To use all functions of this page, please activate cookies in your browser.

my.chemeurope.com

With an accout for my.chemeurope.com you can always see everything at a glance – and you can configure your own website and individual newsletter.

- My watch list
- My saved searches
- My saved topics
- My newsletter

## Planck's law
In physics, -
^{[1]}
As a function of wavelength λ it is written (for infinitesimal solid angle) as: -
^{[2]}
Note that this formula will vary slightly by a constant factor (often a pi) depending on the field of view assumed in the derivation.
Note also that the two functions have different units - the first is radiance per unit frequency interval while the second is radiance per unit wavelength interval. Hence, the quantities The following table provides the definition and SI units of measure for each symbol: Symbol Meaning SI units *spectral radiance*or energy per unit time per unit surface area per unit solid angle per unit frequency or wavelength (as specified)J•s ^{-1}•m^{-2}•sr^{-1}•Hz^{-1},*or*J•s^{-1}•m^{-2}•sr^{-1}•m^{-1}frequency hertz wavelength meter temperature of the black body kelvin Planck's constant joules per hertz speed of light meters per second base of the natural logarithm, 2.718281... *dimensionless*Boltzmann's constant joules per kelvin
## Additional recommended knowledge
## OverviewThe wavelength is related to the frequency by -
^{[3]}
The law is sometimes written in terms of the spectral energy density which has units of energy per unit volume per unit frequency (joule per cubic meter per hertz). Integrated over frequency, this expression yields the total energy density. The radiation field of a black body may be thought of as a photon gas, in which case this energy density would be one of the thermodynamic parameters of that gas. The spectral energy density can also be expressed as a function of wavelength: as shown in the derivation below. Max Planck originally produced this law in 1900 (published Planck made this quantization assumption five years Although Planck's formula predicts that a black body will radiate energy at all frequencies, the formula is only applicable when many photons are being measured. For example, a black body at room temperature (300 kelvin) with one square meter of surface area will emit a photon in the visible range once about every thousand years or so, meaning that for most practical purposes, a black body at room temperature does not emit in the visible range. Ultimately, Planck's assumption of energy quantization and Einstein's photon hypothesis became the fundamental basis for the development of Quantum Mechanics. ## Derivation
Consider a cube of side where the The quantum number According to statistical mechanics, the probability distribution over the energy levels of a particular mode is given by: Here The denominator Here we have defined which is the energy of a single photon. As explained here, the average energy in a mode can be expressed in terms of the partition function: This formula is a special case of the general formula for particles obeying Bose-Einstein statistics. Since there is no restriction on the total number of photons, the chemical potential is zero. The total energy in the box now follows by summing over all allowed single photon states. This can be done exactly in the thermodynamic limit as To calculate the density of states we rewrite equation (1) as follows: where For every vector Inserting this in Eq. (2) gives: From this equation one easily derives the spectral energy density as a function of frequency where:
And: where This is also a spectral energy density function with units of energy per unit wavelength per unit volume. Integrals of this type for Bose and Fermi gases can be expressed in terms of polylogarithms. In this case, however, it is possible to calculate the integral in closed form using only elementary functions. Substituting in Eq. (3), makes the integration variable dimensionless giving: where We prove this result in the Appendix below. The total electromagnetic energy inside the box is thus given by: where which yields ## HistoryMany popular science accounts of quantum theory, as well as some physics textbooks, contain some serious errors in their discussions of the history of Planck's Law. Although these errors were pointed out over forty years ago by historians of physics, they have proved to be difficult to eradicate. An article by Helge Kragh Contrary to popular opinion, Planck did not quantize light. This is evident from his original 1901 paper Contrary to another myth, Planck did not derive his law in an attempt to resolve the "ultraviolet catastrophe", the name given by Paul Ehrenfest to the paradoxical result that the total energy in the cavity tends to infinity when the equipartition theorem of classical statistical mechanics is applied to black body radiation. Planck did not consider the equipartion theorem to be universally valid, so he never noticed any sort of "catastrophe" — it was only discovered some five years later by Einstein, Lord Rayleigh, and Sir James Jeans. ## AppendixA simple way to calculate the integral is to calculate the general case first and then compute the answer at the end. Consider the integral Multiplying the numerator and denominator by exp( − Since the denominator is always less than one, we can expand the denominator in powers of exp( − Then we have Since each term in the sum represents a convergent integral, remove the summation out from under the integral sign. In addition, by a change of variable such that The summation on the left is the Riemann zeta function ζ( For our problem, the numerator contains Here we have used that is the Riemann zeta function evaluated for the argument 4, which is given by π Where we see that the residue at zero is − π ## Notes**^**Rybicki, p. 22.**^**Rybicki, p. 22.**^**Rybicki, p. 1.- ^
^{a}^{b}Brehm, J.J. and Mullin, W.J., "Introduction to the Structure of Matter: A Course in Modern Physics," (Wiley, New York, 1989)__ISBN 047160531X__. - ^
^{a}^{b}Planck, Max, "On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik, vol. 4, p. 553 ff (1901) Planck's original 1901 paper. **^**Kragh, Helge Max Planck: The reluctant revolutionary Physics World, December 2000
## References- Rybicki, G. B., A. P. Lightman (1979).
*Radiative Processes in Astrophysics*. New York: John Wiley & Sons.__ISBN 0-471-82759-2__. - Thornton, Stephen T., Andrew Rex (2002).
*Modern Physics*. USA: Thomson Learning.__ISBN 0-03-006049-4__.
## Further readingPeter C. Milonni (1994). Categories: Statistical mechanics | Foundational quantum physics |
||||||||||||||||||||||||||||

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Planck's_law". A list of authors is available in Wikipedia. |