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Planck constant
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A commemoration plaque forMax Planck on his discovery of Planck‘s constant, in front ofHumboldt University,Berlin. English translation: "Max Planck, discoverer of the elementary quantum of action h, taught in this building from 1889 to 1928."
The Planck constant (denoted h) is aphysical constant that is used to describe the sizes ofquanta. It plays a central part in the theory ofquantum mechanics, and is named afterMax Planck, one of the founders of quantum theory. A closely related quantity is the reduced Planck constant (also known asDirac‘s constant and denoted, pronounced "h-bar"). The Planck constant is also used in measuring energy emitted asphotons, such as in the equation E = hν, where E is energy, h is Planck‘s constant, and ν (Greek letternu) is frequency.
The Planck constant and the reduced Planck constant are used to describe quantization, a phenomenon occurring in subatomicparticles such aselectrons andphotons in which certain physical properties occur in fixed amounts rather than assuming a continuous range of possible values.
Contents
[hide]
1 Significance of the size of Planck‘s constant
2 Units, value and symbols2.1 More recent values
3 Origins of Planck‘s constant
4 Usage
5 Dirac constant
6 See also
7 References
8 External links
[edit] Significance of the size of Planck‘s constant
Expressed in theSI units ofjoule seconds(J·s), the Planck constant is one of the smallest constants used inphysics. The significance of this is that it reflects the extremelysmall scales at which quantum mechanical effects are observed, andhence why we are not familiar with quantum physics in our everydaylives in the way that we are withclassical physics. Indeed, classical physics can essentially be defined as the limit of quantum mechanics as the Planck constant tends to zero.
Innatural units, the Dirac constant is taken as 1 (i.e., the Planck constant is 2·π), as is convenient for describing physics at the atomic scale dominated by quantum effects.
[edit] Units, value and symbols
The Planck constant has dimensions ofenergy multiplied bytime, which are also the dimensions ofaction. InSI units, the Planck constant is expressed injoule seconds (J·s). The dimensions may also be written asmomentum timesdistance (N·m·s), which are also the dimensions ofangular momentum. Often the unit of choice iseV·s, because of the small energies that are often encountered in quantum physics.
The value of the Planck constant is:
[1][2]
The two digits between theparentheses denote the standard uncertainty in the last two digits of the value.
The value of the Dirac constant is:

The figures cited here are the 2006CODATA-recommendedvalues for the constants and their uncertainties. The 2006 CODATAresults were made available in March 2007 and represent the best-known,internationally-accepted values for these constants, based on all dataavailable as of 31 December 2006. New CODATA figures are scheduled tobe published approximately every four years.
Unicode reserves codepoints U+210E (?) for the Planck constant, and U+210F (?) for the Dirac constant.
[edit] More recent values
In October 2005, the National Physical Laboratory reported initialmeasurements of the Planck constant using a newly improved wattbalance. They report a value of:

which is not consistent with the current CODATA value above (Robinson and Kibble 2007).
Many scientists hope this does not mean the Planck constant is increasing as the Universe expands.
[edit] Origins of Planck‘s constant
The Planck constant,, was proposed in reference to the problem ofblack-bodyradiation. The underlying assumption toPlanck‘s law of black body radiation was that the electromagnetic radiation emitted by a black body could be modeled as a set ofharmonic oscillators with quantized energy of the form:

is the quantized energy of thephotons of radiation havingfrequency (Hz) of(nu) orangular frequency (rad/s) of(omega).
This model proved extremely accurate, but it provided anintellectual stumbling block for theoreticians who did not understandwhere the quantization of energy arose — Planck himself only consideredit "a purely formal assumption"[citation needed]. This line of questioning helped lead to the formation ofquantum mechanics.
In addition to some assumptions underlying the interpretation ofcertain values in the quantum mechanical formulation, one of thefundamental corner-stones to the entire theory lies in thecommutator relationship between the position operatorand the momentum operator:

whereis theKronecker delta. For more information, see themathematical formulation of quantum mechanics.
[edit] Usage
The Planck constant is used to describe quantization. For instance, theenergy (E) carried by a beam oflight with constantfrequency () can only take on the values

It is sometimes more convenient to use theangular frequency, which gives

Many such "quantization conditions" exist. A particularly interesting condition governs the quantization ofangular momentum. Let J be the total angular momentum of a system with rotational invariance, and Jz the angular momentum measured along any given direction. These quantities can only take on the values

Thus,may be said to be the "quantum of angular momentum".
The Planck constant also occurs in statements ofHeisenberg‘suncertainty principle. Given a large number of particles prepared in the same state, theuncertainty in their position, Δx, and the uncertainty in their momentum (in the same direction), Δp, obey

where the uncertainty is given as thestandard deviation of the measured value from itsexpected value.
There are a number of other such pairs of physically measurable values which obey a similar rule.
[edit] Dirac constant
The Dirac constant or the "reduced Planck constant",, differs only from the Planck constant by a factor of 2π. The Planck constant is stated inSI units of measurement, joules perhertz, or joules per (cycle per second), while the Dirac constant is the same value stated in joules per (radian per second).
In essence, the Dirac constant is a conversion factor betweenphase (in radians) andaction (in joule-seconds) as seen in theSchrödinger equation.The Planck constant is similarly a conversion factor between phase (incycles) and action. All other uses of Planck‘s constant and Dirac‘sconstant follow from that.
[edit] See also
Planck units
Electromagnetic radiation
Natural units
Schrödinger equation
Wave–particle duality
Quantum Hall effect
[edit] References
^Planck constant. 2006CODATA recommended values.NIST. Retrieved on2007-08-08.
^Planck constant in eV s. 2006CODATA recommended values.NIST. Retrieved on2007-08-08.
Barrow, John D. (2002). The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe (in English). Pantheon Books.ISBN 0-375-42221-8.
Robinson, I. A.; B. P. Kibble (October 2007). "An initial measurement of Planck’s constant using the NPL Mark II watt balance". Metrologia 44: 427-440.doi:10.1088/0026-1394/44/6/001. Retrieved on2007-11-25.
[edit] External links
Planck‘s original 1901 paper