Quantum theory of light
Blackbody spectra, the failure of classical theory, Planck's quantisation, and Compton scattering with its derivation from conservation laws.
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Before you begin
Two experiments broke the classical picture of light as a continuous wave. The blackbody spectrum showed that energy is exchanged in quanta, and Compton scattering showed that photons carry momentum. The lessons follow that story: the blackbody spectrum, why classical theory fails to explain it, Planck’s solution, and then the Compton shift and its derivation. Light still behaves as a wave too; the topic shows why it also needs a quantum description. This is an optional extension beyond the H3 Physics syllabus.
Be comfortable with: quantum physics and waves at H2, and relativistic momentum and energy for the Compton derivation.
Learning goals
- Connect blackbody evidence, Planck quantisation, photon momentum, and Compton scattering.
Lessons
Work through them in order.
- Blackbody radiationRead blackbody spectra and use the Stefan–Boltzmann and Wien laws.
- The failure of classical theoryExplain why the classical Rayleigh–Jeans law fails at short wavelengths.
- Planck's hypothesisUse quantised oscillator energies to explain the blackbody spectrum and its classical limit.
- The Compton shiftUse the Compton equation to find the wavelength shift of scattered X-rays.
- Deriving the Compton shift equationDerive the Compton shift from conservation of momentum and energy.
Practise and check
Topic reference
Quick reference
| Quantity | Equation | Notes |
|---|---|---|
| Power radiated (Stefan–Boltzmann) | P = eσ AT⁴ | σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴; T in kelvin |
| Net power radiated | Pₙₑₜ = eσ A(T⁴-Tₛ⁴) | surroundings at Tₛ |
| Peak wavelength (Wien) | λₘₐₓT = b | b = 2.90 × 10⁻³ m K; wavelength spectrum only |
| Classical spectrum (Rayleigh–Jeans) | uλ = 8π kT/λ⁴ | correct only when hf≪ kT |
| Planck’s oscillator energies | Eₙ = nhf | n = 0,1,2,… |
| Average energy of a mode | ⟨E⟩ = hf/(e^(hf/kT)-1) | → kT when hf≪ kT |
| Photon energy and momentum | E = hf = hc/λ, p = h/λ | |
| Compton shift | Δλ = (h/mₑc)(1- cos θ) | h/(mₑc) = 2.43 pm; θ is the photon’s angle |
Core knowledge to remember
- A blackbody absorbs all the radiation falling on it and, at a given temperature, emits the most possible at every wavelength.
- A hotter blackbody emits more at every wavelength; the total power rises as T⁴ and the peak moves to shorter wavelengths as 1/T.
- Classical equipartition gives every mode the energy kT; with no limit on the number of high-frequency modes, the predicted energy is infinite (the ultraviolet catastrophe).
- Planck’s quantisation makes modes with hf≫ kT almost never excited, which keeps the spectrum finite, while agreeing with classical physics where hf≪ kT.
- In Compton scattering, a photon behaves as a particle with momentum h/λ; conserving energy and momentum gives a shift that depends only on the scattering angle.
Common mistakes
- Treating the Rayleigh–Jeans law as wrong at every wavelength. It agrees with experiment at long wavelengths.
- Taking the peak height of a spectrum as the total power; the total power is the area.
- Converting a wavelength peak to a frequency peak with f = c/λₘₐₓ.
- Starting Planck’s energies at n = 1; the zero-energy state has n = 0.
- Saying Compton scattering proves light is only a particle. It shows that photons carry momentum, as well as behaving as waves.
- Thinking the Compton shift depends on the intensity or the incident wavelength. For a free electron, it depends only on the scattering angle.