\[\lambda \nu = c\]

• Speed of light: \(c = 3 \cdot 10^8 \, m/s\) (fundamental constant)

• Frequency \(\nu\): cycles per second, \(1\,Hz = 1\,s^{-1}\)

• Wavelength \(\lambda\): distance between successive peaks

• Key question: how does energy \(E\) relate to frequency \(\nu\)?

In 1900 Planck postulated that only discrete energy values are allowed:

\[\boxed{E= h\nu}\]

• Planck’s constant: \(h = 6.63 \cdot 10^{-34} \, \text{J} \cdot \text{s}\)

• Atoms absorb and emit radiation in quanta, multiples of \(h\nu\)

• \(h\) is tiny, so quantization is invisible at the macro scale (classical limit \(h \to 0\))

Quantized oscillators give a frequency-dependent average energy: \[\langle E \rangle = \frac{h\nu}{e^{\frac{h\nu}{ kT}} - 1}\]

This yields a distribution that vanishes at high frequency: \[\rho_{\nu}(T) = \frac{8\pi \nu^2}{c^3} \cdot \frac{1}{e^{\frac{h\nu}{kT}} - 1}\]

• Integrating over all \(\nu\) gives a finite total: \(\int^{\infty}_0 \rho_{\nu}(T)d\nu = \sigma T^4\)
• No more catastrophe!