Blackbody radiation puzzled scientists in the late 19th century. Classical physics couldn't explain why hot objects didn't emit infinite energy at short wavelengths, leading to the "ultraviolet catastrophe."

Max Planck solved this mystery by proposing that energy is emitted in discrete packets called quanta. This revolutionary idea laid the foundation for quantum mechanics and changed our understanding of light and matter forever.

Blackbody Radiation Laws

Blackbody Characteristics and Thermal Radiation

• Blackbody absorbs all incident electromagnetic radiation, regardless of frequency or wavelength
• Ideal blackbody emits thermal radiation based solely on its temperature
• Real-world examples approximate blackbody behavior (cavity with small hole, carbon black)
• Blackbody radiation spectrum shows characteristic shape with peak at specific wavelength

Fundamental Laws Governing Blackbody Radiation

• Stefan-Boltzmann law relates total energy radiated by blackbody to its temperature
  • Expressed mathematically as $E = σT^4$
  • E represents total radiant energy, σ is Stefan-Boltzmann constant, T is absolute temperature
  • Demonstrates fourth-power dependence of energy on temperature
• Wien's displacement law describes relationship between blackbody's temperature and wavelength of peak emission
  • Formulated as $λ_max = b/T$
  • λ_max is wavelength of maximum emission, b is Wien's displacement constant, T is absolute temperature
  • Explains why hotter objects emit radiation at shorter wavelengths (blue-shift)

Rayleigh-Jeans Law and Its Limitations

• Rayleigh-Jeans law attempts to describe spectral radiance of blackbody radiation
• Derived using classical physics principles and equipartition theorem
• Accurately predicts blackbody radiation at long wavelengths
• Fails catastrophically at short wavelengths, leading to "ultraviolet catastrophe"
• Expressed mathematically as $B_λ(λ,T) = (2ckT)/λ^4$
• B_λ represents spectral radiance, c is speed of light, k is Boltzmann constant, T is temperature, λ is wavelength

The Ultraviolet Catastrophe

The Discrepancy Between Theory and Observation

• Ultraviolet catastrophe refers to significant disagreement between classical physics predictions and observed blackbody radiation spectrum
• Classical theory (Rayleigh-Jeans law) predicts infinite energy emission at short wavelengths
• Observed blackbody spectrum shows finite energy emission across all wavelengths
• Discrepancy most pronounced in ultraviolet region of spectrum

Limitations of Classical Physics in Explaining Blackbody Radiation

• Rayleigh-Jeans law based on classical assumptions of continuous energy emission
• Assumes equipartition theorem applies to all wavelengths of electromagnetic radiation
• Predicts spectral radiance inversely proportional to fourth power of wavelength
• Results in "catastrophe" as wavelength approaches zero, with energy approaching infinity
• Demonstrates fundamental flaw in classical physics' ability to describe atomic-scale phenomena

Implications for the Development of Quantum Mechanics

• Ultraviolet catastrophe highlighted need for new theoretical framework
• Revealed limitations of classical physics in describing microscopic systems
• Prompted scientists to explore alternative explanations for blackbody radiation
• Led to development of quantum mechanics as new branch of physics
• Served as crucial turning point in understanding nature of light and matter

Planck's Quantum Solution

Planck's Revolutionary Hypothesis

• Planck's law introduces concept of energy quantization to resolve ultraviolet catastrophe
• Assumes energy can only be emitted or absorbed in discrete packets called quanta
• Energy of quantum expressed as $E = hν$
• E represents energy, h is Planck's constant, ν is frequency of radiation
• Planck's law accurately describes entire blackbody radiation spectrum

Mathematical Formulation and Implications

• Planck's law expressed as $B_λ(λ,T) = (2hc^2)/λ^5 1/(e^(hc/λkT) - 1)$
• B_λ represents spectral radiance, h is Planck's constant, c is speed of light, λ is wavelength, k is Boltzmann constant, T is temperature
• Reduces to Rayleigh-Jeans law at long wavelengths
• Avoids ultraviolet catastrophe by limiting energy of high-frequency oscillators
• Introduces concept of zero-point energy, fundamental to quantum mechanics

Impact on Physics and Resolution of the Ultraviolet Catastrophe

• Planck's quantum hypothesis successfully explains observed blackbody radiation spectrum
• Resolves ultraviolet catastrophe by limiting high-frequency energy emission
• Marks beginning of quantum era in physics
• Leads to development of quantum mechanics and modern understanding of atomic structure
• Influences later work by Einstein on photoelectric effect and Bohr's atomic model