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Difference between revisions of "Planck constant"

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An absolute physical constant, having the dimension of action (energy<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p0727801.png" />time) in the CGS system. Planck's constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p0727802.png" /> is
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An absolute physical constant, having the dimension of action (energy$\times$time) in the CGS system. Planck's constant $h$ is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p0727803.png" /></td> </tr></table>
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$$6.626\,070\,040(81)\times10^{-34}\,\mathrm J\,\mathrm s$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p0727804.png" /> is the possible error in the measurement). It was first introduced by M. Planck (1900) in a paper on the radiation of light, in which he suggested that the energy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p0727805.png" /> of a photon (an electromagnetic wave) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p0727806.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p0727807.png" /> is the frequency of the wave. Later, when quantum mechanics arose, Planck's constant was used in the definition of major quantum-mechanical quantities (momentum and energy operators, etc.) and appeared in almost all equations of quantum mechanics.
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(the number in parentheses is the standard uncertainty referred to the corresponding last digits of the quoted numerical value). It was first introduced by M. Planck (1900) in a paper on the radiation of light, in which he suggested that the energy $E$ of a photon (an electromagnetic wave) is $E=h\nu$, where $\nu$ is the frequency of the wave. Later, when quantum mechanics arose, Planck's constant was used in the definition of major quantum-mechanical quantities (momentum and energy operators, etc.) and appeared in almost all equations of quantum mechanics.
  
Planck's constant characterizes in a certain sense the limits of the use of classical mechanics: The laws of quantum mechanics deviate substantially from those of classical mechanics only for physical systems for which the characteristic distances, velocities and masses are such that the corresponding characteristic action is of the same order as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p0727808.png" />. In a formal mathematical treatment this means that the equations of quantum mechanics go over to the corresponding classical equations as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p0727809.png" />.
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Planck's constant characterizes in a certain sense the limits of the use of classical mechanics: The laws of quantum mechanics deviate substantially from those of classical mechanics only for physical systems for which the characteristic distances, velocities and masses are such that the corresponding characteristic action is of the same order as $h$. In a formal mathematical treatment this means that the equations of quantum mechanics go over to the corresponding classical equations as $h\to0$.
  
The constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p07278010.png" /> may be replaced by the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072780/p07278011.png" />, which is also called Planck's constant.
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The constant $h$ may be replaced by the constant $\hbar=h/(2\pi)$, which is also called Planck's constant.
  
  

Revision as of 23:31, 30 November 2018

An absolute physical constant, having the dimension of action (energy$\times$time) in the CGS system. Planck's constant $h$ is

$$6.626\,070\,040(81)\times10^{-34}\,\mathrm J\,\mathrm s$$

(the number in parentheses is the standard uncertainty referred to the corresponding last digits of the quoted numerical value). It was first introduced by M. Planck (1900) in a paper on the radiation of light, in which he suggested that the energy $E$ of a photon (an electromagnetic wave) is $E=h\nu$, where $\nu$ is the frequency of the wave. Later, when quantum mechanics arose, Planck's constant was used in the definition of major quantum-mechanical quantities (momentum and energy operators, etc.) and appeared in almost all equations of quantum mechanics.

Planck's constant characterizes in a certain sense the limits of the use of classical mechanics: The laws of quantum mechanics deviate substantially from those of classical mechanics only for physical systems for which the characteristic distances, velocities and masses are such that the corresponding characteristic action is of the same order as $h$. In a formal mathematical treatment this means that the equations of quantum mechanics go over to the corresponding classical equations as $h\to0$.

The constant $h$ may be replaced by the constant $\hbar=h/(2\pi)$, which is also called Planck's constant.


Comments

References

[a1] L.I. Schiff, "Quantum mechanics" , McGraw-Hill (1968)
[a2] A. Messiah, "Quantum mechanics" , I-II , North-Holland (1961)
[a3] Th.T. Taylor, "Mechanics: classical and quantum" , Pergamon (1976) pp. 124ff
How to Cite This Entry:
Planck constant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Planck_constant&oldid=12737
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article