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Assume dilute, nondegenerate ideal electrons, protons and neutral atoms in thermal and chemical equilibrium, with charge neutrality $n_p=n_e$. Set ${\mu }_H={\mu }_p+{\mu }_e$, using each ideal species' translational factor $n{\lambda }_{\mathrm{th}}^3/g$ and hydrogen ground-state binding energy $-R$. Neglect excited states and take the heavy proton/atom thermal-wavelength ratio as one. With electron spin and neutral-atom degeneracies treated in the convention of the source, the remaining factor isDifferent partition-degeneracy conventions must be propagated consistently. Screening, dense-plasma level dissolution and departures from local thermodynamic equilibrium lie outside this derivation.

To distinguish $2^b$ equal transverse bins across width $w$, $\boxed{d=w/2^b}$. Six bits gives one-nanometre pixels; ten gives 0.0625 nm, below the helium wavelength. In the stipulated fixed-energy freely propagating beam, the latter entails transverse diffraction momenta comparable to or exceeding the available total momentum and lies outside the paraxial aperture model. Six-bit localization is less demanding, although fine detector resolution and losses remain practical constraints. This is not a universal theorem that an atom can carry at most six bits in every encoding: accessible modes, energy, aperture, measurements and success probabilities define the channel.

A heated cathode emits electrons into low-pressure atomic vapor, and a variable accelerating voltage gives them kinetic energy. A grid and small retarding region precede the collecting anode. Below the first excitation threshold most collisions are elastic; above it, electrons can transfer a definite excitation energy to atoms and may then fail to overcome the collector's retarding field. Collector current develops repeated minima as accelerating voltage increases, separated by roughly the same excitation voltage, approximately 4.9 V for mercury. Contact potentials and energy spreads shift or smear the minima without removing their characteristic spacing. A smooth loss mechanism would not produce these repeated fixed-energy thresholds.^[ref]

An electric field shifts and can split the energy eigenvalues of an atom. This is the Stark effect; its hydrogen atom ground-state case is quadratic in a weak field.

The observable in Stark effect spectroscopy is a change in a spectral line, rather than a direct reading of an individual energy eigenvalue. If an upper energy eigenstate $u$ emits a photon into a lower energy eigenstate $l$, the Planck constant $h$ relates its frequency to the difference:Here $\mathcal{E}$ denotes the applied electric field component along a chosen axis. Both levels can move, and electric-dipole selection rules for hydrogen decide which pairs produce observable spectral lines. This makes Stark effect spectroscopy a probe of energy eigenstates and their electric dipole moments: the slope $-\partial E/\partial \mathcal{E}$ is the corresponding dipole component by the Hellmann–Feynman theorem. The comparison with Zeeman splitting is useful: there an applied magnetic field supplies the perturbation.