Planck’s Hypothesis

Planck resolved the ultraviolet catastrophe by changing how matter exchanges energy with the electromagnetic field. Oscillators of frequency f can occupy only discrete energies separated by hf, so high-frequency excitation becomes unlikely when hf greatly exceeds the thermal energy scale kT.

1. Hypothesis and assumptions

For an oscillator of frequency f,

Eₙ = nhf, n = 0,1,2,…

The allowed energies are discrete, with adjacent levels separated by

Δ E = hf

Planck originally quantised the energy exchange of material oscillators in the cavity walls. The later photon model interprets a quantum of electromagnetic radiation as carrying energy hf. Keep those historical steps distinct even though modern calculations use the photon language naturally.

2. Mean energy and spectrum

Applying Boltzmann weighting to the allowed oscillator energies gives the mean thermal energy per mode

⟨E⟩ = hf/(e^(hf/(kT))-1)

Multiplying by the electromagnetic mode density gives the spectral energy density per unit frequency:

u_f(f,T) = ((8π hf³)/c³)1/(e^(hf/(kT))-1)

Here u_f df is the energy per unit volume in the frequency interval [f,f + df].

The wavelength form is not obtained by merely substituting f = c/λ; the interval also transforms. The result is

u_λ(λ,T) = ((8π hc)/λ⁵)1/(e^(hc/(λ kT))-1)

3. Why quantisation fixes the spectrum

Define the dimensionless ratio

x = hf/kT

• If x≪1, then e^x-1 ≈ x. Hence ⟨E⟩ ≈ kT, recovering classical equipartition and the Rayleigh–Jeans limit.
• If x≫1, then ⟨E⟩ ≈ hf e^(-x). The exponential factor overwhelms the increasing number of modes and suppresses the high-frequency spectrum.

Thus Planck’s law agrees with classical physics where classical physics works, but remains finite where the classical model diverges.

4. Photon relations

For a photon in vacuum,

E = hf = hc/λ

and the relativistic massless-particle relation E = pc gives

p = E/c = h/λ

This photon momentum is the key input to Compton scattering.

5. Common mistakes

• Starting n at one and excluding the zero-energy state in Planck’s original oscillator model.
• Saying all energies in nature must be integer multiples of one universal amount. The spacing is hf and depends on oscillator frequency.
• Treating hf as the energy of the oscillator at every temperature. It is the spacing between adjacent levels; the mean energy depends on T.
• Substituting f = c/λ into a spectral density without transforming the interval.
• Describing quantisation as an arbitrary high-frequency cutoff. The suppression follows continuously from the Boltzmann factor.

6. Worked Examples

7. Model check

Next: Compton Shift.