Failure of Classical Theory

The Rayleigh–Jeans model correctly describes the long-wavelength part of a blackbody spectrum, but it predicts unbounded energy at short wavelength. The failure is not merely a poor fitted constant: it exposes a wrong classical assumption about how thermal energy is shared among electromagnetic modes.

1. Classical prediction

For spectral energy density per unit wavelength, the Rayleigh–Jeans law is

u_λ(λ,T) = (8π kT)/λ⁴

where u_λ dλ is energy per unit volume in the wavelength interval [λ,λ + dλ].

At fixed T,

u_λ ∝ λ⁻⁴

so the prediction diverges as λ → 0.

Blackbody spectra and the Rayleigh–Jeans failure

Hotter and cooler blackbody spectra each rise to a peak and fall at short wavelength. The hotter peak is at shorter wavelength and has greater area. A dashed Rayleigh–Jeans curve diverges toward zero wavelength.

Scroll across the graph to read all labels.

Open full-size graph

View figure data

Values and uncertainty for Blackbody spectra and the Rayleigh–Jeans failure

Series	Wavelength (relative units)	Wavelength uncertainty	Spectral intensity (relative units)	Spectral intensity uncertainty
Hotter body	0.5		0.01
Hotter body	1		0.12
Hotter body	1.5		0.45
Hotter body	2		0.78
Hotter body	2.5		0.96
Hotter body	3		1
Hotter body	3.5		0.94
Hotter body	4		0.83
Hotter body	5		0.62
Hotter body	6		0.46
Hotter body	7		0.34
Hotter body	8		0.25
Hotter body	9		0.18
Hotter body	10		0.13
Cooler body	0.5		0
Cooler body	1		0.01
Cooler body	1.5		0.04
Cooler body	2		0.1
Cooler body	2.5		0.2
Cooler body	3		0.32
Cooler body	3.5		0.44
Cooler body	4		0.53
Cooler body	5		0.6
Cooler body	6		0.57
Cooler body	7		0.49
Cooler body	8		0.4
Cooler body	9		0.32
Cooler body	10		0.25
Rayleigh–Jeans prediction	0.5		1.45
Rayleigh–Jeans prediction	1		0.9
Rayleigh–Jeans prediction	1.5		0.58
Rayleigh–Jeans prediction	2		0.4
Rayleigh–Jeans prediction	2.5		0.29
Rayleigh–Jeans prediction	3		0.22
Rayleigh–Jeans prediction	4		0.14
Rayleigh–Jeans prediction	5		0.1
Rayleigh–Jeans prediction	6		0.07
Rayleigh–Jeans prediction	8		0.04
Rayleigh–Jeans prediction	10		0.025

Values and uncertainty for Blackbody spectra and the Rayleigh–Jeans failure

Series	Wavelength (relative units)	Wavelength uncertainty	Spectral intensity (relative units)	Spectral intensity uncertainty
Hotter body	0.5		0.01
Hotter body	1		0.12
Hotter body	1.5		0.45
Hotter body	2		0.78
Hotter body	2.5		0.96
Hotter body	3		1
Hotter body	3.5		0.94
Hotter body	4		0.83
Hotter body	5		0.62
Hotter body	6		0.46
Hotter body	7		0.34
Hotter body	8		0.25
Hotter body	9		0.18
Hotter body	10		0.13
Cooler body	0.5		0
Cooler body	1		0.01
Cooler body	1.5		0.04
Cooler body	2		0.1
Cooler body	2.5		0.2
Cooler body	3		0.32
Cooler body	3.5		0.44
Cooler body	4		0.53
Cooler body	5		0.6
Cooler body	6		0.57
Cooler body	7		0.49
Cooler body	8		0.4
Cooler body	9		0.32
Cooler body	10		0.25
Rayleigh–Jeans prediction	0.5		1.45
Rayleigh–Jeans prediction	1		0.9
Rayleigh–Jeans prediction	1.5		0.58
Rayleigh–Jeans prediction	2		0.4
Rayleigh–Jeans prediction	2.5		0.29
Rayleigh–Jeans prediction	3		0.22
Rayleigh–Jeans prediction	4		0.14
Rayleigh–Jeans prediction	5		0.1
Rayleigh–Jeans prediction	6		0.07
Rayleigh–Jeans prediction	8		0.04
Rayleigh–Jeans prediction	10		0.025

2. Why the divergence occurs

Classical equipartition assigns an average energy of order kT to every allowed electromagnetic standing-wave mode in the cavity. The number of modes per unit frequency interval grows as ν². With no upper limit on frequency, infinitely many high-frequency modes each receive finite average energy.

The total predicted energy therefore diverges:

U ∝ ∫₀^∞ ν² kT dν → ∞

This unphysical prediction is the ultraviolet catastrophe.

3. Comparison with experiment

Measured spectra have a finite peak and fall toward zero at high frequency or short wavelength. The Rayleigh–Jeans result remains useful when hν≪ kT, which is the low-frequency, long-wavelength limit. A strong answer therefore says both where the model works and where it fails.

Planck’s resolution changes the energy-exchange assumption. An oscillator of frequency f can exchange energy only in quanta hf. When hf≫ kT, thermal excitation of that mode becomes exponentially unlikely, suppressing the high-frequency contribution.

4. Common mistakes

• Saying classical physics predicts infinite intensity at every wavelength. The divergence occurs in the short-wavelength limit.
• Treating the catastrophe as an experimental divergence. Experiment shows the opposite: intensity falls.
• Claiming quantisation removes high-frequency modes. The modes remain; their average thermal occupation is suppressed.
• Mixing wavelength and frequency densities without changing variables correctly.

6. Worked Examples

7. Limiting-case check

Next: Planck’s Hypothesis.