Hold a flame to table salt and it glows orange; heat copper and it burns green; a neon sign shines red-orange. Each element always makes its own exact set of colours — a fingerprint of light. Hydrogen, the simplest atom of all, glows a soft pink-red. The puzzle for a century was: why does an atom emit only a few precise colours instead of a smooth rainbow?
Picture the electron as standing on a staircase, not a ramp. It can rest on step 1, step 2, step 3 — labelled by the principal quantum number $n$ — but never in between. Each step has a fixed energy. For hydrogen the rule is beautifully simple:
When the electron drops from a high step to a low one, the leftover energy leaves as a single particle of light — a photon — of one exact colour: $\Delta E = hf = hc/\lambda$. Drop from $n=3$ to $n=2$ and the energy released is $E_3-E_2 = (-1.51)-(-3.40) = 1.89$ eV, which comes out as red light at $656$ nm — that pink-red hydrogen glow. A different jump, a different colour. The few allowed jumps are exactly why an atom emits only a few sharp lines.
Why only certain steps? In 1913 Bohr proposed that an electron's angular momentum can take only whole-number multiples of $\hbar$: $m_e v r = n\hbar$. That single quantum condition forces the orbits — and so the energies — to be discrete. Combining the jumps gives the formula Rydberg had already guessed from data:
The steps are not evenly spaced: they bunch together as $n$ grows and pile up at $E=0$, where the electron breaks free (ionization). The nuclear charge $Z$ enters squared, so a one-electron ion like He⁺ ($Z=2$) has every energy multiplied by four and every wavelength shrunk by four. In the sim, the n slider picks the upper step, n' the landing step, and Z the nucleus.
Set the lower level $n'=2$ (the Balmer series) and step $n$ up from 3 to 6: watch the emission lines march from red toward violet, crowding together as they approach the $364.6$ nm series limit. Switch $n'=1$ (Lyman) and notice every line jumps into the ultraviolet — invisible to the eye. Finally push $Z$ from 1 to 2: the whole pattern keeps its shape but slides to four-times-higher energy, the signature of hydrogen-like ions.
| Symbol | Quantity | Value / Unit |
|---|---|---|
| n | Principal quantum number | 1, 2, 3, … |
| Z | Nuclear charge number | dimensionless |
| E_n | Energy of level n | eV or J |
| R_∞ | Rydberg constant | 1.097×10⁷ m⁻¹ |
| a₀ | Bohr radius | 0.529 Å |
| λ | Photon wavelength | nm |
| v_n | Orbital velocity | m/s |
| r_n | Orbital radius | n²a₀/Z (Å) |
The electrostatic attraction between electron and nucleus provides the centripetal force needed for circular orbit.
Angular momentum is quantized in integer multiples of ℏ = h/2π.
As n₂→∞, λ approaches the series limit. For Lyman (n₁=1): λ_lim = 91.2 nm. For Balmer (n₁=2): λ_lim = 364.6 nm.
$$\frac{1}{\lambda} = (1.097\times10^7)\left(\frac{1}{4}-\frac{1}{9}\right) = 1.524\times10^6\,\text{m}^{-1}$$ $$\lambda = 656.3\,\text{nm}\quad(\text{red, H}_\alpha\text{ line})$$ $$\Delta E = \frac{hc}{\lambda} = \frac{(6.626\times10^{-34})(3\times10^8)}{656.3\times10^{-9}} = 1.89\,\text{eV}$$
Atkins, P. W. & De Paula, J. (2014). Physical Chemistry, 10th ed., Ch. 9. Oxford University Press.