Checkpoint: Quanta and Photons

Module 5 introduced the chain of experiments and ideas that forced physics to abandon continuous energy in favour of discrete quanta. This checkpoint ties those threads together by asking you to apply several results in one calculation.

The key results

Planck's quantization hypothesis. A harmonic oscillator of frequency $\nu$ exchanges energy with the radiation field only in whole multiples of

$E = h\nu,$

where $h \approx 6.626 \times 10^{-34}\,\text{J·s}$ is Planck's constant. This single idea fixed the ultraviolet catastrophe and introduced the quantum of action.

Einstein's photon. Light of frequency $\nu$ consists of photons each carrying energy $h\nu$ and momentum $h/\lambda$. The photoelectric effect confirmed that energy arrives in indivisible lumps: below the threshold frequency $\nu_0 = \phi/h$ (where $\phi$ is the work function) no electrons are emitted regardless of intensity.

Compton scattering. When a photon of wavelength $\lambda$ scatters off a nearly-free electron at angle $\theta$, the wavelength increases by

$\Delta\lambda = \frac{h}{m_e c}\,(1-\cos\theta).$

This is only possible if the photon carries momentum $p = h/\lambda$ — it behaves as a particle in the collision.

The Bohr model. Niels Bohr proposed in 1913 that electrons in hydrogen orbit the nucleus only on discrete circular orbits that satisfy $m_e v r = n\hbar$ (integer angular momentum). The total mechanical energy on orbit $n$ is

$E_n = -\frac{m_e e^4}{8\varepsilon_0^2 h^2}\,\frac{1}{n^2} \approx -\frac{13.6\,\text{eV}}{n^2},$

where the numerical value $13.6\,\text{eV}$ is the Rydberg energy $R_\infty hc$. The ground state ($n = 1$) sits at $-13.6\,\text{eV}$; the ionization threshold is $E = 0$.

de Broglie's matter waves. Louis de Broglie unified the photon relations by assigning a wavelength to any particle of momentum $p$:

$\lambda_{dB} = \frac{h}{p}.$

This explains Bohr's quantization rule: a stable orbit is one whose circumference fits exactly $n$ de Broglie wavelengths, $2\pi r = n\lambda_{dB}$.

Connecting the ideas

The module's central theme is that both radiation and matter are simultaneously wave-like and particle-like. Each experiment added a facet: Planck showed energy is discrete; Einstein showed light is a particle; Compton showed that particle carries momentum; Bohr showed atomic orbits are quantized; de Broglie unified them by treating matter as waves.

The H-alpha line

The most famous line in the hydrogen spectrum is H-alpha, produced when an electron falls from $n = 3$ to $n = 2$. The photon energy is

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