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| {{redirect|Planck's relation|the law governing black body radiation|Planck's law}}
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| {| class="wikitable" style="float:right; margin:0em 1em 1em 1em"
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| ! Values of ''h''
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| | align=right |{{val|6.62606957|(29)|e=-34}} ||align=center | [[joule|J]]·[[second|s]] || align=center | <ref name="2010 CODATA">P.J. Mohr, B.N. Taylor, and D.B. Newell (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 6.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: [http://physics.nist.gov/cgi-bin/cuu/Results?search_for=planck http://physics.nist.gov] [Thursday, 02-Jun-2011 21:00:12 EDT]. National Institute of Standards and Technology, Gaithersburg, MD 20899.</ref>
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| | align=right |{{val|4.135667516|(91)|e=-15}} || align=center |[[electron-volt|eV]]·[[second|s]] || align=center | <ref name="2010 CODATA" />
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| | align=center |2π|| align=center | [[Planck Energy|''E''<sub>P</sub>]]·[[Planck Time|''t<sub>P</sub>'']] || align=center |
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| ! Values of ''ħ''
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| | align=right |{{val|1.054571726|(47)|e=-34}} || align=center |[[joule|J]]·[[second|s]] || align=center | <ref name="2010 CODATA" />
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| | align=right |{{val|6.58211928|(15)|e=-16}} || align=center | [[electron-volt|eV]]·[[second|s]] || align=center | <ref name="2010 CODATA" />
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| | align=center |{{val|1|}} || align=center | [[Planck Energy|''E''<sub>P</sub>]]·[[Planck Time|''t<sub>P</sub>'']] || align=center | def
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| | align=right | {{val|1.98644568||e=-25}} || align=center | [[joule|J]]·[[metre|m]] ||
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| | align=right |{{val|1.23984193}} || align=center |[[electron volt|eV]]·[[micrometre|μm]] || align=center |
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| | align=center |2π|| align=center | [[Planck Energy|''E''<sub>P</sub>]]·[[Planck Length|<big>ℓ</big><sub>P</sub>]] || align=center |
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| [[File:MaxPlanckWirkungsquantums20050815 CopyrightKaihsuTai.jpg|thumb|right|Plaque at the [[Humboldt University of Berlin]]: "Max Planck, discoverer of the elementary quantum of action ''h'', taught in this building from 1889 to 1928."]]
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| The '''Planck constant''' (denoted '''''h''''', also called '''Planck's constant''') is a [[physical constant]] that is the [[quantum]] of [[action (physics)|action]] in [[quantum mechanics]]. The Planck constant was first described as the [[proportionality constant]] between the [[energy]] (''E'') of a charged atomic oscillator in the wall of a [[black body]], and the [[frequency]] ([[Nu (letter)|''ν'']]) of its associated [[electromagnetic wave]]. This relation between the energy and frequency is called the '''Planck relation''':
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| :<math>E = h\nu \,.</math>
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| In 1905 the value (E), the energy of a charged atomic oscillator, was theoretically associated with the energy of the electromagnetic wave itself, representing the minimum amount of energy required to form an electromagnetic field (a "quantum"). Further investigation of quanta revealed behaviour associated with an independent unit ("particle") as opposed to an electromagnetic wave and was eventually given the term [[photon]]. The Planck relation now describes the energy of each photon in terms of the photon's frequency. This energy is extremely small in terms of ordinary experience.
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| Since the [[frequency]] ''<math>\nu</math>'', [[wavelength]] ''λ'', and [[speed of light]] ''c'' are related by {{nowrap|''λν'' {{=}} ''c''}}, the Planck relation for a photon can also be expressed as
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| :<math>E = \frac{hc}{\lambda}.\,</math> | |
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| The above equation leads to another relationship involving the Planck constant. Given ''p'' for the linear [[momentum]] of a particle (not only a photon, but other particles as well), the [[de Broglie wavelength]] λ of the particle is given by
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| :<math>\lambda = \frac{h}{p}.</math>
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| In applications where frequency is expressed in terms of [[radian]]s per second ("[[angular frequency]]") instead of [[cycle per second|cycles per second]], it is often useful to absorb a factor of 2[[Pi|π]] into the Planck constant. The resulting constant is called the '''reduced Planck constant''' or '''Dirac constant'''. It is equal to the Planck constant divided by 2π, and is denoted ''ħ'' (or "'''h-bar'''", as it is often also called):
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| :<math>\hbar = \frac{h}{2 \pi}.</math>
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| The energy of a photon with angular frequency ''ω'', where ''ω'' = 2π''ν'', is given by
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| :<math>E = \hbar \omega.</math> | |
| The reduced Planck constant is the quantum of [[angular momentum]] in quantum mechanics.
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| The Planck constant is named after [[Max Planck]], the founder of [[quantum mechanics|quantum theory]], who discovered it in 1900, and who coined the term "Quantum". Classical [[statistical mechanics]] requires the existence of ''h'' (but does not define its value).<ref>{{Citation|title=Statistical mechanics: an intermediate course
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| |author= Giuseppe Morandi, F. Napoli, E. Ercolessi|quote=See page 85|url=http://books.google.com/?id=MhInFlnNsREC&pg=PA51&lpg=PA51&dq=celestial+mechanics+planck+constant#v=onepage&q=celestial%20mechanics%20planck%20constant&f=false|isbn=978-981-02-4477-4|year=2001}}</ref> Planck discovered that physical [[Action (physics)|action]] could not take on any indiscriminate value. Instead, the action must be some multiple of a very small quantity (later to be named the "[[quantum]] of action" and now called Planck constant). This inherent [[granularity]] is counterintuitive in the everyday world, where it is possible to "make things a little bit hotter" or "move things a little bit faster". This is because the quanta of action are very, very small in comparison to everyday [[macroscopic scale|macroscopic]] human experience. Hence, the granularity of nature appears smooth to us.
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| Thus, on the macroscopic scale, quantum mechanics and classical physics converge at the [[classical limit]]. Nevertheless, it is impossible, as Planck discovered, to explain some phenomena without accepting the fact that action is quantized. In many cases, such as for monochromatic light or for atoms, this quantum of action also implies that only certain energy levels are allowed, and values in-between are forbidden.<ref>{{Citation|last=Einstein|title=Physics and Reality|page= 24|first=Albert|quote=The question is first: How can one assign a discrete succession of energy value H<sub>σ</sub> to a system specified in the sense of classical mechanics (the energy function is a given function of the coordinates q<sub>r</sub> and the corresponding momenta p<sub>r</sub>)? Planck constant ''h'' relates the frequency ''H<sub>σ</sub>/h'' to the energy values ''H<sub>σ</sub>''. It is therefore sufficient to give to the system a succession of discrete frequency values.|url=http://www.kostic.niu.edu/Physics_and_Reality-Albert_Einstein.pdf|doi=10.1162/001152603771338742|year=2003|journal=Daedalus|volume=132|issue=4}}</ref> In 1923, [[Louis de Broglie]] generalized the Planck relation by postulating that the Planck constant represents the proportionality between the momentum and the quantum wavelength of not just the photon, but the quantum wavelength of any particle. This was confirmed by experiments soon afterwards.
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| ==Value==
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| {{See also|New SI definitions}}
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| The Planck constant of [[action (physics)|action]] has the [[dimensional analysis|dimensionality]] of [[specific relative angular momentum]] (areal momentum) or [[angular momentum]]'s intensity. In [[International System of Units|SI units]], the Planck constant is expressed in [[joule second]]s ({{nowrap|J·s}}) or ({{nowrap|[[Newton (unit)|N]]·[[metre|m]]·[[second|s]]}}).
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| The value of the Planck constant is:<ref name="2010 CODATA" />
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| :<math>h = 6.626\ 069\ 57(29)\times 10^{-34}\ \mathrm{J \cdot s} = 4.135\ 667\ 516(91)\times 10^{-15}\ \mathrm{eV \cdot s}.</math>
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| The value of the reduced Planck constant is:
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| :<math>\hbar = {{h}\over{2\pi}} = 1.054\ 571\ 726(47)\times 10^{-34}\ \mathrm{J \cdot s} = 6.582\ 119\ 28(15)\times 10^{-16}\ \mathrm{eV \cdot s}.</math>
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| The two digits inside the parentheses denote the [[measurement uncertainty|standard uncertainty]] in the last two digits of the value. The figures cited here are the 2010 [[CODATA]] recommended values for the constants and their uncertainties. The 2010 CODATA results were made available in June 2011<ref>{{cite web|url=http://physics.nist.gov/cuu/Reference/versioncon.shtml|title=CODATA recommended values}}</ref> and represent the best-known, internationally-accepted values for these constants, based on all data available as of 2010. New CODATA figures are scheduled to be published approximately every four years.
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| ==Significance of the value==
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| The Planck constant is related to the quantization of light and matter. Therefore, the Planck constant can be seen as a [[subatomic]]-scale constant. In a unit system adapted to subatomic scales, the [[electronvolt]] is the appropriate unit of energy and the [[hertz|petahertz]] the appropriate unit of frequency. [[Atomic unit]] systems are based (in part) on the Planck constant.
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| The numerical value of the Planck constant depends entirely on the system of units used to measure it. When it is expressed in SI units, it is one of the smallest constants used in physics. This reflects the fact that ''on a scale adapted to humans'', where energies are typically of the order of kilojoules and times are typically of the order of seconds or minutes, Planck constant (the quantum of action) is very small.
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| Equivalently, the smallness of Planck constant reflects the fact that everyday objects and systems are made of a ''large'' number of particles. For example, green light with a [[wavelength]] of 555 [[nanometre]]s (the approximate wavelength to which human eyes are most sensitive) has a frequency of 540 THz (540{{e|12}} [[Hertz|Hz]]). Each [[photon]] has an energy ''E'' of ''hν'' = 3.58{{e|−19}} J. That is a very small amount of energy in terms of everyday experience, but everyday experience is not concerned with individual photons any more than with individual atoms or molecules. An amount of light compatible with everyday experience
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| is the energy of one [[mole (unit)|mole]] of photons; its energy can be calculated by multiplying the photon energy by the [[Avogadro constant]], ''N''<sub>A</sub> ≈ {{nowrap|6.022{{e|23}} mol<sup>−1</sup>}}. The result is that green light of wavelength 555 nm has an energy of 216 kJ/mol, a typical energy of everyday life.
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| This energy is comparable in strength to some of the weaker [[bond dissociation energy|chemical bonds]] such as with [[iodine]], while that of ultraviolet light at 254 nm from a [[mercury-vapor lamp]] is a stronger 472 kJ/mol suiting it to the [[homolysis|homolytic]] action of [[ultraviolet germicidal irradiation]].
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| When the reduced Planck constant is treated as a conversion factor between [[phase (waves)|phase]], in radians, and [[action (physics)|action]], in joule-seconds (as seen in the [[Schrödinger equation]]), it may be written with units J·s/rad.
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| ==Origins==
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| ===Black-body radiation===
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| {{Main|Planck's law}}
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| [[File:Wiens law.svg|thumb|right|250px|Intensity of light emitted from a [[black body]] at any given frequency. Each color is a different temperature. Planck was the first to explain the shape of these curves.]]
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| In the last years of the nineteenth century, Planck was investigating the problem of [[Black body|black-body radiation]] first posed by [[Gustav Kirchhoff|Kirchhoff]] some forty years earlier. It is well known that hot objects glow, and that hotter objects glow brighter than cooler ones. The reason is that the electromagnetic field obeys laws of motion just like a mass on a spring, and can come to thermal equilibrium with hot atoms. When a hot object is in equilibrium with light, the amount of light it absorbs is equal to the amount of light it emits. If the object is black, meaning it absorbs all the light that hits it, then it emits the maximum amount of thermal light too.
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| The assumption that blackbody radiation is thermal leads to an accurate prediction: the total amount of emitted energy goes up with the temperature according to a definite rule, the [[Stefan–Boltzmann law]] (1879–84). But it was also known that the colour of the light given off by a hot object changes with the temperature, so that "white hot" is hotter than "red hot". Nevertheless, [[Wilhelm Wien]] discovered the mathematical relationship between the peaks of the curves at different temperatures, by using the principle of [[adiabatic invariant|adiabatic invariance]]. At each different temperature, the curve is moved over by [[Wien's displacement law]] (1893). Wien also proposed an [[Wien approximation|approximation]] for the spectrum of the object, which was correct at high frequencies (short wavelength) but not at low frequencies (long wavelength).<ref name="bowleysanchez1999" >{{citation
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| | author=R. Bowley, M. Sánchez
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| | year=1999
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| | title=Introductory Statistical Mechanics
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| | edition=2nd
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| | publisher=Clarendon Press
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| | location=Oxford
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| | isbn=0-19-850576-0}}</ref> It still was not clear ''why'' the spectrum of a hot object had the form that it has (see diagram).
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| Planck hypothesized that the equations of motion for light are a set of [[harmonic oscillator]]s, one for each possible frequency. He examined how the [[entropy]] of the oscillators varied with the temperature of the body, trying to match Wien's law, and was able to derive an approximate mathematical function for black-body spectrum.<ref name="Planck01">{{citation | first = Max | last = Planck | author-link = Max Planck | title = Ueber das Gesetz der Energieverteilung im Normalspectrum | url = http://www.physik.uni-augsburg.de/annalen/history/historic-papers/1901_309_553-563.pdf | journal = [[Annalen der Physik|Ann. Phys.]] | year = 1901 | volume = 309 | issue = 3 | pages = 553–63 | doi = 10.1002/andp.19013090310|bibcode = 1901AnP...309..553P }}. English translation: "[http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html On the Law of Distribution of Energy in the Normal Spectrum]".</ref>
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| However, Planck soon realized that his solution was not unique. There were several different solutions, each of which gave a different value for the entropy of the oscillators.<ref name="Planck01"/> To save his theory, Planck had to resort to using the then controversial theory of [[statistical mechanics]],<ref name="Planck01" /> which he described as "an act of despair … I was ready to sacrifice any of my previous convictions about physics."<ref name="Kragh">{{citation | first = Helge | last = Kragh | url = http://physicsworld.com/cws/article/print/373 | title = Max Planck: the reluctant revolutionary | publisher = PhysicsWorld.com | date = 1 December 2000}}</ref> One of his new boundary conditions was
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| {{quote|text=to interpret ''U''<sub>N</sub> [''the vibrational energy of N oscillators''] not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts. Let us call each such part the energy element ε;|sign=Planck|source=On the Law of Distribution of Energy in the Normal Spectrum<ref name="Planck01" />}}
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| With this new condition, Planck had imposed the quantization of the energy of the oscillators, "a purely formal assumption … actually I did not think much about it…" in his own words,<ref>{{citation | title = Quantum Generations: A History of Physics in the Twentieth Century | first = Helge | last = Kragh | year = 1999 | publisher = Princeton University Press | isbn = 0-691-09552-3 | page = 62 | url = http://books.google.com/?id=ELrFDIldlawC&printsec=frontcover}}</ref> but one which would revolutionize physics. Applying this new approach to Wien's displacement law showed that the "energy element" must be proportional to the frequency of the oscillator, the first version of what is now termed "Planck's relation":
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| :<math>E = h\nu.\,</math>
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| Planck was able to calculate the value of ''h'' from experimental data on black-body radiation: his result, {{nowrap|6.55 × 10<sup>−34</sup> J·s}}, is within 1.2% of the currently accepted value.<ref name="Planck01" /> He was also able to make the first determination of the [[Boltzmann constant]] ''k''<sub>B</sub> from the same data and theory.<ref name="PlanckNobel">{{citation | first = Max | last = Planck | author-link = Max Planck | title = The Genesis and Present State of Development of the Quantum Theory (Nobel Lecture) | url = http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html | date = 2 June 1920}}</ref>
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| [[File:Black body.svg|350px|thumb|Note that the (black) Raleigh-Jeans curve never touches the Planck curve.]]
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| Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a [[Continuous function|continuous variable]]. The [[Rayleigh-Jeans law]] makes close predictions for a narrow range of values at one limit of temperatures, but the results diverge more and more strongly as temperatures increase. To make [[Planck's law]], which correctly predicts blackbody emissions, it was necessary to multiply the classical expression by a complex factor that involves '''h''' in both the numerator and the denominator. The influence of '''h''' in this complex factor would not disappear if it were set to zero or to any other value. Making an equation out of Planck's law that would reproduce the Rayleigh-Jeans law could not be done by changing the values of '''h''', of the Boltzmann constant, or of any other constant or variable in the equation. In this case the picture given by classical physics is not duplicated by a range of results in the quantum picture.
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| The black-body problem was revisited in 1905, when [[John Strutt, 3rd Baron Rayleigh|Rayleigh]] and [[James Hopwood Jeans|Jeans]] (on the one hand) and [[Einstein]] (on the other hand) independently proved that classical electromagnetism could ''never'' account for the observed spectrum. These proofs are commonly known as the "[[ultraviolet catastrophe]]", a name coined by [[Paul Ehrenfest]] in 1911. They contributed greatly (along with Einstein's work on the [[photoelectric effect]]) to convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The very first [[Solvay Conference]] in 1911 was devoted to "the theory of radiation and quanta".<ref>{{citation | url = http://www.solvayinstitutes.be/Conseils%20Solvay/PreviousPhysics.html | title = Previous Solvay Conferences on Physics | accessdate = 12 December 2008 | publisher = International Solvay Institutes}}</ref> Max Planck received the 1918 [[Nobel Prize in Physics]] "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".
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| ===Photoelectric effect===
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| {{Main|Photoelectric effect}}
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| The photoelectric effect is the emission of electrons (called "photoelectrons") from a surface when light is shone on it. It was first observed by [[Alexandre Edmond Becquerel]] in 1839, although credit is usually reserved for [[Heinrich Hertz]],<ref name="Nobel21">See, e.g., {{citation | title = Presentation speech of the 1921 Nobel Prize for Physics | url = http://nobelprize.org/nobel_prizes/physics/laureates/1921/press.html | first = Svante | last = Arrhenius | author-link = Svante Arrhenius | date = 10 December 1922}}</ref> who published the first thorough investigation in 1887. Another particularly thorough investigation was published by [[Philipp Lenard]] in 1902.<ref name="Lenard">{{citation | first = P. | last = Lenard | author-link = Philipp Lenard | title = Ueber die lichtelektrische Wirkung | journal = [[Annalen der Physik|Ann. Phys.]] | volume = 313 | issue = 5 | pages = 149–98 | year = 1902 | doi = 10.1002/andp.19023130510|bibcode = 1902AnP...313..149L }}</ref> Einstein's 1905 paper<ref>{{Citation |last = Einstein | first = Albert | author-link = Albert Einstein | year = 1905 | title = Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt | url = http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_132-148.pdf | journal = [[Annalen der Physik|Ann. Phys.]] | volume = 17 |issue = 6 | pages = 132–48 | doi = 10.1002/andp.19053220607|bibcode = 1905AnP...322..132E }}</ref> discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921,<ref name="Nobel21" /> when his predictions had been confirmed by the experimental work of [[Robert Andrews Millikan]].<ref name="Millikan">{{citation | first = R. A. | last = Millikan |author-link = Robert Andrews Millikan | title = A Direct Photoelectric Determination of Planck's '''h''' | journal = [[Physical Review|Phys. Rev.]] | year = 1916 | volume = 7 | issue = 3 | pages = 355–88 | doi = 10.1103/PhysRev.7.355|bibcode = 1916PhRv....7..355M }}</ref> The Nobel committee awarded the prize for his work on the photo-electric effect, rather than relativity, both because of a bias against purely theoretical physics not grounded in discovery or experiment, and dissent amongst its members as to the actual proof that relativity was real.<ref>{{Citation|url=http://books.google.com/books/about/Einstein.html?id=cdxWNE7NY6QC|title=Einstein: His Life and Universe|isbn=1416539328|author1=Isaacson|first1=Walter|date=2007-04-10}}, pp. 309–314.</ref>
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| Prior to Einstein's paper, electromagnetic radiation such as visible light was considered to behave as a wave: hence the use of the terms "frequency" and "wavelength" to characterise different types of radiation. The energy transferred by a wave in a given time is called its [[intensity (physics)|intensity]]. The light from a theatre spotlight is more ''intense'' than the light from a domestic lightbulb; that is to say that the spotlight gives out more energy per unit time (and hence consumes more electricity) than the ordinary bulb, even though the colour of the light might be very similar. Other waves, such as sound or the waves crashing against a seafront, also have their own intensity. However the energy account of the photoelectric effect didn't seem to agree with the wave description of light.
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| The "photoelectrons" emitted as a result of the photoelectric effect have a certain [[kinetic energy]], which can be measured. This kinetic energy (for each photoelectron) is ''independent'' of the intensity of the light,<ref name="Lenard" /> but depends linearly on the frequency;<ref name="Millikan" /> and if the frequency is too low (corresponding to a kinetic energy for the photoelectrons of zero or less), no photoelectrons are emitted at all, unless a plurality of photons, whose energetic sum is greater than the energy of the photoelectrons, acts virtually simultaneously (multiphoton effect) <ref>{{Citation|last=Smith |title=Two Photon Photoelectric Effect|doi=10.1103/PhysRev.128.2225 |volume=128|page=2225|year=1962|postscript=.|first1=Richard|journal=Physical Review|issue=5|bibcode = 1962PhRv..128.2225S }}{{Citation|doi=10.1103/PhysRev.130.2599.4|title=Two-Photon Photoelectric Effect|year=1963|last1=Smith|first1=Richard|journal=Physical Review|volume=130|issue=6|pages=2599|postscript=.|bibcode = 1963PhRv..130.2599S }}</ref> Assuming the frequency is high enough to cause the photoelectric effect, a rise in intensity of the light source causes more photoelectrons to be emitted with the same kinetic energy, rather than the same number of photoelectrons to be emitted with higher kinetic energy.<ref name="Lenard" />
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| Einstein's explanation for these observations was that light itself is quantized; that the energy of light is not transferred continuously as in a classical wave, but only in small "packets" or quanta. The size of these "packets" of energy, which would later be named [[photon]]s, was to be the same as Planck's "energy element", giving the modern version of Planck's relation:
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| :<math>E = h\nu.\,</math>
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| Einstein's postulate was later proven experimentally: the constant of proportionality between the frequency of incident light (''ν'') and the kinetic energy of photoelectrons (''E'') was shown to be equal to the Planck constant (''h'').<ref name="Millikan" />
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| ===Atomic structure===
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| {{Main|Bohr model}}
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| [[File:Bohr-atom-PAR.svg|thumb|right|A schematization of the Bohr model of the hydrogen atom. The transition shown from the ''n''=3 level to the ''n''=2 level gives rise to visible light of wavelength 656 nm (red), as the model predicts.]]
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| [[Niels Bohr]] introduced the first quantized model of the atom in 1913, in an attempt to overcome a major shortcoming of [[Ernest Rutherford|Rutherford's]] classical model.<ref name="Bohr">{{citation | first = Niels | last = Bohr | author-link = Niels Bohr | title = On the Constitution of Atoms and Molecules | journal = [[Philosophical Magazine|Phil. Mag.]], Ser. 6 | year = 1913 | volume = 26 | issue = 153 | pages = 1–25 | doi = 10.1080/14786441308634993 }}</ref> In classical electrodynamics, a charge moving in a circle should radiate electromagnetic radiation. If that charge were to be an electron orbiting a [[Atomic nucleus|nucleus]], the radiation would cause it to lose energy and spiral down into the nucleus. Bohr solved this paradox with explicit reference to Planck's work: an electron in a Bohr atom could only have certain defined energies ''E<sub>n</sub>''
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| :<math>E_n = -\frac{h c_0 R_{\infty}}{n^2}</math>
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| where c<sub>0</sub> is the speed of light in vacuum, ''R''<sub>∞</sub> is an experimentally-determined constant (the [[Rydberg constant]]) and ''n'' is any integer (''n'' = 1, 2, 3, …). Once the electron reached the lowest energy level ({{nowrap|''n'' {{=}} 1}}), it could not get any closer to the nucleus (lower energy). This approach also allowed Bohr to account for the [[Rydberg formula]], an empirical description of the atomic spectrum of hydrogen, and to account for the value of the Rydberg constant ''R''<sub>∞</sub> in terms of other fundamental constants.
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| Bohr also introduced the quantity ''h''/2π, now known as the '''reduced Planck constant''', as the quantum of angular momentum. At first, Bohr thought that this was the angular momentum of each electron in an atom: this proved incorrect and, despite developments by Sommerfeld and others, an accurate description of the electron angular momentum proved beyond the Bohr model. The correct quantization rules for electrons – in which the energy reduces to the Bohr-model equation in the case of the hydrogen atom – were given by Heisenberg's [[matrix mechanics]] in 1925 and the [[Schrödinger wave equation]] in 1926: the reduced Planck constant remains the fundamental quantum of angular momentum. In modern terms, if ''J'' is the total angular momentum of a system with rotational invariance, and ''J<sub>z</sub>'' the angular momentum measured along any given direction, these quantities can only take on the values
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| :<math>
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| \begin{align}
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| J^2 = j(j+1) \hbar^2,\qquad & j = 0, \tfrac{1}{2}, 1, \tfrac{3}{2}, \ldots, \\
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| J_z = m \hbar, \qquad\qquad\quad & m = -j, -j+1, \ldots, j.
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| \end{align}
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| </math>
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| ===Uncertainty principle===
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| {{Main|Uncertainty principle}}
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| The Planck constant also occurs in statements of [[Werner Heisenberg]]'s uncertainty principle. Given a large number of particles prepared in the same state, the [[uncertainty]] in their position, Δ''x'', and the uncertainty in their momentum (in the same direction), Δ''p'', obey
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| :<math> \Delta x\, \Delta p \ge \frac{\hbar}{2}</math>
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| where the uncertainty is given as the [[standard deviation]] of the measured value from its [[expected value]]. There are a number of other such pairs of physically measurable values which obey a similar rule. One example is time vs. energy. The either-or nature of uncertainty forces measurement attempts to choose between trade offs, and given that they are quanta, the trade offs often take the form of either-or (as in [[Fourier analysis]]), rather than the compromises and gray areas of [[time series]] analysis.
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| In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental cornerstones to the entire theory lies in the [[commutator]] relationship between the position operator <math>\hat{x}</math> and the momentum operator <math>\hat{p}</math>:
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| :<math>[\hat{p}_i, \hat{x}_j] = -i \hbar \delta_{ij}</math>
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| where δ<sub>ij</sub> is the [[Kronecker delta]].
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| ==Dependent physical constants==
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| The following list is based on the 2006 CODATA evaluation;<ref name="CODATA">{{CODATA2006|url=http://physics.nist.gov/cgi-bin/cuu/Value?h}}</ref> for the constants listed below, more than 90% of the uncertainty is due to the uncertainty in the value of the Planck constant, as indicated by the square of the [[Pearson product-moment correlation coefficient|correlation coefficient]] (''r''<sup>2</sup> > 0.9, ''r'' > 0.949). The Planck constant is (with one or two exceptions)<ref>The main exceptions are the [[Newtonian constant of gravitation]] ''G'' and the [[gas constant]] ''R''. The uncertainty in the value of the gas constant also affects those physical constants which are related to it, such as the [[Boltzmann constant]] and the [[Loschmidt constant]].</ref> the fundamental physical constant which is known to the lowest level of precision, with a [[measurement uncertainty|relative uncertainty]] ''u''<sub>r</sub> of 5.0{{e|−8}}.
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| ===Rest mass of the electron===
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| The normal textbook derivation of the [[Rydberg constant]] ''R''<sub>∞</sub> defines it in terms of the electron mass ''m''<sub>e</sub> and a variety of other physical constants.
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| :<math>R_\infty = \frac{m_{\rm e} e^4}{8 \epsilon_0^2 h^3 c_0} = \frac{m_{\rm e} c_0 \alpha^2}{2 h}</math>
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| However, the Rydberg constant can be determined very accurately (''u''<sub>r</sub> = 6.6{{e|−12}}) from the atomic spectrum of hydrogen, whereas there is no direct method to measure the mass of a stationary electron in SI units. Hence the equation for the calculation of ''m''<sub>e</sub> becomes
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| :<math>m_{\rm e} = \frac{2 R_{\infty} h}{c_0 \alpha^2}</math>
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| where ''c''<sub>0</sub> is the speed of light and ''α'' is the [[fine-structure constant]]. The speed of light has an exactly defined value in SI units, and the fine-structure constant can be determined more accurately (''u''<sub>r</sub> = 6.8{{e|−10}}) than the Planck constant: the uncertainty in the value of the electron rest mass is due entirely to the uncertainty in the value of the Planck constant (''r''<sup>2</sup> > 0.999).
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| ===Avogadro constant===
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| {{Main|Avogadro constant}}
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| The Avogadro constant ''N''<sub>A</sub> is determined as the ratio of the mass of one mole of electrons to the mass of a single electron: The mass of one mole of electrons is the "[[relative atomic mass]]" of an electron ''A''<sub>r</sub>(e), which can be measured in a [[Penning trap]] (''u''<sub>r</sub> = 4.2{{e|−10}}), multiplied by the [[molar mass constant]] ''M''<sub>u</sub>, which is defined as 0.001 kg/mol.
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| :<math>N_{\rm A} = \frac{M_{\rm u} A_{\rm r}({\rm e})}{m_{\rm e}} = \frac{M_{\rm u} A_{\rm r}({\rm e}) c_0 \alpha^2}{2 R_{\infty} h}</math>
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| The dependence of the Avogadro constant on the Planck constant (''r''<sup>2</sup> > 0.999) also holds for the physical constants which are related to amount of substance, such as the [[atomic mass constant]]. The uncertainty in the value of the Planck constant limits the knowledge of the masses of atoms and subatomic particles when expressed in SI units. It is possible to measure the masses more precisely in [[atomic mass unit]]s, but not to convert them more precisely into [[kilogram]]s.
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| ===Elementary charge===
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| {{Main|Elementary charge}}
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| [[Arnold Sommerfeld|Sommerfeld]] originally defined the fine-structure constant ''α'' as:
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| :<math>\alpha\ =\ \frac{e^2}{\hbar c_0 \ 4 \pi \epsilon_0}\ =\ \frac{e^2 c_0 \mu_0}{2 h}</math>
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| where ''e'' is the [[elementary charge]], ''ε''<sub>0</sub> is the [[electric constant]] (also called the [[permittivity]] of free space), and ''μ''<sub>0</sub> is the [[magnetic constant]] (also called the [[permeability (electromagnetism)|permeability]] of free space). The latter two constants have fixed values in the International [[SI|System of Units]]. However, ''α'' can also be determined experimentally, notably by measuring the [[g-factor (physics)|electron spin g-factor]] ''g''<sub>e</sub>, then comparing the result with the value predicted by [[quantum electrodynamics]].
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| At present, the most precise value for the elementary charge is obtained by rearranging the definition of ''α'' to obtain the following definition of ''e'' in terms of ''α'' and ''h'':
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| :<math>e = \sqrt{\frac{2\alpha h}{\mu_0 c_0}} = \sqrt{{2\alpha h \epsilon_0 c_0}}.</math>
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| ===Bohr magneton and nuclear magneton===
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| {{Main|Bohr magneton|Nuclear magneton}}
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| The Bohr magneton and the nuclear magneton are units which are used to describe the magnetic properties of the electron and atomic nuclei respectively. The Bohr magneton is the [[magnetic moment]] which would be expected for an electron if it behaved as a spinning charge according to classical electrodynamics. It is defined in terms of the reduced Planck constant, the elementary charge and the electron mass, all of which depend on the Planck constant: the final dependence on ''h''<sup>½</sup> (''r''<sup>2</sup> > 0.995) can be found by expanding the variables.
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| :<math>\mu_{\rm B} = \frac{e \hbar}{2 m_{\rm e}} = \sqrt{\frac{c_0 \alpha^5 h}{32 \pi^2 \mu_0 R_{\infty}^2}}</math>
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| The nuclear magneton has a similar definition, but corrected for the fact that the proton is much more massive than the electron. The ratio of the electron relative atomic mass to the proton relative atomic mass can be determined experimentally to a high level of precision (''u''<sub>r</sub> = 4.3{{e|−10}}).
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| :<math>\mu_{\rm N} = \mu_{\rm B} \frac{A_{\rm r}({\rm e})}{A_{\rm r}({\rm p})}</math>
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| ==Determination==
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| {| class="wikitable" style="float:left; width:50%;"
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| |-
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| ! Method
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| ! Value of ''h''<br />(10<sup>−34</sup> J·s)
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| ! Relative<br />uncertainty
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| ! Ref.
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| |-
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| | Watt balance
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| | {{val|6.62606889|(23)}}
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| | align=center | 3.4{{e|−8}}
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| | <ref>{{citation | year = 1990 | title = A Realization of the SI Watt by the NPL Moving-coil Balance | journal = [[Metrologia]] | volume = 27 | issue = 4 | pages = 173–92 | doi = 10.1088/0026-1394/27/4/002|bibcode = 1990Metro..27..173K | last1 = Kibble | first1 = B P | last2 = Robinson | first2 = I A | last3 = Belliss | first3 = J H }}</ref><ref>{{citation | author = Steiner, R.; Newell, D.; Williams, E. | year = 2005 | title = Details of the 1998 Watt Balance Experiment Determining the Planck Constant | url = http://nvl.nist.gov/pub/nistpubs/jres/110/1/j110-1ste.pdf | journal = Journal of Research|publisher= National Institute of Standards and Technology | volume = 110 | issue = 1 | pages = 1–26}}</ref><ref name="NIST">{{citation | year = 2007 | title = Uncertainty Improvements of the NIST Electronic Kilogram | volume = 56 | issue = 2 | pages = 592–96 | doi = 10.1109/TIM.2007.890590 | last1 = Steiner | first1 = Richard L. | last2 = Williams | first2 = Edwin R. | last3 = Liu | first3 = Ruimin | last4 = Newell | first4 = David B. | journal = IEEE Transactions on Instrumentation and Measurement}}</ref>
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| |-
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| | X-ray crystal density
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| | {{val|6.6260745|(19)}}
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| | align=center | 2.9{{e|−7}}
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| | <ref>{{citation| year = 2005 | title = Present state of the avogadro constant determination from silicon crystals with natural isotopic compositions | journal = IEEE Transactions on Instrumentation and Measurement| volume = 54 | issue = 2 | pages = 854–59 | doi = 10.1109/TIM.2004.843101| last1 = Fujii| first1 = K.| last2 = Waseda| first2 = A.| last3 = Kuramoto| first3 = N.| last4 = Mizushima| first4 = S.| last5 = Becker| first5 = P.| last6 = Bettin| first6 = H.| last7 = Nicolaus| first7 = A.| last8 = Kuetgens| first8 = U.| last9 = Valkiers| first9 = S.|displayauthors=333| last10 = Taylor| first10 = P.| last11 = Debievre| first11 = P.| last12 = Mana| first12 = G.| last13 = Massa| first13 = E.| last14 = Matyi| first14 = R.| last15 = Kessler| first15 = E.G.| last16 = Hanke| first16 = M.}}</ref>
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| |-
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| | Josephson constant
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| | {{val|6.6260678|(27)}}
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| | align=center | 4.1{{e|−7}}
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| | <ref>{{citation | author = Sienknecht, Volkmar; Funck, Torsten | year = 1985 | title = Determination of the SI Volt at the PTB | journal = IEEE Trans. Instrum. Meas. | volume = 34 | issue = 2 | pages = 195–98 | doi = 10.1109/TIM.1985.4315300}}. {{citation | year = 1986 | title = Realization of the SI Unit Volt by Means of a Voltage Balance |
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| journal = [[Metrologia]] | volume = 22 | issue = 3 | pages = 209–12 | doi = 10.1088/0026-1394/22/3/018|bibcode = 1986Metro..22..209S | last1 = Sienknecht | first1 = V | last2 = Funck | first2 = T }}. {{citation | year = 1991 | title = Determination of the volt with the improved PTB voltage balance | journal = IEEE Transactions on Instrumentation and Measurement| volume = 40 | issue = 2 | pages = 158–61 | doi = 10.1109/TIM.1990.1032905 | last1 = Funck | first1 = T. | last2 = Sienknecht | first2 = V. }}</ref><ref>{{citation | author = Clothier, W. K.; Sloggett, G. J.; Bairnsfather, H.; Currey, M. F.; Benjamin, D. J. | year = 1989 | title = A Determination of the Volt | journal = [[Metrologia]] | volume = 26 | issue = 1 | pages = 9–46 | doi = 10.1088/0026-1394/26/1/003|bibcode = 1989Metro..26....9C }}</ref>
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| |-
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| | Magnetic resonance
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| | {{val|6.6260724|(57)}}
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| | align=center | 8.6{{e|−7}}
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| |<ref>{{citation | year = 1979 | title = A Measurement of the Gyromagnetic Ratio of the Proton in a Strong Magnetic Field | journal = [[Metrologia]] | volume = 15 | issue = 1 | pages = 5–30 | doi = 10.1088/0026-1394/15/1/002|bibcode = 1979Metro..15....5K | last1 = Kibble | first1 = B P | last2 = Hunt | first2 = G J }}</ref><ref>{{citation | author = Liu Ruimin; Liu Hengji; Jin Tiruo; Lu Zhirong;Du Xianhe; Xue Shouqing; Kong Jingwen; Yu Baijiang;Zhou Xianan; Liu Tiebin; Zhang Wei | year = 1995 | title = A Recent Determination for the SI Values of ''γ′''<sub>p</sub> and 2''e''/''h'' at NIM | journal = Acta Metrologica Sinica | volume = 16 | issue = 3|url=http://en.cnki.com.cn/Article_en/CJFDTOTAL-JLXB503.000.htm | pages = 161–68}}</ref>
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| |-
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| | Faraday constant
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| | {{val|6.6260657|(88)}}
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| | align=center | 1.3{{e|−6}}
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| | <ref>{{citation | author = Bower, V. E.; Davis, R. S. | year = 1980 | title = The Electrochemical Equivalent of Pure Silver: A Value of the Faraday Constant | journal =Journal of Research|publisher= National Bureau Standards | volume = 85 | issue = 3 | pages = 175–91|url=http://cdm16009.contentdm.oclc.org/cdm/compoundobject/collection/p13011coll6/id/58310/rec/14|doi=10.6028/jres.085.009 }}</ref>
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| |-
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| | '''CODATA 2010<br />recommended value'''
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| | '''{{val|6.62606957|(29)}}'''
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| | align=center | '''4.4{{e|−8}}'''
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| | <ref name="2010 CODATA" />
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| |-
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| | colspan=4 |The nine recent determinations of the Planck constant cover five separate methods. Where there is more than one recent determination for a given method, the value of ''h'' given here is a weighted mean of the results, as calculated by CODATA.
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| |}
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| In principle, the Planck constant could be determined by examining the spectrum of a black-body radiator or the kinetic energy of photoelectrons, and this is how its value was first calculated in the early twentieth century. In practice, these are no longer the most accurate methods. The CODATA value quoted here is based on three watt-balance measurements of ''K''<sub>J</sub><sup>2</sup>''R''<sub>K</sub> and one inter-laboratory determination of the molar volume of silicon,<ref name="CODATA" /> but is mostly determined by a 2007 watt-balance measurement made at the U.S. [[National Institute of Standards and Technology]] (NIST).<ref name="NIST" /> Five other measurements by three different methods were initially considered, but not included in the final refinement as they were too imprecise to affect the result.
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| There are both practical and theoretical difficulties in determining ''h''. The practical difficulties can be illustrated by the fact that the two most accurate methods, the [[watt balance]] and the X-ray crystal density method, do not appear to agree with one another. The most likely reason is that the measurement uncertainty for one (or both) of the methods has been estimated too low – it is (or they are) not as precise as is currently believed – but for the time being there is no indication which method is at fault.
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| The theoretical difficulties arise from the fact that all of the methods ''except'' the X-ray crystal density method rely on the theoretical basis of the [[Josephson effect]] and the quantum Hall effect. If these theories are slightly inaccurate – though there is no evidence at present to suggest they are – the methods would not give accurate values for the Planck constant. More importantly, the values of the Planck constant obtained in this way cannot be used as tests of the theories without falling into a circular argument. Fortunately, there are other statistical ways of testing the theories, and the theories have yet to be refuted.<ref name="CODATA" />
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| ===Josephson constant===
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| The Josephson constant ''K''<sub>J</sub> relates the potential difference ''U'' generated by the [[Josephson effect]] at a "Josephson junction" with the frequency ''ν'' of the microwave radiation. The theoretical treatment of Josephson effect suggests very strongly that ''K''<sub>J</sub> = 2''e''/''h''.
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| :<math>K_{\rm J} = \frac{\nu}{U} = \frac{2e}{h}\,</math>
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| The Josephson constant may be measured by comparing the potential difference generated by an array of Josephson junctions with a potential difference which is known in SI [[volt]]s. The measurement of the potential difference in SI units is done by allowing an electrostatic force to cancel out a measurable gravitational force. Assuming the validity of the theoretical treatment of the Josephson effect, ''K''<sub>J</sub> is related to the Planck constant by
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| :<math>h = \frac{8\alpha}{\mu_0 c_0 K_{\rm J}^2}.</math>
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| ===Watt balance===
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| {{Main|Watt balance}}
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| A watt balance is an instrument for comparing two [[power (physics)|powers]], one of which is measured in SI [[watt]]s and the other of which is measured in [[conventional electrical unit]]s. From the definition of the ''conventional'' watt ''W''<sub>90</sub>, this gives a measure of the product ''K''<sub>J</sub><sup>2</sup>''R''<sub>K</sub> in SI units, where ''R''<sub>K</sub> is the [[von Klitzing constant]] which appears in the [[quantum Hall effect]]. If the theoretical treatments of the Josephson effect and the quantum Hall effect are valid, and in particular assuming that ''R''<sub>K</sub> = ''h''/''e''<sup>2</sup>, the measurement of ''K''<sub>J</sub><sup>2</sup>''R''<sub>K</sub> is a direct determination of the Planck constant.
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| :<math>h = \frac{4}{K_{\rm J}^2 R_{\rm K}}</math>
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| ===Magnetic resonance===
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| {{Main|Gyromagnetic ratio}}
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| The gyromagnetic ratio ''γ'' is the constant of proportionality between the frequency ''ν'' of [[nuclear magnetic resonance]] (or [[electron paramagnetic resonance]] for electrons) and the applied magnetic field ''B'': ''ν'' = ''γB''. It is difficult to measure gyromagnetic ratios precisely because of the difficulties in precisely measuring ''B'', but the value for [[proton]]s in [[water (molecule)|water]] at 25 °C is known to better than one [[part per million]]. The protons are said to be "shielded" from the applied magnetic field by the electrons in the water molecule, the same effect that gives rise to [[chemical shift]] in NMR spectroscopy, and this is indicated by a prime on the symbol for the gyromagnetic ratio, ''γ′''<sub>p</sub>. The gyromagnetic ratio is related to the shielded proton magnetic moment ''μ′''<sub>p</sub>, the [[spin number]] ''I'' (''I'' = {{frac|1|2}} for protons) and the reduced Planck constant.
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| :<math>\gamma^{\prime}_{\rm p} = \frac{\mu^{\prime}_{\rm p}}{I \hbar} = \frac{2 \mu^{\prime}_{\rm p}}{\hbar}</math>
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| The ratio of the shielded proton magnetic moment ''μ′''<sub>p</sub> to the electron magnetic moment ''μ''<sub>e</sub> can be measured separately and to high precision, as the imprecisely-known value of the applied magnetic field cancels itself out in taking the ratio. The value of ''μ''<sub>e</sub> in Bohr magnetons is also known: it is half the electron g-factor ''g''<sub>e</sub>. Hence
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| :<math>\mu^{\prime}_{\rm p} = \frac{\mu^{\prime}_{\rm p}}{\mu_{\rm e}} \frac{g_{\rm e} \mu_{\rm B}}{2}</math>
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| :<math>\gamma^{\prime}_{\rm p} = \frac{\mu^{\prime}_{\rm p}}{\mu_{\rm e}} \frac{g_{\rm e} \mu_{\rm B}}{\hbar}.</math>
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| A further complication is that the measurement of ''γ′''<sub>p</sub> involves the measurement of an electric current: this is invariably measured in ''conventional'' amperes rather than in SI [[ampere]]s, so a conversion factor is required. The symbol ''Γ′''<sub>p-90</sub> is used for the measured gyromagnetic ratio using conventional electrical units. In addition, there are two methods of measuring the value, a "low-field" method and a "high-field" method, and the conversion factors are different in the two cases. Only the high-field value ''Γ′''<sub>p-90</sub>(hi) is of interest in determining the Planck constant.
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| :<math>\gamma^{\prime}_{\rm p} = \frac{K_{\rm J-90} R_{\rm K-90}}{K_{\rm J} R_{\rm K}} \Gamma^{\prime}_{\rm p-90}({\rm hi}) = \frac{K_{\rm J-90} R_{\rm K-90} e}{2} \Gamma^{\prime}_{\rm p-90}({\rm hi})</math>
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| Substitution gives the expression for the Planck constant in terms of ''Γ′''<sub>p-90</sub>(hi):
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| :<math>h = \frac{c_0 \alpha^2 g_{\rm e}}{2 K_{\rm J-90} R_{\rm K-90} R_{\infty} \Gamma^{\prime}_{\rm p-90}({\rm hi})} \frac{\mu_{\rm p}^{\prime}}{\mu_{\rm e}}.</math>
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| ===Faraday constant===
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| {{Main|Faraday constant}}
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| The Faraday constant ''F'' is the [[Electric charge|charge]] of one mole of electrons, equal to the Avogadro constant ''N''<sub>A</sub> multiplied by the elementary charge ''e''. It can be determined by careful [[electrolysis]] experiments, measuring the amount of [[silver]] dissolved from an electrode in a given time and for a given electric current. In practice, it is measured in conventional electrical units, and so given the symbol ''F''<sub>90</sub>. Substituting the definitions of ''N''<sub>A</sub> and ''e'', and converting from conventional electrical units to SI units, gives the relation to the Planck constant.
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| :<math>h = \frac{c_0 M_{\rm u} A_{\rm r}({\rm e})\alpha^2}{R_{\infty}} \frac{1}{K_{\rm J-90} R_{\rm K-90} F_{90}}</math>
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| ===X-ray crystal density===
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| The X-ray crystal density method is primarily a method for determining the Avogadro constant ''N''<sub>A</sub> but as the Avogadro constant is related to the Planck constant it also determines a value for ''h''. The principle behind the method is to determine ''N''<sub>A</sub> as the ratio between the volume of the [[unit cell]] of a crystal, measured by [[X-ray crystallography]], and the [[molar volume]] of the substance. Crystals of [[silicon]] are used, as they are available in high quality and purity by the technology developed for the [[semiconductor]] industry. The unit cell volume is calculated from the spacing between two crystal planes referred to as ''d''<sub>220</sub>. The molar volume ''V''<sub>m</sub>(Si) requires a knowledge of the [[density]] of the crystal and the [[atomic weight]] of the silicon used. The Planck constant is given by
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| :<math>h = \frac{M_{\rm u} A_{\rm r}({\rm e}) c_0 \alpha^2}{R_{\infty}} \frac{\sqrt{2}d^3_{220}}{V_{\rm m}({\rm Si})}.</math>
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| ===Particle accelerator===
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| The experimental measurement of the Planck constant in the [[Large Hadron Collider]] laboratory was carried out in 2011. The study called PCC using a giant particle accelerator helped to better understand the relationships between the Planck constant and measuring distances in space. [needs citation]
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| ==Fixation==
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| As mentioned above, the numerical value of the Planck constant depends on the system of units used to describe it. Its value in SI units is known to 50 [[parts per billion]] but its value in atomic units is known ''exactly'', because of the way the scale of atomic units is defined. The same is true of conventional electrical units, where the Planck constant (denoted ''h''<sub>90</sub> to distinguish it from its value in SI units) is given by
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| :<math>h_{90} = \frac{4}{K_{J-90}^2 R_{K-90}}</math>
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| with ''K''<sub>J–90</sub> and ''R''<sub>K–90</sub> being exactly defined constants. Atomic units and conventional electrical units are very useful in their respective fields, because the uncertainty in the final result does not depend on an uncertain conversion factor, only on the uncertainty of the measurement itself.
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| There are a number of proposals to redefine certain of the [[SI base unit]]s in terms of fundamental physical constants.<ref name="94thCIPM">94th Meeting of the [[International Committee for Weights and Measures]] (2005). [http://www.bipm.org/utils/en/pdf/CIPM2005-EN.pdf Recommendation 1: Preparative steps towards new definitions of the kilogram, the ampere, the kelvin and the mole in terms of fundamental constants]</ref> This has already been done for the metre, which is defined in terms of a fixed value of the speed of light. The most urgent unit on the list for redefinition is the [[kilogram]], whose value has been fixed for all science (since 1889) by the mass of a small cylinder of [[platinum]]–[[iridium]] alloy kept in a vault just outside Paris. While nobody knows if the mass of the [[International Prototype Kilogram]] has changed since 1889 – the value 1 kg of its mass expressed in kilograms is by definition unchanged and therein lies one of the problems – it is known that over such a timescale the many similar Pt–Ir alloy cylinders kept in national laboratories around the world, have changed their relative mass by several tens of parts per million, however carefully they are stored, and the more so the more they have been taken out and used as mass standards. A change of several tens of micrograms in one kilogram is equivalent to the current uncertainty in the value of the Planck constant in SI units.
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| The legal process to change the definition of the kilogram is already underway,<ref name="94thCIPM" /> but it had been decided that no final decision would be made before the next meeting of the [[General Conference on Weights and Measures]] in 2011.<ref name="23rdCGPM">23rd [[General Conference on Weights and Measures]] (2007). [http://www.bipm.org/utils/en/pdf/Resol23CGPM-EN.pdf Resolution 12: On the possible redefinition of certain base units of the International System of Units (SI)].</ref> (For more detailed information, see [[Kilogram#Proposed future definitions|kilogram definitions]].) The Planck constant is a leading contender to form the basis of the new definition, although not the only one.<ref name="23rdCGPM" /> Possible new definitions include "the mass of a body at rest whose equivalent energy equals the energy of photons whose frequencies sum to {{val|135639274|e=42|u=Hz}}",<ref>{{citation | title = On the redefinition of the kilogram | author = Taylor, B. N.; Mohr, P. J. | url = http://www.iop.org/EJ/article/0026-1394/36/1/11/me9111.pdf | journal = [[Metrologia]] | volume = 36 | issue = 1 | year = 1999 | pages = 63–64 | doi = 10.1088/0026-1394/36/1/11|bibcode = 1999Metro..36...63T }}</ref> or simply "the kilogram is defined so that the Planck constant equals {{val|6.62606896|e=-34|u=J·s}}".
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| The [[BIPM]] provided ''Draft Resolution A'' in anticipation of the 24th General Conference on Weights and Measures meeting (2011-10-17 through 2011-10-21), detailing the considerations "On the possible future revision of the International System of Units, the SI".<ref name="24thCGPM">{{Citation| url=http://www.bipm.org/utils/common/pdf/24_CGPM_Convocation_Draft_Resolution_A.pdf |title=Draft Resolution A: On the possible future revision of the International System of Units, the SI}}</ref>
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| [[Watt balance]]s already measure mass in terms of the Planck constant: at present, standard mass is taken as fixed and the measurement is performed to determine the Planck constant but, were the Planck constant to be fixed in SI units, the same experiment would be a measurement of the mass. The relative uncertainty in the measurement would remain the same.
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| Mass standards could also be constructed from silicon crystals or by other atom-counting methods. Such methods require a knowledge of the Avogadro constant, which fixes the proportionality between [[atomic mass]] and macroscopic mass but, with a defined value of the Planck constant, ''N''<sub>A</sub> would be known to the same level of uncertainty (if not better) than current methods of comparing macroscopic mass.
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| ==See also==
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| * [[New SI definitions]]
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| * [[Basic concepts of quantum mechanics]]
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| * [[Planck units]]
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| * [[Stigler's law]]
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| * [[Wave–particle duality]]
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| ==Notes==
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| {{Reflist|colwidth=30em}}
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| ==References==
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| * {{Citation
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| | last = Barrow
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| | first = John D.
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| | authorlink = John D. Barrow
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| | title = The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe
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| | year = 2002
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| | publisher = Pantheon Books
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| | location =
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| | isbn = 0-375-42221-8
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| }}
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| ==External links==
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| * [http://www.numericana.com/answer/constants.htm#h Quantum of Action and Quantum of Spin – Numericana]
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| * {{cite web|last=Moriarty|first=Philip|title=h Planck's Constant|url=http://www.sixtysymbols.com/videos/planck.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|coauthors=[[Laurence Eaves{{!}}Eaves, Laurence]]; Merrifield, Michael|year=2009}}
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| [[Category:Fundamental constants]]
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| [[Category:Max Planck|constant]]
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