The book QM Demystified states this about black body radiation spectrum:

An attempt to explain these results using classical theory was codified in the Rayleigh-Jeans formula, which is an expression that attempts to give us the energy density u(ν,T) of radiation in the cavity, where ν is frequency and T is the temperature. Qualitatively, it is formed as a product of two quantities:

u = number degrees of freedom for frequency ν * average energy per degree of freedom"

The * means multiplication here. What does degree of freedom of frequency mean here?


1 Answer 1


I'm assuming that this section of the book is talking about the ultraviolet catastrophe, where an ideal black body in thermal equilibrium will emit an infinite amount of power through radiative means. The source goes on to say:

The ultraviolet catastrophe results from the equipartition theorem of classical statistical mechanics which states that all harmonic oscillator modes (degrees of freedom) of a system at equilibrium have an average energy of kT.

Following that:

According to classical electromagnetism, the number of electromagnetic modes in a 3-dimensional cavity, per unit frequency, is proportional to the square of the frequency. This therefore implies that the radiated power per unit frequency should follow the Rayleigh–Jeans law, and be proportional to frequency squared. Thus, both the power at a given frequency and the total radiated power is unlimited as higher and higher frequencies are considered: this is clearly unphysical as the total radiated power of a cavity is not observed to be infinite, a point that was made independently by Einstein and by Lord Rayleigh and Sir James Jeans in the year 1905.

Essentially, if in a cavity there are an infinite number of electromagnetic modes possible (think standing waves), the Equipartition Theorem says that a system in equilibrium has an average of $k_{b}T$ worth of energy per mode. This is not what's actually observed, since we don't see an infinite amount of power radiated. How was this solved:

Max Planck solved the problem by postulating that electromagnetic energy did not follow the classical description, but could only be emitted in discrete packets of energy proportional to the frequency, as given by Planck's law. The radiated power eventually goes to zero at infinite frequencies, and the total predicted power is finite.(1) The formula for the radiated power for the idealized system (black body) was in line with known experiments, and came to be called Planck's law of black body radiation. Based on past experiments, Planck was also able to determine the value of its parameter, now called Planck's constant. The packets of energy later came to be called photons, and played a key role in the quantum description of electromagnetism.(1)

  • $\begingroup$ "This has the effect of reducing the number of possible excited modes with a given frequency in the cavity described above, and thus the average energy at those frequencies. " This is not correct. The number of modes is the same; what changes is that in first Planck's theory, the material resonators can have only energies that are multiples of $\hbar \omega$. This then leads to the result that their energy (and energy of EM mode) is not the same $k_B T$ for all modes, but depends on the frequency of the mode and goes to zero as $\omega \rightarrow \infty$. $\endgroup$ Dec 19, 2013 at 23:37
  • $\begingroup$ What you've argued is mentioned on the line after what you have a problem with. By limiting the energy of photons to discrete packets of $\hbar \omega$, you reduce the number of resonant modes which could exist within your cavity. Think about it - some classically allowed modes could have been resonant within the cavity, but are no longer allowed by discretization. I believe that there are still an infinite amount of allowed modes which are discretized, but you're excluding many classically allowed modes which contribute to the total power radiated. $\endgroup$
    – astromax
    Dec 20, 2013 at 0:37
  • 2
    $\begingroup$ You seem to confuse the concept of mode with energy stored in this mode. The modes are indexed discretely always, both in classical and quantum theory. Mode is defined by three numbers $n_1,n_2,n_3 \in N$ and one number $\lambda \in \{1,2\}$ - node numbers + polarization state of standing wave. It is the energy associated with the mode that is restricted to discrete multiples of $\hbar \omega$ in quantum theory, while it is continuous in classical theory. $\endgroup$ Dec 20, 2013 at 2:37
  • $\begingroup$ Ah yes, I was confusing the two. I think that sentence should then be deleted from the wiki article as well as my answer (I'll edit). Thanks for the correction. $\endgroup$
    – astromax
    Dec 20, 2013 at 14:43

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