Black-body radiation revolutionized physics, revealing classical theory's limitations. It showed that energy is quantized, not continuous, leading to the birth of quantum mechanics and challenging our understanding of light and matter.

Planck's distribution law accurately describes radiation from ideal absorbers and emitters. It explains the spectrum of stars, predicts total energy output, and laid the foundation for quantum concepts like photons and wave-particle duality.

Black-Body Radiation and Planck Distribution

Concept of black-body radiation

• Electromagnetic radiation emitted by an idealized physical body that absorbs all incident radiation and emits radiation at all wavelengths
  • Perfect absorber and emitter of radiation (black body)
  • Radiation emitted depends only on temperature, not composition or shape (ideal radiator)
• Led to the development of quantum mechanics by revealing limitations of classical physics
  • Observed spectrum of black-body radiation could not be explained by classical physics (ultraviolet catastrophe)
  • Introduced the concept of quantized energy levels (photons)

Classical physics failure for radiation

• Rayleigh-Jeans law based on classical physics predicted intensity of black-body radiation should increase infinitely with increasing frequency (shorter wavelengths)
  • Prediction known as the "ultraviolet catastrophe" inconsistent with experimental observations (infinite energy)
  • Failed to explain observed spectrum of black-body radiation, particularly at high frequencies (X-rays, gamma rays)
• Predicted an infinite total energy emitted by a black body, which is physically impossible
  • Violated conservation of energy principle (energy cannot be created or destroyed)
  • Highlighted need for a new theory to explain black-body radiation (quantum mechanics)

Derivation of Planck distribution law

• Max Planck proposed energy of electromagnetic radiation is quantized, with each quantum having energy $E = hν$
  • $h$ is Planck's constant ($6.626 \times 10^{-34}$ J⋅s)
  • $ν$ is frequency of radiation (Hz)
• Using quantization of energy, Planck derived distribution law for black-body radiation:
  • $B_ν(T) = \frac{2hν^3}{c^2} \frac{1}{e^{hν/kT} - 1}$
    • $B_ν(T)$ is spectral radiance (power per unit area per unit solid angle per unit frequency) at frequency $ν$ and temperature $T$ (W⋅sr$^{−1}$⋅m$^{−2}$⋅Hz$^{−1}$)
    • $c$ is speed of light ($3 \times 10^8$ m/s)
    • $k$ is Boltzmann's constant ($1.380649 \times 10^{-23}$ J/K)

Applications of Planck distribution

• Calculate spectral radiance $B_ν(T)$ at given frequency $ν$ and temperature $T$ using Planck distribution equation
  • Determines power emitted per unit area, solid angle, and frequency at specific wavelength (color) and temperature
• Calculate total radiated power per unit area (intensity) of black body by integrating Planck distribution over all frequencies:
  • $I(T) = \int_0^∞ B_ν(T) dν = \frac{2π^5k^4}{15c^2h^3}T^4 = σT^4$
    • $σ$ is Stefan-Boltzmann constant ($5.670374419 \times 10^{-8}$ W⋅m$^{-2}$⋅K$^{-4}$)
    • Relates total energy emitted by black body to its temperature (hotter objects emit more energy)
• Explains color of stars based on their surface temperatures (Hertzsprung-Russell diagram)
  • Cool stars appear red (Betelgeuse), hot stars appear blue (Rigel)

Implications for quantum mechanics

• Success of Planck distribution in describing black-body radiation indicated energy is quantized
  • Challenged foundations of classical physics based on continuous energy (wave theory of light)
  • Led to development of quantum mechanics describing behavior of matter and energy at atomic and subatomic scales (particles)
• Introduced concept of energy quanta (photons) laying groundwork for wave-particle duality
  • Einstein extended concept to explain photoelectric effect, further supporting quantum nature of light and matter (photons ejecting electrons)
  • Paved way for other quantum phenomena like Compton scattering and Bohr model of the atom (discrete energy levels)