Table of Contents
- Introduction
- What Is a Blackbody?
- Classical Approach and the Ultraviolet Catastrophe
- Rayleigh–Jeans Law and Its Limitations
- Wien’s Empirical Law
- Planck’s Quantum Hypothesis
- Derivation of Planck’s Law
- Energy Quantization and the Planck Constant
- Spectral Radiance and Distribution
- Peak Wavelength and Wien’s Displacement Law
- Stefan–Boltzmann Law
- Photon Viewpoint and Quantum Interpretation
- Experimental Validation
- Applications in Astrophysics and Technology
- Blackbody Radiation in Modern Quantum Physics
- Conclusion
1. Introduction
Blackbody radiation played a pivotal role in the development of quantum mechanics. The failure of classical physics to accurately model blackbody spectra led to the groundbreaking realization that energy must be quantized. This marked the beginning of the quantum revolution.
2. What Is a Blackbody?
A blackbody is an idealized object that absorbs all electromagnetic radiation incident on it and re-emits it with a characteristic spectrum that depends only on its temperature.
- Perfect emitter and absorber
- Emission spectrum is thermal and continuous
- Used as a model in thermodynamics and quantum theory
3. Classical Approach and the Ultraviolet Catastrophe
According to classical electrodynamics, the energy density \( u(\nu, T) \) of blackbody radiation should increase indefinitely with frequency:
\[
u(\nu, T) \propto \nu^2
\]
This prediction leads to infinite energy as \( \nu \to \infty \), a divergence known as the ultraviolet catastrophe.
4. Rayleigh–Jeans Law and Its Limitations
Based on classical equipartition and electromagnetic theory:
\[
u(\nu, T) = \frac{8\pi \nu^2 kT}{c^3}
\]
Valid at low frequencies, but fails badly at high frequencies — energy density increases without bound.
5. Wien’s Empirical Law
Wien fitted experimental data with:
\[
u(\nu, T) \propto \nu^3 e^{-h\nu / kT}
\]
Good match at high frequencies, poor at low frequencies. Lacked theoretical basis, but hinted at the exponential suppression of high-energy modes.
6. Planck’s Quantum Hypothesis
In 1900, Max Planck proposed that energy is emitted in discrete packets (quanta):
\[
E_n = n h \nu, \quad n = 1, 2, 3, \dots
\]
This quantization led to a successful formula for spectral energy density:
\[
u(\nu, T) = \frac{8\pi h \nu^3}{c^3} \cdot \frac{1}{e^{h\nu / kT} – 1}
\]
Planck introduced the Planck constant \( h = 6.626 \times 10^{-34} \ \text{Js} \).
7. Derivation of Planck’s Law
Consider a cavity with standing electromagnetic waves. For each mode:
- Energy levels: \( E_n = n h \nu \)
- Average energy per mode:
\[
\langle E \rangle = \frac{\sum_n E_n e^{-E_n / kT}}{\sum_n e^{-E_n / kT}} = \frac{h\nu}{e^{h\nu / kT} – 1}
\]
Multiply by number of modes per unit volume per unit frequency:
\[
u(\nu, T) = \frac{8\pi \nu^2}{c^3} \cdot \langle E \rangle
\]
Yields Planck’s law.
8. Energy Quantization and the Planck Constant
The success of Planck’s theory required abandoning the assumption of continuous energy.
- Energy levels are quantized
- The Planck constant \( h \) sets the scale for quantum effects
- A foundational principle in quantum mechanics
9. Spectral Radiance and Distribution
The spectral radiance \( B(\nu, T) \) gives energy emitted per unit surface area, per unit solid angle, per unit frequency:
\[
B(\nu, T) = \frac{2h\nu^3}{c^2} \cdot \frac{1}{e^{h\nu / kT} – 1}
\]
Alternately, in terms of wavelength \( \lambda \):
\[
B(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{hc / \lambda kT} – 1}
\]
10. Peak Wavelength and Wien’s Displacement Law
The peak of the blackbody spectrum occurs at:
\[
\lambda_{\text{max}} T = b, \quad b = 2.897 \times 10^{-3} \ \text{m K}
\]
Indicates that hotter bodies emit radiation at shorter wavelengths.
11. Stefan–Boltzmann Law
Total power radiated per unit area of a blackbody:
\[
P = \sigma T^4, \quad \sigma = \frac{2\pi^5 k^4}{15h^3 c^2}
\]
Where \( \sigma \approx 5.670 \times 10^{-8} \ \text{W m}^{-2} \text{K}^{-4} \)
Integral of Planck’s law over all frequencies.
12. Photon Viewpoint and Quantum Interpretation
Modern interpretation: blackbody radiation is a gas of photons in thermal equilibrium.
- Number of photons not conserved
- Follows Bose–Einstein statistics
- Photon occupation number:
\[
\langle n \rangle = \frac{1}{e^{h\nu/kT} – 1}
\]
13. Experimental Validation
- Blackbody cavities and furnaces
- Sun and stars approximate blackbodies
- Planck’s law matches experimental curves perfectly
- Confirmed across infrared, visible, and ultraviolet spectra
14. Applications in Astrophysics and Technology
- Stellar classification (color–temperature relation)
- Cosmic Microwave Background (CMB) as a near-perfect blackbody
- Infrared thermography
- Radiation detectors and calibration sources
- Climate modeling and thermal emission analysis
15. Blackbody Radiation in Modern Quantum Physics
- Introduced energy quantization
- Paved way for Einstein’s photon theory
- Foundation for quantum statistical mechanics
- Helps derive Planck units and natural constants
- Connected to Hawking radiation and cosmological horizons
16. Conclusion
Blackbody radiation was the problem that shattered the classical worldview and sparked the quantum era. Planck’s quantization of energy not only resolved the ultraviolet catastrophe but opened the door to a new physics where probability, discreteness, and wave–particle duality reign. It remains one of the most important turning points in the history of science.
