Atomic Spectra & Energy Levels

Send light from a low-pressure hydrogen discharge through a grating and you do not get a rainbow. You get four sharp lines in the visible — 656.3, 486.1, 434.0 and 410.2 nm — nothing between them, and more lines crowding towards 364.7 nm. Sodium gives a different set, mercury another again. The pattern is a fingerprint: it names the element, and it does not shift when you change the temperature or the pressure of the source. Classical electromagnetism cannot produce this. An electron on a circular orbit is accelerating, so it must radiate at its orbital frequency, lose energy, spiral inwards and raise that frequency continuously. The prediction is a continuous smear sweeping upwards in frequency, and a hydrogen atom that survives about 10⁻¹¹ s. Both are spectacularly wrong, and the sharpness of the lines is the clue: something about the atom's internal energy comes in steps.

One rule does the translation. An atom changes its internal energy by emitting or absorbing exactly one photon, and that photon carries the whole difference: hf = Eᵢ − E_f. In electronvolts and nanometres hc = 1240 eV nm, so ΔE = (1240 eV nm)/λ. Hydrogen's red line at 656.3 nm is therefore a gap of 1.889 eV, and its blue-green line at 486.1 nm a gap of 2.551 eV. Now the arithmetic starts paying. Those two lines share a lower level, so the difference of their gaps, 2.551 − 1.889 = 0.662 eV, must itself be the gap between the two upper levels, and a line should exist near 1240/0.662 = 1.87 × 10³ nm. It does: the infrared Paschen-α line at 1875 nm. That is the Ritz combination principle, and it is strong evidence that the lines are differences drawn from a single list of numbers rather than the frequencies of independent oscillators.

Bohr kept classical mechanics for the orbit and added one rule. For an electron of mass mₑ on a circular orbit of radius r about a charge +Ze, Coulomb's law supplies the centripetal force: Ze²/(4πε₀r²) = mₑv²/r, so mₑv² = Ze²/(4πε₀r). The kinetic energy is then half the magnitude of the potential energy, K = Ze²/(8πε₀r) against U = −Ze²/(4πε₀r), and the total is E = −Ze²/(8πε₀r), bound and fixed by r alone. Classically r may take any value at all. Bohr's postulate is that the angular momentum is quantized, mₑvr = nħ with n a positive integer. Eliminating v between the two relations gives rₙ = 4πε₀n²ħ²/(mₑZe²) = n²a₀/Z with a₀ = 5.29 × 10⁻¹¹ m, and putting that back into E gives Eₙ = −(13.6 eV)Z²/n². Nothing was fitted to the spectrum: the 13.6 eV is assembled out of e, mₑ, ε₀ and ħ.

Draw the levels as horizontal lines with energy increasing up the page, and put E = 0 at the top, where the electron is free and at rest. Every bound level is then negative, and Eₙ = −13.6/n² eV places hydrogen's at −13.60, −3.40, −1.51, −0.850 and −0.544 eV. Two features follow at once. The spacing collapses: the 1 → 2 gap is 10.2 eV while the 5 → 6 gap is only 0.166 eV, so the levels pile up against the limit and every family of lines converges. And the depth of a level is the energy needed to ionize from it, 13.6 eV from the ground state but only 3.40 eV from n = 2. Lines are grouped by their lower level: n_f = 1 gives the ultraviolet Lyman series, n_f = 2 the Balmer series that reaches the visible, n_f = 3 the infrared Paschen series, with limits at 91.2, 364.7 and 820.6 nm.

Absorption is the rule run backwards. A photon whose energy matches a gap can be swallowed, lifting the atom up the ladder; a photon that does not match passes straight through. So a cool gas in front of a continuum source cuts dark lines at precisely the wavelengths it would emit if hot, which is what Fraunhofer's dark lines in sunlight are. What decides which lines appear is which levels are occupied. At room temperature kT = 0.0259 eV, hopelessly short of the 10.2 eV needed to reach n = 2, so cold hydrogen can absorb only from n = 1 and its absorption spectrum is pure ultraviolet. The Balmer absorption lines that dominate an A-type star need atoms already sitting in n = 2. With a Boltzmann factor e^(−ΔE/kT) and level degeneracies gₙ = 2n², the n = 2 population at 9.5 × 10³ K is 4e^(−12.46) ≈ 1.6 × 10⁻⁵ of the ground state: tiny, yet enough to carve deep lines through a thick atmosphere.

The Bohr ladder holds for one electron bound to a nucleus, and nowhere else. It predicts hydrogen's lines to better than a part in a thousand, and to a part in 10⁵ once the finite nuclear mass is folded in; it handles He⁺ and Li²⁺ through the Z² factor; it even explains the isotope shift between hydrogen and deuterium. Add a second electron and it fails outright, because the electrons repel one another as well as feeling the nucleus: there is no Bohr formula for neutral helium. It is silent on how bright a line is and how long an excited state lives, since it supplies no transition rates. Its choice of orbits stays unexplained until de Broglie rereads mₑvr = nħ as the standing-wave condition 2πrₙ = nλ. And two of its pictures are simply wrong: it gives the ground state angular momentum ħ where measurement gives zero, and it asserts a sharp radius and a sharp speed together, which the uncertainty principle forbids.