View source for Planck constant

{{For|the law governing black body radiation|Planck's law}}

{{Planck constant|right}}

[[File:Max Planck Wirkungsquantums 20050815.jpg|thumb|right|Plaque at the [[Humboldt University of Berlin]]: "Max Planck, discoverer of the elementary quantum of action <math>h </math>, taught in this building from 1889 to 1928."]]

The '''Planck constant''', or '''Planck's constant''', denoted <math>h </math> is a [[physical constant]] that is the [[quantum]] of [[Electromagnetism|electromagnetic]] [[action (physics)|action]], which relates the energy carried by a [[photon]] to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant. The Planck constant is of fundamental importance in [[quantum mechanics]], and in [[metrology]] it is the basis for the definition of the [[kilogram]].

The Planck constant is defined to have the exact value {{nowrap|<math>h=</math> {{physconst|h|after=.}}}}<ref>{{cite web|url=https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf|date=2018-11-16|work=BIPM|access-date=2018-11-20|language=en-GB|title=Resolutions of the 26th CGPM}}</ref>

At the end of the 19th century, physicists were unable to explain why the observed spectrum of [[black body]] radiation, which is still considered to have been accurately measured, diverged significantly at higher frequencies from that predicted by existing theories. In 1900, [[Max Planck]] empirically derived a formula for the observed spectrum. He assumed that a hypothetical electrically charged [[Oscillation|oscillator]] in a cavity that contained black-body radiation could only change its [[energy]] in a minimal increment, <math>E </math>, that was proportional to the [[frequency]] of its associated [[electromagnetic wave]].<ref name="Planck01" /> He was able to calculate the [[proportionality constant]], <math>h </math>, from the experimental measurements, and that constant is named in his honor. In 1905, the value <math>E </math> was associated by [[Albert Einstein]] with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle, as opposed to an electromagnetic wave. It was eventually called a [[photon]]. 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".

Since [[Mass–energy equivalence|energy and mass are equivalent]], the Planck constant also relates mass to frequency.

==Origin of the constant==

{{Main|Planck's law}}
[[File:Wiens law.svg|thumb|right|250px|Intensity of light emitted from a [[black body]] at any given wavelength. Each curve represents behaviour at a different body temperature. Max Planck was the first to explain the shape of these curves.]]
In the last years of the 19th century, Max Planck was investigating the problem of [[Black body|black-body radiation]] first posed by [[Gustav Kirchhoff|Kirchhoff]] some 40 years earlier.

Every [[physical body]] spontaneously and continuously emits [[electromagnetic radiation]]. At low frequencies, Planck's law tends to the [[Rayleigh–Jeans law]], while in the limit of high frequencies (i.e. small wavelengths) it tends to the [[Wien approximation]], but there was no overall expression or explanation for the shape of the observed emission spectrum.

Approaching this problem, Planck hypothesized that the equations of motion for light describe 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 the 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] {{webarchive|url=https://web.archive.org/web/20080418002757/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html |date=2008-04-18 }}".{{cite web |url=http://theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf |title=Archived copy |accessdate=2011-10-13 |url-status=dead |archiveurl=https://web.archive.org/web/20111006162543/http://theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf |archivedate=2011-10-06 |df= }}</ref>
To create [[Planck's law]], which correctly predicts blackbody emissions by fitting the observed curves, he multiplied the classical expression by a factor that involves a constant, <math>h </math>, in both the numerator and the denominator, which subsequently became known as the Planck Constant.

The [[Radiance|spectral radiance]] of a body, <math>B_{\nu} </math>, describes the amount of energy it emits at different radiation frequencies. It is the power emitted per unit area of the body, per unit solid angle of emission, per unit frequency.

Planck showed that the spectral radiance of a body for [[frequency]] {{math|''ν''}} at [[absolute temperature]] {{math|''T''}} is given by
:<math>B_\nu(\nu, T) = \frac{ 2 h \nu^3}{c^2} \frac{1}{e^\frac{h\nu}{k_\mathrm B T} - 1}</math>

where <math>k_B </math> is the [[Boltzmann constant]], <math>h </math> is the Planck constant, and <math>c </math> is the [[speed of light]] in the medium, whether material or vacuum.<ref name="Planck 1914 6 168">{{harvnb|Planck|1914|pp=6,&nbsp;168}}</ref><ref name="Chan8">{{harvnb|Chandrasekhar|1960|p=8}}</ref><ref name="Rybicki 1979 22">{{harvnb|Rybicki|Lightman|1979|p=22}}</ref>

The spectral radiance can also be expressed per unit [[wavelength]] <math>\lambda </math> instead of per unit frequency. In this case, it is given by

:<math>B_\lambda(\lambda, T) =\frac{2hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{\lambda k_\mathrm B T}} - 1},</math>

showing how radiated energy emitted at shorter wavelengths increases more rapidly with temperature than energy emitted at longer wavelengths.<ref>{{cite journal | last1=Shao| first1=Gaofeng|display-authors=etal| title= Improved oxidation resistance of high emissivity coatings on fibrous ceramic for reusable space systems | journal= Corrosion Science | year=2019 | volume=146| pages= 233–246 | doi= 10.1016/j.corsci.2018.11.006 }}</ref>

The law may also be expressed in other terms, such as the number of photons emitted at a certain wavelength, or the energy density in a volume of radiation. The [[Radiance|SI units]] of <math>B_{\nu} </math> are {{nobreak|[[Watt|W]]·[[steradian|sr]]<sup>−1</sup>·[[metre|m]]<sup>−2</sup>·[[Hertz|Hz]]<sup>−1</sup>}}, while those of <math>B_{\lambda} </math> are {{nobreak|W·sr<sup>−1</sup>·m<sup>−3</sup>}}.

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 resorted 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
{{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" />}}
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 = 978-0-691-09552-3 | page = 62 | url = https://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 sometimes termed the "Planck–Einstein relation":
:<math>E = hf.</math>
Planck was able to calculate the value of <math>h </math> from experimental data on black-body radiation: his result, {{val|6.55|e=-34|u=J.s}}, is within 1.2% of the currently accepted value.<ref name="Planck01" /> He also made the first determination of the [[Boltzmann constant]] <math>k_B </math> 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>

[[File:Black body.svg|350px|thumb|The divergence of the theoretical Rayleigh–Jeans (black) curve from the observed Planck curves at different temperatures.]]

==Development and application==
The black-body problem was revisited in 1905, when [[John William Strutt, 3rd Baron Rayleigh|Rayleigh]] and [[James Hopwood Jeans|Jeans]] (on the one hand) and [[Albert Einstein|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]]) in 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 | url-status=dead | archiveurl = https://web.archive.org/web/20081216120021/http://www.solvayinstitutes.be/Conseils%20Solvay/PreviousPhysics.html | archivedate = 16 December 2008 | df =  }}</ref>

=== Photoelectric effect ===
{{Main|Photoelectric effect}}
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 | url = https://zenodo.org/record/1424009 }}</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 | doi-access = free }}</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=https://books.google.com/books/about/Einstein.html?id=cdxWNE7NY6QC|title=Einstein: His Life and Universe|isbn=978-1-4165-3932-2|last1=Isaacson|first1=Walter|date=2007-04-10}}, pp. 309–314.</ref><ref>{{cite web|url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1921/ |title=The Nobel Prize in Physics 1921 |publisher=Nobelprize.org |accessdate=2014-04-23}}</ref>

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 per unit space (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.

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 photon energy that is less than the [[work function]] of the material), 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|page=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" />

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 the Planck–Einstein relation:
:<math>E = hf .</math>

Einstein's postulate was later proven experimentally: the constant of proportionality between the frequency of incident light <math>f</math> and the kinetic energy of photoelectrons <math>E</math> was shown to be equal to the Planck constant <math>h</math>.<ref name="Millikan" />

=== Atomic structure ===
{{Main|Bohr model}}
[[File:Bohr atom model.svg|thumb|right|A schematization of the Bohr model of the hydrogen atom. The transition shown from the {{nowrap|1=''n'' = 3}} level to the {{nowrap|1=''n'' = 2}} level gives rise to visible light of wavelength 656 nm (red), as the model predicts.]]
[[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.]] |series=6th Series | year = 1913 | volume = 26 | issue = 153 | pages = 1–25 | doi = 10.1080/14786441308634993 | url = https://zenodo.org/record/1430916 }}</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 <math>E_n</math>

:<math>E_n = -\frac{h c R_{\infty}}{n^2} ,</math>

where <math>c</math> is the speed of light in vacuum, <math>R_{\infty}</math> is an experimentally determined constant (the [[Rydberg constant]]) and <math>n \in \{1, 2, 3, ... \}</math>. Once the electron reached the lowest energy level (<math>n = 1</math>), 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 <math>R_{\infty}</math> in terms of other fundamental constants.

Bohr also introduced the quantity <math>\frac{h}{2\pi}</math>, now known as the [[#Value|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 <math>J</math> is the total angular momentum of a system with rotational invariance, and <math>J_z</math> the angular momentum measured along any given direction, these quantities can only take on the values
:<math>
\begin{align}
J^2 = j(j+1) \hbar^2,\qquad  & j = 0, \tfrac{1}{2}, 1, \tfrac{3}{2}, \ldots, \\
J_z = m \hbar, \qquad\qquad\quad & m = -j, -j+1, \ldots, j.
\end{align}
</math>

=== Uncertainty principle ===
{{Main|Uncertainty principle}}
The Planck constant also occurs in statements of [[Werner Heisenberg]]'s uncertainty principle. Given numerous particles prepared in the same state, the [[uncertainty]] in their position, <math>\Delta x</math>, and the uncertainty in their momentum, <math>\Delta p_{x}</math>, obey

:<math> \Delta x\, \Delta p_{x} \ge \frac{\hbar}{2} ,</math>

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 [[conjugate variable]]s which obey a similar rule. One example is time vs. energy. The inverse relationship between the uncertainty of the two conjugate variables forces a tradeoff in quantum experiments, as measuring one quantity more precisely results in the other quantity becoming imprecise.

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>:
:<math>[\hat{p}_i, \hat{x}_j] = -i \hbar \delta_{ij} ,</math>
where <math> \delta_{ij}</math> is the [[Kronecker delta]].

== Photon energy==

The [[Planck–Einstein relation]] connects the particular [[photon energy]] {{math|''E''}} with its associated wave frequency {{math|''f''}}:

:<math>E = hf</math>

This energy is extremely small in terms of ordinarily perceived everyday objects.

Since the frequency {{math|''f''}}, [[wavelength]] {{math|''λ''}}, and [[speed of light]] {{math|''c''}} are related by <math>f= \frac{c}{\lambda} </math>, the relation can also be expressed as

:<math>E = \frac{hc}{\lambda} .</math>

The [[de Broglie wavelength]] {{math|''λ''}} of the particle is given by

:<math>\lambda = \frac{h}{p}</math>

where {{math|''p''}} denotes the linear [[momentum]] of a particle, such as a photon, or any other [[elementary particle]].

In applications where it is natural to use the [[angular frequency]] (i.e. where the frequency is expressed in terms of [[radian]]s per second instead of [[Turn (geometry)|cycle]]s per second or [[hertz]]) it is often useful to absorb a factor of {{math|2''π''}} into the Planck constant. The resulting constant is called the [[#Value|reduced Planck constant]].  It is equal to the Planck constant divided by {{math|2''π''}}, and is denoted {{math|''ħ''}} (pronounced "h-bar"):

:<math>\hbar = \frac{h}{2 \pi} .</math>

The [[photon energy|energy of a photon]] with angular frequency {{math|1=''ω'' = 2''πf''}} is given by

:<math>E = \hbar \omega ,</math>

while its linear momentum relates to

:<math>p = \hbar k ,</math>

where {{math|''k''}} is an [[wavenumber|angular wavenumber]]. In 1923, [[Louis de Broglie]] generalized the Planck–Einstein 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. This holds throughout quantum theory, including [[electrodynamics]].

Problems can arise when dealing with frequency or the Planck constant because the units of angular measure (cycle or radian) are omitted in SI.<ref>{{cite journal | last1 = Mohr | first1 = J. C. | last2 = Phillips | first2 = W. D. | year = 2015 | title = Dimensionless Units in the SI | url = | journal = Metrologia | volume = 52 | issue = 1 | pages = 40–47 | doi = 10.1088/0026-1394/52/1/40 | bibcode = 2015Metro..52...40M | arxiv = 1409.2794 }}</ref><ref>{{cite journal | last1 = Mills | first1 = I. M. | year = 2016 | title = On the units radian and cycle for the quantity plane angle | url = | journal = Metrologia | volume = 53 | issue = 3 | pages = 991–997 | doi = 10.1088/0026-1394/53/3/991 | bibcode = 2016Metro..53..991M }}</ref><ref>Nature (2017) ‘’A Flaw in the SI system,’' Volume 548, Page 135</ref> In the language of quantity calculus,<ref>Maxwell J.C. (1873) A Treatise on Electricity and Magnetism, Oxford University Press</ref> the expression for the "value" of the Planck constant, or of a frequency, is the product of a "numerical value" and a "unit of measurement". When we use the symbol {{math|''f''}}  (or {{math|''ν''}}) for the value of a frequency it implies the units cycles per second or hertz, but when we use the symbol {{math|''ω''}} for its value it implies the units radians per second; the numerical values of these two ways of expressing the value of a frequency have a [[ratio]] of {{math|2''π''}}, but their values are equal. Omitting the units of angular measure "cycle" and "radian" can lead to an error of {{math|2''π''}}. A similar state of affairs occurs for the Planck constant. We use the symbol {{math|''h''}} when we express the value of the Planck constant in J⋅s/cycle, and we use the symbol {{math|''ħ''}} when we express its value in J⋅s/rad. Since both represent the value of the Planck constant, but in different units, we have {{math|1=''h'' = ''ħ''}}. Their "values" are equal but, as discussed below, their "numerical values" have a ratio of {{math|2''π''}}. In this Wikipedia article the word "value" as used in the tables means "numerical value", and the equations involving the Planck constant and/or frequency actually involve their numerical values using the appropriate implied units. The distinction between "value" and "numerical value" as it applies to frequency and the Planck constant is explained in more detail in this pdf file [https://chemphys.ca/pbunker/Planckpdffile.html Link].

These two relations are the temporal and spatial component parts of the special relativistic expression using [[Four-Vector|4-vectors]].

:<math>P^\mu = \left(\frac{E}{c}, \vec{p}\right) = \hbar K^\mu = \hbar\left(\frac{\omega}{c}, \vec{k}\right) </math>

Classical [[statistical mechanics]] requires the existence of {{math|''h''}} (but does not define its value).<ref>{{Citation|title=Statistical mechanics: an intermediate course |author1=Giuseppe Morandi |author2=F. Napoli |author3=E. Ercolessi |page=84|url=https://books.google.com/?id=MhInFlnNsREC&pg=PA51 |isbn=978-981-02-4477-4|year=2001}}</ref> Eventually, following upon Planck's discovery, it was recognized that physical [[Action (physics)|action]] cannot take on an arbitrary value.  Instead, it must be some integer multiple of a very small quantity, the "[[quantum]] of action", now called the ''reduced Planck constant'' or the ''natural unit of action''. This is the so-called "old quantum theory" developed by [[Niels Bohr|Bohr]] and [[Arnold Sommerfeld|Sommerfeld]], in which particle trajectories exist but are [[Hidden variable theory|hidden]], but quantum laws constrain them based on their action. This view has been largely replaced by fully modern quantum theory, in which definite trajectories of motion do not even exist, rather, the particle is represented by a wavefunction spread out in space and in time. Thus there is no value of the action as classically defined. Related to this is the concept of energy quantization which existed in old quantum theory and also exists in altered form in modern quantum physics. Classical physics cannot explain either quantization of energy or the lack of a classical particle motion.

In many cases, such as for monochromatic light or for atoms, quantization of energy 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 {{math|''H<sub>σ</sub>''}} to a system specified in the sense of classical mechanics (the energy function is a given function of the coordinates {{math|''q<sub>r</sub>''}} and the corresponding momenta {{math|''p<sub>r</sub>''}})? The Planck constant {{math|''h''}} relates the frequency {{math|''H<sub>σ</sub>''/''h''}} to the energy values {{math|''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|url-status=dead|archiveurl=https://web.archive.org/web/20120415132339/http://www.kostic.niu.edu/Physics_and_Reality-Albert_Einstein.pdf|archivedate=2012-04-15|df=}}</ref>

== Value<!--'Reduced Planck constant' and 'Dirac constant' redirect here--> ==
The Planck constant has [[dimensional analysis|dimensions]] of physical [[action (physics)|action]]; i.e., [[energy]] multiplied by [[time]], or [[momentum]] multiplied by [[distance]], or [[angular momentum]]. In [[International System of Units|SI units]], the Planck constant is expressed in [[joule-second]]s (J⋅s or [[Newton (unit)|N]]⋅[[metre|m]]⋅[[second|s]] or  [[Kilogram|kg]]⋅m<sup>2</sup>⋅s<sup>&minus;1</sup>). Implicit in the dimensions of the Planck constant is the fact that the SI unit of frequency, the [[Hertz]], represents one complete cycle, 360 degrees or {{math|2''π''}} radians, per second. An [[angular frequency]] in radians per second is often more natural in mathematics and physics and many formulas use a '''reduced Planck constant'''<!--boldface per WP:R#PLA--> (pronounced ''h-bar'')
:<math>h = 6.626\ 070\ 15\times 10^{-34}\ \text{J}{\cdot}\text{s}</math>
:<math>\hbar={{h}\over{2\pi}} = 1.054\ 571\ 817...\times 10^{-34}\ \text{J}{\cdot}\text{s} = 6.582\ 119\ 569...\times 10^{-16}\ \text{eV}{\cdot}\text{s}</math>
:The above values are recommended by 2018 [[Committee on Data for Science and Technology|CODATA]].
In [[Hartree atomic units|atomic units]],
:<math>h = 2 \pi \text{ a.u. of action}</math>
:<math>\hbar = ~~ 1 \text{ a.u. of action}</math>

=== Understanding the 'fixing' of the value of ''h'' ===

Since 2019, the numerical value of the Planck constant has been ''fixed, with infinite significant figures''.  Under the present definition of the [[kilogram]], which states that "The kilogram [...] is defined by taking the fixed numerical value of {{math|''h''}} to be {{val|6.62607015|e=-34}} when expressed in the unit J⋅s, which is equal to kg⋅m<sup>2</sup>⋅s<sup>−1</sup>, where the [[metre]] and the [[second]] are defined in terms of [[speed of light]] {{Mvar|c}} and duration of [[Hyperfine structure|hyperfine transition]] of the [[ground state]] of an unperturbed [[Cesium-133]] atom {{Math|Δ''ν''<sub>Cs</sub>}}."<ref>{{Cite web|url=https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9.pdf|title=BIPM|last=9th edition|first=SI BROCHURE|date=|website=BIPM|archive-url=|archive-date=|url-status=|access-date=}}</ref>  This implies that mass [[metrology]] is now aimed to find the value of one kilogram, and thus it is kilogram which is ''compensating''.  Every experiment aiming to measure the kilogram (such as the [[Kibble balance]] and the X-ray crystal density method), will essentially refine the value of a kilogram.

As an illustration of this, suppose the decision of making {{math|''h''}} to be exact was taken in 2010, when its measured value was {{val|6.62606957|e=-34|u=J.s}}, thus the present definition of kilogram was also enforced. In future, the value of one kilogram must have become refined to {{nowrap|{{sfrac|{{val|6.62607015}}|{{val|6.62606957}}}} ≈ {{val|1.0000001}}}} times the mass of the [[International Prototype of the Kilogram]] (IPK), neglecting the metre and second units' share, for sake of simplicity.

== Significance of the value ==

The Planck constant is related to the quantization of light and matter. It 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. The physical meaning of the Planck's constant could suggest some basic features of our physical world.

The Planck constant 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, the Planck constant (the quantum of action) is very small. One can regard the Planck constant to be only relevant to the microscopic scale instead of the macroscopic scale in our everyday experience.

Equivalently, the order of the Planck constant reflects the fact that everyday objects and systems are made of a ''large'' number of microscopic particles. For example, green light with a [[wavelength]] of 555&nbsp;[[nanometre]]s (a wavelength that can be perceived by the human eye to be [[green]]) has a frequency of {{val|540|u=THz}} ({{val|540|e=12|ul=Hz}}). Each [[photon]] has an energy {{nowrap|1=''E'' = ''hf'' = {{val|3.58|e=-19|u=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 more typical in everyday experience (though much larger than the smallest amount perceivable by the human eye) is the energy of one [[mole (unit)|mole]] of photons; its energy can be computed by multiplying the photon energy by the [[Avogadro constant]], {{nowrap|1=''N''<sub>A</sub> = {{val|6.02214076||e=23|u=mol<sup>−1</sup>}}}}, with the result of {{val|216|u=kJ/mol}}, about the food energy in three apples.

:

== Determination ==
In principle, the Planck constant can 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.

Since the value of the Planck constant is fixed now, it is no longer determined or calculated in laboratories. Some of the practices given below to determine the Planck constant are now used to determine the mass of the kilogram. The below given methods ''except'' the X-ray crystal density method rely on the theoretical basis of the [[Josephson effect]] and the [[quantum Hall effect]].

=== Josephson constant ===
{{main|Magnetic flux quantum}}
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 {{nowrap|1=''K''<sub>J</sub> = 2''e''/''h''}}.
:<math>K_{\rm J} = \frac{\nu}{U} = \frac{2e}{h}\,</math>
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, in a Kibble balance. Assuming the validity of the theoretical treatment of the Josephson effect, ''K''<sub>J</sub> is related to the Planck constant by
:<math>h = \frac{8\alpha}{\mu_0 c_0 K_{\rm J}^2}.</math>

=== Kibble balance ===
{{Main|Kibble balance}}
A Kibble balance (formerly known as a watt balance)<ref>{{Cite web|url=https://www.nist.gov/si-redefinition/kilogram-kibble-balance|title=Kilogram: The Kibble Balance|last=Materese|first=Robin|date=2018-05-14|work=NIST|access-date=2018-11-13|language=en}}</ref> 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 {{nowrap|1=''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.
:<math>h = \frac{4}{K_{\rm J}^2 R_{\rm K}} .</math>

=== Magnetic resonance ===
{{Main|Gyromagnetic ratio}}
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 {{nowrap|1=''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 {{val|25|u=°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'' ({{nowrap|1=''I'' = {{frac|1|2}}}} for protons) and the reduced Planck constant.
:<math>\gamma^{\prime}_{\rm p} = \frac{\mu^{\prime}_{\rm p}}{I \hbar} = \frac{2 \mu^{\prime}_{\rm p}}{\hbar}</math>
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
:<math>\mu^{\prime}_{\rm p} = \frac{\mu^{\prime}_{\rm p}}{\mu_{\rm e}} \frac{g_{\rm e} \mu_{\rm B}}{2}</math>
:<math>\gamma^{\prime}_{\rm p} = \frac{\mu^{\prime}_{\rm p}}{\mu_{\rm e}} \frac{g_{\rm e} \mu_{\rm B}}{\hbar}.</math>
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.
:<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>
Substitution gives the expression for the Planck constant in terms of ''Γ''′<sub>p-90</sub>(hi):
:<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>

=== Faraday constant ===
{{Main|Faraday constant}}
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.
:<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>

=== X-ray crystal density ===
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
:<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>

=== Particle accelerator ===
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.{{citation needed|date=October 2014}}

== See also ==
* [[Introduction to quantum mechanics]]
* [[Planck units]]
* [[Wave–particle duality]]
* [[CODATA 2018]]

== Notes ==
{{NoteFoot}}

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{refbegin}}
* {{Citation
 | last = Barrow
 | first = John D.
 | author-link = John D. Barrow
 | title = The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe
 | year = 2002
 | publisher = Pantheon Books
 | location =
 | isbn = 978-0-375-42221-8
 }}
{{refend}}

== External links ==
* [http://www.numericana.com/answer/constants.htm#h Quantum of Action and Quantum of Spin – Numericana]
* {{cite web|last=Moriarty|first=Philip|title=h Planck's Constant|url=http://www.sixtysymbols.com/videos/planck.htm|website=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|author2=Eaves, Laurence|last3= Merrifield|first3= Michael|year=2009|author2-link=Laurence Eaves}}
* A pdf file explaining the relation between {{math|''h''}} and {{math|''ħ''}}, their units, and the history of their introduction [https://chemphys.ca/pbunker/Planckpdffile.html Link]

===Videos===
* [https://www.youtube.com/thebipm The BIPM YouTube channel]
* [https://www.youtube.com/watch?v=yhgb23tAFFs&list=PL-vj-3_a7wTDeKEupZSX7Tw42yReNgJLl "The role of the Planck constant in physics" - presentation at 26th CGPM meeting at Versailles, France, November 2018 when voting took place.]

{{Planck's natural units}}
{{Scientists whose names are used in physical constants}}

[[Category:Fundamental constants]]
[[Category:1900 in science]]
[[Category:Planck units]]