Blackbody radiation and Planck

The blackbody problem

A blackbody is an ideal object that absorbs all radiation incident on it and emits thermal radiation determined only by its temperature. A small hole in a heated cavity is a good approximation: light entering the hole bounces around and is almost certainly absorbed, while radiation escaping from the hole has a characteristic spectrum.

Classical physics could explain some parts of the spectrum, but not all of it. Experiments showed that a hot object emits radiation over a range of wavelengths, with a peak that shifts toward shorter wavelengths as temperature increases. Everyday examples include a stove element glowing red, then orange, then white as it gets hotter.

Classical predictions

Two partial classical laws were known. Wien's displacement law described the peak wavelength:

$lambda_{max}T=b$

where $b$ is Wien's constant. This captured the shift of the peak with temperature.

The Stefan-Boltzmann law described total radiated power per area:

$j^*=sigma T^4$

But the full shape of the spectrum remained a problem.

The Rayleigh-Jeans law, based on classical equipartition, predicted that the energy density at high frequency should grow without bound. In wavelength form, it implied far too much radiation at short wavelengths. This failure became known as the ultraviolet catastrophe.

Planck's idea

Max Planck solved the spectrum in 1900 by making a radical assumption. He modeled the cavity walls as microscopic oscillators and proposed that they could exchange energy with radiation only in discrete amounts:

$E_n=nhf$

where $n=0,1,2,ldots$, $f$ is oscillator frequency, and $h$ is Planck's constant.

The energy spacing is

$Delta E=hf$

High-frequency oscillators require larger energy packets. At ordinary temperatures, these high-frequency modes are hard to excite, so they do not contribute unlimited energy. This removes the ultraviolet catastrophe.

Planck distribution

Planck's spectral energy density can be written in frequency form as

u(f,T)=rac{8pi h f^3}{c^3}rac{1}{e^{hf/(k_BT)}-1}

This formula matched experiments across the entire spectrum. At low frequencies where $hfll k_BT$, it reduces approximately to the Rayleigh-Jeans result. At high frequencies, the exponential term suppresses the energy density.

Meaning of quantization

At first, Planck treated quantization as a mathematical trick, not necessarily a statement that nature was fundamentally discontinuous. But the success of the formula was too striking to ignore. It suggested that microscopic energy exchange might not be continuous.

This was a profound break from classical physics. In classical mechanics, an oscillator can have any energy. In Planck's theory, only certain energies are allowed.

Planck's constant

Planck's constant $h$ sets the scale of quantum effects:

$h=6.626 imes10^{-34} J,s$

Because $h$ is so small, quantum discreteness is usually hidden in macroscopic life. For atomic systems, however, $h$ is large enough to dominate behavior.

The big idea

Blackbody radiation forced physicists to abandon the assumption that energy exchange is always continuous. Planck's quantization rule $E_n=nhf$ explained the observed spectrum and introduced a new constant of nature. Quantum mechanics began when the thermal glow of matter revealed that classical physics could not describe microscopic energy.