stimulated-emission cross section convention in part (a). With $r=g_2/g_1$, the rate equations areAt steady state, $N_2=W_{a}N/[A+(1+1/r)W_a]$ for $W_a={\sigma }_{a}I/(\hslash \omega )$. Hence the positive absorption coefficient isFor the given sodium fine-structure transition, the statistical weights are $g_1=2$, $g_2=4$, so $r=2$ and $1+1/r=3/2$. The small-signal transmission gives the optical depthTo account for the finite decrease of intensity through the sample, integrate $dI/dz=-\kappa I$ rather than treating the stronger beam as exactly uniform. If $T=I_{\mathrm{out}}/I_{\mathrm{in}}$, this givesTaking $270\mathrm{W}{\mathrm{m}}^{-2}$ as the incident intensity and $T=0.967$,Using the photon energy $hc/\lambda$ then givesIgnoring the small variation of the stronger intensity would instead give about $1.11\times 10^{-13}{\mathrm{m}}^2$; the integrated value uses the stated transmissions more consistently. The length is not needed to eliminate $N$, but gives the check $N\simeq 9.3\times 10^{14}{\mathrm{m}}^{-3}$.

The resonant absorption cross section is of optical-wavelength-squared size, much larger than a typical atomic geometric collision area. Pressure broadening cross sections involve phase-changing interatomic encounters and need not equal hard-sphere collision areas; their numerical value depends on collision partners and speed. No precise ratio follows from the supplied data. A useful distinction is that resonant absorption involves coherent response on the optical scale, so its cross section need not resemble an chemical atom's geometric area.

For homogeneous spectral broadening, all participating chemical atoms must share the same resonance and single-atom response width. In a dilute sample dominated by natural linewidth, the weak-beam requirement isUsing the sodium atomic mass, equality occurs near $T=0.017\mathrm{K}$, so $T\ll 17\mathrm{mK}$ is a sufficient thermal condition. Spatial Zeeman energy shifts, light shifts and transit effects should also be small, the laser should address the same selected component, and the interaction should last long enough for steady populations. Uniform impact pressure broadening can also be homogeneous, but would change the width and effective cross section; it is not included in the lifetime-only numerical model.

Real sodium also has hyperfine structure and optical pumping among magnetic sublevels. The inferred ${\sigma }_a$ is the effective cross section of the stated $g_J=2J+1$ two-level system model, assuming statistical sublevel populations and no loss into unaddressed states. A measurement on a resolved, polarized closed hyperfine structure component requires its actual transition dipole moment and selection rules; a universal polarization-independent sodium cross section cannot be inferred from this simplified model. The dependence of resonant cross sections and saturation intensities on sublevel preparation is tabulated explicitly for the real sodium D lines.^[ref]

The rotational state populations start at $N_0/N_0=1$. Their $2J+1$ statistical weight factor initially favors higher $J$, while the Boltzmann factor eventually suppresses them, producing one maximum followed by a tail tending to zero. The physically allowed values occur at integer $J$; a continuous curve only guides the eye.

Use the stipulated common Hückel Coulomb parameter $\alpha$, an equal Hückel resonance integral $\beta$ along each cube edge, and zero coupling elsewhere; take an orthonormal site basis. Each lithium vertex has three sodium neighbors. The totally symmetric combinations therefore have the blockFor a coordinate-like $T_2$ combination, the sum of the three neighboring sodium coordinate signs equals the lithium coordinate sign. Thus each of the three equivalent blocks isTheir eigenvalues give the cube nearest-neighbour Hückel model:For the bonding convention $\beta <0$ these are in increasing energy order, with statistical weights $1,3,3,1$. The eight valence electrons occupy $(a_1{)}^2(t_2{)}^6$, filling the four bonding spatial molecular orbitals. Their total one-electron energy is $8\alpha +12\beta$. This is a deliberately simplified model: unequal atomic energies or unequal overlaps require a generalized calculation.

Using mass $7\mathrm{u}$ for the dominant lithium isotope gives a transition wavenumber ${\tilde{\nu }}_0=14925\mathrm{c}{\mathrm{m}}^{-1}$. Inserting the thermal speed into the stated Doppler broadening expression givesThe supplied formula is the Gaussian half width at half maximum; the corresponding full width is about $0.256\mathrm{c}{\mathrm{m}}^{-1}$. Meanwhile exponential population decay with lifetime $\tau$ produces a natural linewidth expressed as a full widthHere $c$ must be in $\mathrm{cm}{\mathrm{s}}^{-1}$ in the second expression. Equivalently the fine-structure separation is about $9.0$ GHz, the Doppler half width about $3.83$ GHz, and the natural linewidth expressed as a full width about $5.89$ MHz. The natural linewidth is therefore negligible compared with thermal broadening; low density also makes collision broadening less important.

The doublet is resolvable in principle, but only marginally against the Doppler width. The separation exceeds one Doppler full width, so two components can survive, although their profiles overlap strongly. The $J=3/2$ and $J=1/2$ upper levels have different statistical weights, so the weaker member need not make an equally conspicuous maximum. A spectral resolving power $R\gtrsim {\tilde{\nu }}_0/0.3\simeq 5\times 10^4$ is the basic instrumental requirement; a width well below $0.1\mathrm{c}{\mathrm{m}}^{-1}$ and adequate signal would make the test more convincing. An arbitrarily fine spectrometer cannot remove the physical Doppler overlap. This conclusion uses the stipulated thermal, optically thin picture; additional photospheric velocity fields or optical-depth effects would require more information.

A local $d$-dimensional quantum field at finite temperature can often be represented by an anisotropic classical-looking action in $d+1$ dimensions, with a finite extra imaginary-time extent. The zero-temperature limit makes that extent infinite. A real nonnegative statistical weight is an additional requirement: fermionic signs and imaginary terms can obstruct interpretation as an ordinary classical probability distribution.

For a linear rigid rotor with moment of inertia $I$, the energy eigenvalue at angular momentum quantum number $J$ is ${\hslash }^{2}J(J+1)/(2I)$. Each level has statistical weight $2J+1$ before additional symmetry or nuclear-spin restrictions.

For unpolarized incident particles, the resonant cross section sums over the resonance's magnetic substates and averages over the incident spin states. This gives the displayed statistical weight under the conventional Breit-Wigner resonance cross section normalization. A spin-zero resonance formed from two spin-half particles has $g=1/4$, whereas a spin-one resonance has $g=3/4$. This counting is distinct from final-state color charge multiplicities.