Atomic Spectra and the Quantum Atom

Light Comes in Pieces

Electromagnetic radiation is characterised by wavelength $\lambda$, frequency $\nu$ and speed $c$, tied together by $c = \lambda\nu$. Classical physics treated radiation as a continuous wave, and then failed twice.

The photoelectric effect was the first failure. Shining light on a metal ejects electrons, but only above a threshold frequency; below it, no electrons appear no matter how intense the beam. Einstein resolved this by treating light as photons of energy

$$E = h\nu$$

with $h = 6.626 \times 10^{-34}$ J s. Ejection needs one photon energetic enough to overcome the work function $\phi$; the leftover becomes kinetic energy, $KE = h\nu - \phi$. Intensity sets how many photons arrive, not how energetic each one is.

The second failure was atomic spectra. A heated gas emits only certain sharp wavelengths, and absorbs exactly those same ones. For hydrogen the lines obey the Rydberg formula

$$\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$$

with $R_H = 1.097 \times 10^7$ m$^{-1}$. Bohr explained this by quantising the electron's energy:

$$E_n = -\frac{2.18 \times 10^{-18}}{n^2} \text{ J}$$

A photon is emitted only when an electron drops between two allowed levels, so its energy, and therefore its wavelength, can take only particular values. The line spectrum is a direct fingerprint of the ladder of allowed energies.

Bohr's model works beautifully for hydrogen and fails for every atom with more than one electron, because it kept classical orbits while bolting quantisation on by hand.

Common pitfall: believing brighter light carries more energy per photon. Intensity multiplies the number of photons; only frequency changes what a single photon can do. A blinding red lamp will never eject an electron that a faint blue one ejects easily.