UV Catastrophe (Black-Body Radiation)

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.

Physical Basis of Quantum Mechanics

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

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

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

Motion in One Dimension

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

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

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 $u(\lambda ,T)\!$:

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

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 $g(\nu /T)=Ce^{h\nu /k_{\text{B}}T}\!$.

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.