UV Catastrophe (Black-Body Radiation)

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
The Correspondence Principle
The Philosophy of Quantum Theory
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
Commutation Relations and Simultaneous Eigenvalues
The Schrödinger Equation in Dirac Notation
Transformations of Operators and Symmetry
Time Evolution of Expectation Values and Ehrenfest's Theorem
One-Dimensional Bound States
Oscillation Theorem
The Dirac Delta Function Potential
Scattering States, Transmission and Reflection
Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

To begin an overview of the evolution of quantum mechanics, one must first examine its birthplace, the black-body radiation problem. It is simple to understand that emission of radiation from an object occurs for all temperatures greater than absolute zero. As the temperature of the object rises, the maximum in the spectral distribution of the black-body radiation shifts away from the long-wavelength (infrared) region, to shorter-wavelength regions, including the visible spectrum and finally the UV and X-ray regions. The total power radiated also increases with temperature.

Imagine a perfect absorber cavity (i.e. it absorbs all radiation at all wavelengths, so that its spectral radiance only depends on temperature). From Kirchoff's law it follows that such a body would not only be a perfect absorber of radiation, but also a perfect emitter. Such a body is called a black body, and the radiation emitted by such a material is called black-body radiation. Lord Rayleigh (John William Strutt) and Sir James Jeans applied classical physics and assumed that the radiation from this perfect absorber could be represented by standing waves. Although the Rayleigh-Jeans result does approach the experimentally recorded values for large values of wavelength, the trend line vastly differs as the wavelength is allowed to tend towards zero. The result predicts that the spectral intensity will increase quadratically with increasing frequency, and would diverge as the wavelength went to zero. This result is known as the "ultraviolet catastrophe." Black-body radiation thus illustrates an important failure of classical physics. The Rayleigh-Jeans law is as follows:

u_{\text{RJ}}(\nu, T) = \frac{8\pi \nu^2}{c^3}~k_{\text{B}} T

where  c \! is the speed of light,  k_{\text{B}}\! is Boltzmann's constant and  T\! is the temperature in Kelvin. There is a relation between the frequency distribution  u(\nu, T) \! and the wavelength distribution  u(\lambda, T) \! :

 
\begin{align}
u(\nu, T) &= u(\lambda, T) \left|\frac{d\lambda}{d\nu} \right| \\
&= \frac{c}{\nu^2} u(\lambda, T).
\end{align}

Based on a thermodynamic argument, Wien found that, under adiabatic expansion, the energy of a mode of light, the frequency of the mode, and the total temperature of the light change together in the same way, so that their ratios are constant. This implies that, for each mode at thermal equilibrium, the adiabatic invariant energy/frequency should only be a function of the adiabatic invariant frequency/temperature. The Wien law is:


u_{\text{Wien}}(\nu,T) = \nu^3 g\left(\frac{\nu}{T}\right)

and Wien predicted that  g(\nu / T) = C e^{h\nu/k_{\text{B}} T} \!.

In 1900, Max Planck offered a successful explanation for black-body radiation. He postulated that the energies of the standing waves in Rayleigh's model were quantized, with energies equal to integer multiples of  h\nu \! , where  h=6.626*10^{-34} \text{J}\cdot\text{s} \! is Planck's constant and \nu\! is the frequency of the photon. This is known as the Quantum Hypothesis. Using this assumption, one obtains an energy distribution that approaches zero as the frequency tends to infinity, thus eliminating the UV catastrophe. Planck's law of black-body radiation is as follows:

u_(\nu,T) = \frac{8 \pi h}{c^3}\frac{\nu^3}{e^\frac{h\nu}{k_{\text{B}} T}-1}

In the limits, \nu  \to 0 and \nu  \to \infty , we can easily recover the Rayleigh-Jeans and Wien formulas, respectively.

Before leaving the subject of black-body radiation, it is important to look at one fundamental realization that has come out of the mathematics. In 1964, A. Penzias and R. Wilson discovered a radio signal of suspected cosmic origin, with an intensity corresponding to approximately 3 K. Upon application of Planck's theorem for said radiation, it soon became evident that the spectrum seen corresponded to that of a black body at 3 K, and since this radiation was incident on Earth evenly from all directions, space itself was deemed to be the emitting black body. This cosmic background radiation gave credence to the Big Bang theory, and upon analysis of an expanding system, allowed for proof that Planck's theorem holds for black bodies of changing size. The results of this particular proof even allow for a fair estimation into the rate of expansion of the universe since the time the black-body radiation was emitted.

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