1: Blackbody radiation

Section 1: Historical Foundations

1: Blackbody Radiation

Definition and Key Problem
Blackbody radiation refers to the electromagnetic radiation emitted by an idealized object (a "blackbody") that absorbs all incident radiation and re-emits energy solely based on its temperature. Real-world approximations include heated metals or stars. By the late 19th century, experiments revealed a universal emission spectrum: intensity versus wavelength curves depended only on temperature, peaking at specific wavelengths (e.g., red glow at 500°C vs. blue-white at 6000°C).

Classical physics, combining thermodynamics and electromagnetism, failed to explain this spectrum. The Rayleigh-Jeans law, derived from classical wave theory, predicted that radiation intensity should increase indefinitely as wavelength decreased (proportional to $\lambda^{-4}$). This implied infinite energy at ultraviolet wavelengths—a nonsensical result termed the ultraviolet catastrophe.

Planck’s Quantum Hypothesis
In 1900, Max Planck resolved the catastrophe by introducing a radical assumption: the oscillating charges in the blackbody’s walls emit or absorb energy in discrete packets called quanta, not continuously. He proposed that energy $E$ for a frequency $\nu$ is:

$$E = nh\nu \quad (n = 1, 2, 3, \dots)$$

where $h$ is Planck’s constant ($6.626 \times 10^{-34} \text{J·s}$) and $n$ is an integer. This quantization meant energy exchange could only occur in multiples of $h\nu$.

Planck’s Radiation Law
Using this hypothesis, Planck derived the blackbody spectrum formula:

$$I(\lambda, T) = \frac{2\pi h c^2}{\lambda^5} \frac{1}{e^{hc / (\lambda k_B T)} - 1}$$

where:

• $I(\lambda, T)$ = radiation intensity at wavelength $\lambda$ and temperature $T$,
• $c$ = speed of light,
• $k_B$ = Boltzmann constant.

Key Implications

1. Resolution of Catastrophe: At short wavelengths ($\lambda \to 0$), the exponential term $e^{hc/(\lambda k_B T)}$ dominates, forcing $I \to 0$, avoiding infinite energy.
2. Wien’s Displacement Law: Planck’s law confirmed $\lambda_{\text{max}} T = \text{constant}$ (peak wavelength shifts inversely with temperature).
3. Stefan-Boltzmann Law: Integrating Planck’s law over all wavelengths gives total emitted power $\propto T^4$, matching prior experiments.

Significance
Planck’s work marked the birth of quantum theory by demonstrating that energy quantization was necessary to describe nature. Though initially a mathematical trick, it laid the foundation for quantum mechanics, directly influencing Einstein’s explanation of the photoelectric effect (1905).