In the Bohr model of the hydrogen atom, an electron can only exist in discrete energy levels given by the formula:

E_n = −13.6 / n² eV

where n is the principal quantum number (n = 1, 2, 3, …). When an electron transitions from a higher energy level (n₂) to a lower one (n₁), it emits a photon whose energy equals the difference between the two levels:

ΔE = E_n₂ − E_n₁ = 13.6 (1/n₁² − 1/n₂²) eV

The wavelength (and hence the position) of the corresponding spectral line is related to this energy difference through ΔE = hc/λ, so a larger ΔE produces a shorter wavelength line, and a smaller ΔE produces a longer wavelength line.

Why the levels crowd together at higher n: because energy varies as −1/n², the magnitude of E_n changes very rapidly for small n but changes only slightly for large n. For example, the gap between n = 1 and n = 2 is very large, while the gap between n = 10 and n = 11 is extremely small. Mathematically, as n increases, successive energy levels become closer together, converging toward a limiting value of 0 eV as n → ∞ (which corresponds to the electron being completely removed from the atom, i.e., ionisation).

Because the energy levels converge (get closer together) at higher n, the energy differences ΔE between adjacent transitions also shrink. Smaller ΔE means the emitted photons have progressively lower energy and longer wavelength, so successive spectral lines within a series (such as the Balmer or Lyman series) appear increasingly closer to one another as n increases, eventually merging into a continuum near the series limit.

The alternative descriptions do not match this physics: the levels are not equally spaced (if they were, spectral lines within a series would be evenly spaced too, which is not observed); they do not become farther apart with increasing n (the −1/n² dependence guarantees the opposite trend); and the energy levels do not disappear — they simply approach (but do not exceed) the zero-energy ionisation limit, remaining well-defined right up to that point.