A guided walk through the quantum and relativity revolution — every idea comes alive in the live panel on the right. Glowing iron changes colour, photons knock electrons loose, a spaceship clock runs slow, and an electron behaves like a wave.
1 — Black-body radiation & the UV catastrophe
Push a horseshoe into a blacksmith's fire and it glows dull red, then orange, then white-hot. Every hot body radiates a smooth spectrum that depends only on its temperature — a black body is the perfect absorber and emitter.
- Wien's law — the peak wavelength shifts shorter as it gets hotter: λmax T = constant. That is why the bar climbs the rainbow as it heats.
- The UV catastrophe — classical (Rayleigh–Jeans) physics predicted the intensity should rise without limit at short wavelengths. It would mean every warm object blazes with infinite ultraviolet — clearly nonsense.
Why it matters: the failure of classical physics to explain a simple glowing object forced the invention of the quantum.
2 — Planck's quantum: E = hf
In 1900 Planck rescued the spectrum with one radical idea: oscillators can only emit or absorb energy in discrete packets, never in a smooth stream. Energy is quantised.
Planck's relationE = h f = h c / λ
h = 6.63 × 10⁻³⁴ J·s (Planck's constant)
Think of energy as coins rather than a flowing tap: a blue (high-frequency) coin is worth far more than a red (low-frequency) one. Big packets are too expensive to make at short wavelength, so the curve turns back down — the catastrophe vanishes.
worked — energy of one photon
Green light, f = 5.0 × 10¹⁴ Hz?
E = 6.63×10⁻³⁴ × 5.0×10¹⁴ = 3.3 × 10⁻¹⁹ J ≈ 2.1 eV
3 — The photoelectric effect
Shine light on a clean metal plate and electrons are knocked out — the photoelectric effect. The surprises that broke classical physics:
- Threshold frequency f₀ — below it, no electrons at all, no matter how bright the light. A dim blue lamp works where a blazing red one fails.
- Instant emission — electrons leave the moment light hits; there is no waiting for energy to "build up".
- Brighter ≠ faster — more intensity ejects more electrons but does not give them more speed. Only higher frequency raises their energy.
Particle picture: light arrives as photons; one photon hits one electron. A weak red photon simply cannot pay the price to free it.
4 — Einstein's equation: hf = Φ + KEmax
Einstein (1905) explained it all with energy bookkeeping for a single photon hitting a single electron:
Einstein's photoelectric equationh f = Φ + KEmax
Φ = h f₀ (work function = minimum escape energy)
KEmax = h f − Φ = h(f − f₀)
- Work function Φ — the "entrance fee" the electron must pay to escape the metal surface.
- KEmax — the leftover energy the freed electron carries away. Below f₀ there is no leftover, so nothing escapes.
worked — sodium plate
Φ = 2.3 eV, light of 3.0 eV photons?
KEmax = 3.0 − 2.3 = 0.7 eV
5 — Stopping potential: eVs = KEmax
How do we measure KEmax in the lab? Apply a reverse voltage that pushes the electrons back. Raise it until even the fastest electron just fails to reach the collector — that is the stopping potential Vs.
Stopping potentiale Vs = KEmax = h f − Φ
e = 1.6 × 10⁻¹⁹ C (electron charge)
Picture each electron as a ball rolling up a hill of voltage: the steeper the hill (larger Vs), the more energy was needed to climb it. The hill that stops the fastest ball measures KEmax exactly.
worked — reading the meter
KEmax = 0.7 eV?
Vs = KEmax / e = 0.7 V
6 — KEmax vs frequency: the slope is h
Rearrange Einstein's equation: KEmax = h f − Φ. This is the equation of a straight line y = mx + c, and it lets us measure Planck's constant with a ruler.
The graph (Millikan's experiment)slope = h (Planck's constant)
x-intercept = f₀ (threshold frequency)
y-intercept = −Φ (work function)
Every metal gives a line with the same slope h — only the intercept shifts. This universal slope was the experimental triumph that earned Einstein the Nobel Prize and confirmed the photon.
Exam point: the gradient of a KEmax–f graph is Planck's constant, h ≈ 6.63 × 10⁻³⁴ J·s.
7 — Special relativity: time dilation & E = mc²
Einstein's 1905 relativity says the speed of light is the same for everyone, with strange consequences when things move near c = 3 × 10⁸ m/s.
Key relativistic resultstime dilation: Δt = Δt₀ / √(1 − v²/c²)
mass–energy: E = m c²
- Time dilation — a clock on a fast spaceship runs slow as seen from Earth. A twin who flies off and returns is younger.
- E = mc² — mass is frozen energy. A tiny mass releases enormous energy — the source of the Sun's power and of nuclear energy.
worked — mass to energy
Convert 1 gram of mass fully?
E = 0.001 × (3×10⁸)² = 9 × 10¹³ J — a city's worth of energy
8 — de Broglie matter waves: λ = h/p
de Broglie's bold symmetry (1924): if waves act like particles, then particles act like waves. Every moving object carries a wavelength:
de Broglie wavelengthλ = h / p = h / (m v)
For a cricket ball λ is absurdly tiny — we never notice. But for a fast electron λ is about the size of an atom, so electrons diffract through crystals just like X-rays. Faster electron → larger momentum → shorter wave.
worked — electron wavelength
Electron, p = 1.0 × 10⁻²⁴ kg·m/s?
λ = 6.63×10⁻³⁴ / 1.0×10⁻²⁴ = 6.6 × 10⁻¹⁰ m (≈ atom-sized)
9 — Recap & applications
Modern physics began when one glowing object refused to obey classical rules. The quantum idea then powered the technology around you:
- Solar cells — the photoelectric effect in silicon: each photon frees a charge carrier, turning sunlight straight into electricity.
- Electron microscope — de Broglie's short electron wavelength resolves detail far finer than light ever could, revealing viruses and atoms.
- Nuclear & medical — E = mc² runs reactors and PET scans; LEDs and digital cameras all rely on E = hf.
- Black-body curve + UV catastrophe → classical physics fails for radiation.
- Planck: energy is quantised, E = hf (h = 6.63 × 10⁻³⁴ J·s).
- Photoelectric effect: threshold f₀; brighter = more electrons, not faster.
- Einstein: hf = Φ + KEmax; stopping potential eVs = KEmax.
- KEmax–f graph is a straight line of slope h.
- Relativity: moving clocks slow; E = mc² ties mass to energy.
- de Broglie: matter waves λ = h/p — the basis of the electron microscope.