View source for Physics:Planck constant

{{Short description|Physical constant in quantum mechanics}}
{{Infobox physical quantity
| name = Planck constant
| unit = joule per hertz (joule seconds)
| otherunits = electronvolt per hertz (electronvolt seconds)
| symbols = <math>h</math>
| baseunits = 
| dimension = <math>\mathsf{M} \mathsf{L}^{2} \mathsf{T}^{-1}</math>
| derivations = 
| value = {{physconst|h|ref=no}}{{br}}{{physconst|h_eV/Hz|ref=no}}
}}
{{Infobox physical quantity
| name = Reduced Planck constant
| unit = joule-seconds
| otherunits = electronvolt-seconds
| symbols = <math>\hbar</math>
| baseunits = 
| dimension = <math>\mathsf{M} \mathsf{L}^{2} \mathsf{T}^{-1}</math>
| derivations = {{ublist
  | <math>\hbar {{=}} h/(2 \pi)</math>
}}
| value = {{physconst|hbar|ref=no}}{{br}}{{physconst|hbar_eVs|ref=no}}
}}

The '''Planck constant''', or '''Planck's constant''', denoted by {{nowrap|<math display=inline>h</math>,}}<ref name="NIST h">{{Cite web |date=20 May 2019 |title=Planck constant |url=https://physics.nist.gov/cgi-bin/cuu/Value?h |url-status=live |archive-url=https://web.archive.org/web/20220527011505/https://physics.nist.gov/cgi-bin/cuu/Value?h |archive-date=2022-05-27 |access-date=2023-09-03 |website=The NIST Reference on Constants, Units, and Uncertainty |publisher=[[Organization:National Institute of Standards and Technology|NIST]]}}</ref> is a fundamental [[Physics:Physical constant|physical constant]]<ref name="NIST h" /> of foundational importance in [[Physics:Quantum mechanics|quantum mechanics]]: a [[Physics:Photon|photon]]'s energy is equal to its [[Physics:Frequency|frequency]] multiplied by the Planck constant, and the [[Wavelength|wavelength]] of a [[Physics:Matter wave|matter wave]] equals the Planck constant divided by the associated particle momentum.

The constant was postulated by [[Biography:Max Planck|Max Planck]] in 1900 as a proportionality constant needed to explain experimental black-body radiation.<ref name="Planck01" /> Planck later referred to the constant as the "quantum of [[Physics:Action|action]]".<ref>{{Cite web |title=Max Planck Nobel Lecture |url=https://www.nobelprize.org/prizes/physics/1918/planck/lecture/ |url-status=live |archive-url=https://web.archive.org/web/20230714164215/https://www.nobelprize.org/prizes/physics/1918/planck/lecture/ |archive-date=2023-07-14 |access-date=2023-07-14}}</ref> In 1905, [[Biography:Albert Einstein|Albert Einstein]] associated the "quantum" or minimal element of the energy to the electromagnetic wave itself.  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".

In [[Metrology|metrology]], the Planck constant is used, together with other constants, to define the [[Kilogram|kilogram]], the SI unit of mass.<ref>{{SIbrochure9th|page=131}}</ref> The SI units are defined in such a way that, when the Planck constant is expressed in SI units, it has the exact value {{nowrap|<math>h</math> {{=}} {{physconst|h|after=.}}}}<ref>{{Cite web |date=2018-11-16 |title=Resolutions of the 26th CGPM |url=https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf |url-status=dead |archive-url=https://web.archive.org/web/20181119214326/https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf |archive-date=2018-11-19 |access-date=2018-11-20 |website=BIPM |language=en-GB}}</ref> It is often used with units of [[Physics:Electronvolt|electronvolt]] (eV), which corresponds to the SI unit per [[Physics:Elementary charge|elementary charge]].

== History ==
=== Origin of the constant ===

{{Main|Physics:Planck's law}}
[[File:Max Planck Wirkungsquantums 20050815.jpg|thumb|right|250px|Plaque at the Humboldt University of Berlin: "Max Planck, who discovered the elementary quantum of action ''h'', taught here from 1889 to 1928."]]
[[File:Wiens law.svg|thumb|right|250px|Intensity of light emitted from a [[Astronomy:Black body|black body]]. Each curve represents behavior at different body temperatures. The Planck constant ''h'' is used to explain the shape of these curves.]]
Planck's constant was formulated as part of Max Planck's successful effort to produce a mathematical expression that accurately predicted the observed spectral distribution of thermal radiation from a closed furnace ([[Astronomy:Black-body radiation|black-body radiation]]).<ref name="Bitter">{{Cite book |last1=Bitter |first1=Francis |title=Fields and particles |last2=Medicus |first2=Heinrich A. |date=1973 |publisher=Elsevier |location=New York |pages=137–144 |}}</ref> This mathematical expression is now known as Planck's law.

In the last years of the 19th century, Max Planck was investigating the problem of black-body radiation first posed by [[Biography:Gustav Kirchhoff|Kirchhoff]] some 40 years earlier. Every [[Philosophy:Physical body|physical body]] spontaneously and continuously emits [[Physics:Electromagnetic radiation|electromagnetic radiation]]. There was no expression or explanation for the overall shape of the observed emission spectrum. At the time, [[Physics:Wien approximation|Wien's law]] fit the data for short wavelengths and high temperatures, but failed for long wavelengths.<ref name="Bitter" />{{rp|141}} Also around this time, but unknown to Planck, Lord Rayleigh had derived theoretically a formula, now known as the [[Physics:Rayleigh–Jeans law|Rayleigh–Jeans law]], that could reasonably predict long wavelengths but failed dramatically at short wavelengths.

Approaching this problem, Planck hypothesized that the equations of motion for light describe a set of [[Physics:Harmonic oscillator|harmonic oscillator]]s, one for each possible frequency. He examined how the [[Physics:Entropy|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 |  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 = [[Physics:Annalen der Physik|Ann. Phys.]] | year = 1901 | volume = 309 | issue = 3 | pages = 553–63 | doi = 10.1002/andp.19013090310 | bibcode = 1901AnP...309..553P | doi-access = free | access-date = 2008-12-15 | archive-date = 2012-06-10 | archive-url = https://web.archive.org/web/20120610124128/http://www.physik.uni-augsburg.de/annalen/history/historic-papers/1901_309_553-563.pdf | url-status = live }}. English translation: {{Cite web |title=On the Law of Distribution of Energy in the Normal Spectrum |url=http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html |url-status=dead |archive-url=https://web.archive.org/web/20080418002757/http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html |archive-date=2008-04-18}}".  {{cite web |url=http://theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf |title=On the Law of Distribution of Energy in the Normal Spectrum |access-date=2011-10-13 |url-status=dead |archive-url=https://web.archive.org/web/20111006162543/http://theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf |archive-date=2011-10-06 }}</ref> which gave a simple empirical formula for long wavelengths.

Planck tried to find a mathematical expression that could reproduce Wien's law (for short wavelengths) and the empirical formula (for long wavelengths). This expression included a constant, <math>h </math>, which is thought to be for Hilfsgrösse (auxiliary variable),<ref>{{Cite arXiv |eprint=physics/0402064v1 |first=Luis J. |last=Boya |title=The Thermal Radiation Formula of Planck (1900) |date=2004}}</ref> and subsequently became known as the Planck constant. The expression formulated by Planck showed that the spectral radiance of a body for [[Physics:Frequency|frequency]] {{math|''ν''}} at [[Physics:Absolute temperature|absolute temperature]] {{math|''T''}} is given by
: <big><math>B_\nu(\nu, T) = \frac{ 2 h \nu^3}{c^2} \frac{1}{e^\frac{h\nu}{k_\mathrm B T} - 1}</math></big>,
where <math>k_\text{B}</math> is the [[Physics:Boltzmann constant|Boltzmann constant]], <math>h </math> is the Planck constant, and <math>c </math> is the [[Physics:Speed of light|speed of light]] in the medium, whether material or vacuum.<ref name="Planck 1914 6 168">{{Cite book |last=Planck |first=M. |url=https://archive.org/details/theoryofheatradi00planrich |title=The Theory of Heat Radiation |publisher=P. Blakiston's Son |others=Masius, M. (transl.) |year=1914 |edition=2nd |pages=6,&nbsp;168 |ol=7154661M |author-link=Max Planck}}</ref><ref name="Chan8">{{Cite book |last=Chandrasekhar |first=S. |url=https://archive.org/details/radiativetransfe0000chan |title=Radiative Transfer |publisher=[[Social:Dover Publications|Dover]] |year=1960 |isbn=978-0-486-60590-6 |edition=Revised reprint |page=8 |author-link=Subrahmanyan Chandrasekhar |url-access=registration |orig-year=1950}}</ref><ref name="Rybicki 1979 22">{{Cite book |last1=Rybicki |first1=G. B. |url=https://books.google.com/books?id=LtdEjNABMlsC |title=Radiative Processes in Astrophysics |last2=Lightman |first2=A. P. |publisher={{wipe|John Wiley & Sons}} |year=1979 |isbn=978-0-471-82759-7 |page=22 |author-link2=Alan Lightman |access-date=2020-05-20 |archive-url=https://web.archive.org/web/20200727111701/https://books.google.com/books?id=LtdEjNABMlsC |archive-date=2020-07-27 |url-status=live}}</ref>

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

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

showing how radiated energy emitted at shorter wavelengths increases more rapidly with temperature than energy emitted at longer wavelengths.<ref>{{Cite journal |last=Shao |first=Gaofeng |display-authors=0 |year=2019 |title=Improved oxidation resistance of high emissivity coatings on fibrous ceramic for reusable space systems |journal=Corrosion Science |volume=146 |pages=233–246 |arxiv=1902.03943 |doi=10.1016/j.corsci.2018.11.006 |s2cid=118927116}}</ref>

Planck's 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 [[Physics:Radiance|SI units]] of <math>B_{\nu} </math> are {{nowrap|[[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 {{nowrap|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 [[Physics:Statistical mechanics|statistical mechanics]],<ref name="Planck01" /> which he described as "an act of desperation".<ref name="Kragh">{{Citation |last=Kragh |first=Helge |title=Max Planck: the reluctant revolutionary |date=1 December 2000 |url=https://physicsworld.com/a/max-planck-the-reluctant-revolutionary/ |archive-url=https://web.archive.org/web/20090108204645/http://physicsworld.com/cws/article/print/373 |publisher=PhysicsWorld.com |archive-date=2009-01-08 |}}</ref> One of his new boundary conditions was
{{blockquote|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 |last=Kragh |first=Helge |title=Quantum Generations: A History of Physics in the Twentieth Century |url=https://books.google.com/books?id=ELrFDIldlawC |page=62 |year=1999 |access-date=2021-10-31 |archive-url=https://web.archive.org/web/20211206180414/https://books.google.com/books?id=ELrFDIldlawC |url-status=live |publisher=Princeton University Press |isbn=978-0-691-09552-3 |archive-date=2021-12-06}}</ref> but one that 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 "[[Physics:Planck–Einstein relation|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 defined value.<ref name="Planck01" /> He also made the first determination of the [[Physics:Boltzmann constant|Boltzmann constant]] <math>k_\text{B}</math> from the same data and theory.<ref name="PlanckNobel">{{Citation |last=Planck |first=Max |title=The Genesis and Present State of Development of the Quantum Theory (Nobel Lecture) |date=2 June 1920 |url=http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html |access-date=13 December 2008 |archive-url=https://web.archive.org/web/20110715190331/http://nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html |url-status=live |archive-date=15 July 2011 |}}</ref>

[[File:Black body.svg|350px|thumb|The observed Planck curves at different temperatures, and the divergence of the theoretical Rayleigh–Jeans (black) curve from the observed Planck curve at 5000K.]]

=== Development and application ===
The black-body problem was revisited in 1905, when [[Biography:John William Strutt, 3rd Baron Rayleigh|Lord Rayleigh]] and [[Biography:James Jeans|James Jeans]] (on the one hand) and [[Biography:Albert Einstein|Albert Einstein]] (on the other hand) independently proved that classical electromagnetism could ''never'' account for the observed spectrum. These proofs are commonly known as the "[[Physics:Ultraviolet catastrophe|ultraviolet catastrophe]]", a name coined by Paul Ehrenfest in 1911. They contributed greatly (along with Einstein's work on the [[Physics:Photoelectric effect|photoelectric effect]]) in convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The first Solvay Conference in 1911 was devoted to "the theory of radiation and quanta".<ref>{{Citation |title=Previous Solvay Conferences on Physics |url=http://www.solvayinstitutes.be/Conseils%20Solvay/PreviousPhysics.html |access-date=12 December 2008 |archive-url=https://web.archive.org/web/20081216120021/http://www.solvayinstitutes.be/Conseils%20Solvay/PreviousPhysics.html |url-status=dead |publisher=International Solvay Institutes |archive-date=16 December 2008}}</ref>

==== Photoelectric effect ====
{{Main|Physics: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 [[Biography:Heinrich Hertz|Heinrich Hertz]],<ref name="Nobel21">See, e.g., {{Citation |last=Arrhenius |first=Svante |title=Presentation speech of the 1921 Nobel Prize for Physics |date=10 December 1922 |url=http://nobelprize.org/nobel_prizes/physics/laureates/1921/press.html |access-date=13 December 2008 |archive-url=https://web.archive.org/web/20110904232203/http://www.nobelprize.org/nobel_prizes/physics/laureates/1921/press.html |url-status=live |archive-date=4 September 2011 |author-link=Svante Arrhenius}}</ref> who published the first thorough investigation in 1887. Another particularly thorough investigation was published by [[Biography:Philipp Lenard|Philipp Lenard]] (Lénárd Fülöp) in 1902.<ref name="Lenard">{{Citation |last=Lenard |first=P. |title=Ueber die lichtelektrische Wirkung |url=https://zenodo.org/record/1424009 |journal=[[Physics:Annalen der Physik|Annalen der Physik]] |volume=313 |issue=5 |pages=149–98 |year=1902 |access-date=2019-07-03 |archive-url=https://web.archive.org/web/20190818204108/https://zenodo.org/record/1424009 |url-status=live |bibcode=1902AnP...313..149L |doi=10.1002/andp.19023130510 |archive-date=2019-08-18 |author-link=Philipp Lenard}}</ref> Einstein's 1905 paper<ref>{{Citation |last=Einstein |first=Albert |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=[[Physics:Annalen der Physik|Annalen der Physik]] |volume=17 |issue=6 |pages=132–48 |year=1905 |access-date=2009-12-03 |archive-url=https://web.archive.org/web/20110709180735/http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_132-148.pdf |url-status=live |bibcode=1905AnP...322..132E |doi=10.1002/andp.19053220607 |archive-date=2011-07-09 |author-link=Albert Einstein |doi-access=free}}</ref> discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921,<ref name="Nobel21" /> after his predictions had been confirmed by the experimental work of [[Biography:Robert Andrews Millikan|Robert Andrews Millikan]].<ref name="Millikan">{{Citation |last=Millikan |first=R. A. |title=A Direct Photoelectric Determination of Planck's '''h''' |journal=[[Physics:Physical Review|Physical Review]] |volume=7 |issue=3 |pages=355–88 |year=1916 |bibcode=1916PhRv....7..355M |doi=10.1103/PhysRev.7.355 |author-link=Robert Andrews Millikan |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 |last=Isaacson |first=Walter |title=Einstein: His Life and Universe |date=2007-04-10 |publisher=Simon and Schuster |url=https://books.google.com/books?id=cdxWNE7NY6QC |access-date=2021-10-31 |archive-url=https://web.archive.org/web/20200109170326/https://books.google.com/books?id=cdxWNE7NY6QC |url-status=live |isbn=978-1-4165-3932-2 |archive-date=2020-01-09}}, pp. 309–314.</ref><ref>{{Cite web |title=The Nobel Prize in Physics 1921 |url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1921/ |url-status=live |archive-url=https://web.archive.org/web/20180703190346/https://www.nobelprize.org/nobel_prizes/physics/laureates/1921/ |archive-date=2018-07-03 |access-date=2014-04-23 |publisher=Nobelprize.org}}</ref>

Before 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 characterize different types of radiation. The energy transferred by a wave in a given time is called its [[Physics:Intensity|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 color of the light might be very similar. Other waves, such as sound or the waves crashing against a seafront, also have their intensity. However, the energy account of the photoelectric effect did not seem to agree with the wave description of light.

The "photoelectrons" emitted as a result of the photoelectric effect have a certain [[Physics:Kinetic energy|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 [[Physics:Work function|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>*{{Cite journal |last=Smith |first=Richard |year=1962 |title=Two Photon Photoelectric Effect |journal=[[Physics:Physical Review|Physical Review]] |volume=128 |issue=5 |page=2225 |bibcode=1962PhRv..128.2225S |doi=10.1103/PhysRev.128.2225}}
*{{cite journal|doi=10.1103/PhysRev.130.2599.4|title=Two-Photon Photoelectric Effect|year=1963|last1=Smith|first1=Richard|journal=[[Physics:Physical Review|Physical Review]]|volume=130|issue=6|page=2599|bibcode = 1963PhRv..130.2599S |doi-access=free}}</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 [[Physics:Photon|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|Physics: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.]]

It was John William Nicholson in 1912 who introduced h-bar into the theory of the atom which was the first quantum and nuclear atom and the first to quantize angular momentum as ''h''/2{{pi}}.<ref name="Heilbron">{{Cite journal |last=Heilbron |first=John L. |year=2013 |title=The path to the quantum atom |journal=Nature |volume=498 |issue=7452 |pages=27–30 |doi=10.1038/498027a |pmid=23739408 |s2cid=4355108}}</ref><ref>{{Cite journal |last=Nicholson |first=J. W. |year=1911 |title=The spectrum of Nebulium |journal=[[Organization:Monthly Notices of the Royal Astronomical Society|Monthly Notices of the Royal Astronomical Society]] |volume=72 |page=49 |bibcode=1911MNRAS..72...49N |doi=10.1093/mnras/72.1.49 |doi-access=free}}</ref><ref>*{{Cite journal |last=Nicholson |first=J. W. |year=1911 |title=The Constitution of the Solar Corona I |journal=[[Organization:Monthly Notices of the Royal Astronomical Society|Monthly Notices of the Royal Astronomical Society]] |volume=72 |page=139 |bibcode=1911MNRAS..72..139N |doi=10.1093/mnras/72.2.139 |doi-access=free}}
*{{cite journal |doi=10.1093/mnras/72.8.677 |title=The Constitution of the Solar Corona II |year=1912 |last1=Nicholson |first1=J. W. |journal=Monthly Notices of the Royal Astronomical Society |volume=72 |issue=8 |pages=677–693 |doi-access=free }}
*{{cite journal |doi=10.1093/mnras/72.9.729 |title=The Constitution of the Solar Corona III |year=1912 |last1=Nicholson |first1=J. W. |journal=Monthly Notices of the Royal Astronomical Society |volume=72 |issue=9 |pages=729–740 |doi-access=free }}</ref><ref>{{Cite journal |last=Nicholson |first=J. W. |year=1912 |title=On the new nebular line at λ4353 |journal=[[Organization:Monthly Notices of the Royal Astronomical Society|Monthly Notices of the Royal Astronomical Society]] |volume=72 |issue=8 |page=693 |bibcode=1912MNRAS..72..693N |doi=10.1093/mnras/72.8.693 |doi-access=free}}</ref><ref name="McCormmach">{{Cite journal |last=McCormmach |first=Russell |year=1966 |title=The Atomic Theory of John William Nicholson |journal=Archive for History of Exact Sciences |volume=3 |issue=2 |pages=160–184 |doi=10.1007/BF00357268 |jstor=41133258 |s2cid=120797894}}</ref> [[Biography:Niels Bohr|Niels Bohr]] quoted him in his 1913 paper of the Bohr model of the atom.<ref name="Bohr">{{Cite journal |last=Bohr |first=N. |year=1913 |title=On the constitution of atoms and molecules |url=https://zenodo.org/record/2493915 |url-status=live |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |series=6th series |volume=26 |issue=151 |pages=1–25 |bibcode=1913PMag...26..476B |doi=10.1080/14786441308634955 |archive-url=https://web.archive.org/web/20230307022713/https://zenodo.org/record/2493915 |archive-date=2023-03-07 |access-date=2023-07-23}}</ref> The influence of the work of Nicholson's nuclear quantum atomic model on Bohr's model has been written about by many historians.<ref>{{Cite journal |last1=Hirosige |first1=Tetu |last2=Nisio |first2=Sigeko |year=1964 |title=Formation of Bohr's theory of atomic constitution |journal=Japanese Studies in History of Science |volume=3 |pages=6–28}}</ref><ref>J. L. Heilbron, A History of Atomic Models from the Discovery of the Electron to the Beginnings of Quantum Mechanics, diss. (University of California, Berkeley, 1964).</ref><ref name=McCormmach/>

[[Biography:Niels Bohr|Niels Bohr]] introduced the third quantized model of the atom in 1913, in an attempt to overcome a major shortcoming of [[Biography:Ernest Rutherford|Rutherford's]] classical model. The first quantized model of the atom was introduced in 1910 by Arthur Erich Haas and was discussed at the 1911 Solvay conference.<ref name=Heilbron/><ref name="Bohr" /> In classical electrodynamics, a charge moving in a circle should radiate electromagnetic radiation. If that charge were to be an electron orbiting a [[Physics: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 [[Physics:Rydberg constant|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 [[Physics:Rydberg formula|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>\hbar=\frac{h}{2\pi}</math>, now known as the reduced Planck constant or Dirac constant, as the quantum of [[Physics:Angular momentum|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 [[Physics:Matrix mechanics|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|Physics:Uncertainty principle}}
The Planck constant also occurs in statements of [[Biography:Werner Heisenberg|Werner Heisenberg]]'s uncertainty principle. Given numerous particles prepared in the same state, the [[Physics:Uncertainty|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|standard deviation]] of the measured value from its [[Expected value|expected value]]. There are several other such pairs of physically measurable conjugate variables 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|commutator]] relationship between the [[Physics:Position operator|position operator]] <math>\hat{x}</math> and the [[Physics:Momentum operator|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 [[Physics:Planck relation|Planck relation]] connects the particular [[Physics:Photon energy|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|wavelength]] {{math|''λ''}}, and [[Physics:Speed of light|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>

==== de Broglie wavelength ====

In 1923, [[Biography:Louis de Broglie|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 afterward. This holds throughout the quantum theory, including [[Physics:Electrodynamics|electrodynamics]].  The de Broglie wavelength {{math|''λ''}} of the particle is given by
: <math>\lambda = \frac{h}{p},</math>
where {{math|''p''}} denotes the linear [[Physics:Momentum|momentum]] of a particle, such as a photon, or any other [[Physics:Elementary particle|elementary particle]].

The [[Physics: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 [[Physics:Wavenumber|angular wavenumber]].

These two relations are the temporal and spatial parts of the special relativistic expression using 4-vectors.

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

==== Statistical mechanics ====
Classical [[Physics:Statistical mechanics|statistical mechanics]] requires the existence of {{math|''h''}} (but does not define its value).<ref>{{Citation |last1=Giuseppe Morandi |title=Statistical mechanics: an intermediate course |url=https://books.google.com/books?id=MhInFlnNsREC&pg=PA51 |page=84 |year=2001 |access-date=2021-10-31 |archive-url=https://web.archive.org/web/20211206180408/https://books.google.com/books?id=MhInFlnNsREC&pg=PA51 |url-status=live |isbn=978-981-02-4477-4 |archive-date=2021-12-06 |last2=F. Napoli |last3=E. Ercolessi|publisher=World Scientific }}</ref> Eventually, following upon Planck's discovery, it was speculated that physical [[Physics:Action|action]] could not take on an arbitrary value, but instead was restricted to integer multiples of a very small quantity, the "[elementary] [[Physics:Quantum|quantum]] of action", now called the ''Planck constant''.<ref>{{Cite book |last=ter Haar |first=D. |url=https://archive.org/details/oldquantumtheory0000haar |title=The Old Quantum Theory |publisher=Pergamon Press |year=1967 |isbn=978-0-08-012101-7 |page=[https://archive.org/details/oldquantumtheory0000haar/page/133 133] |url-access=registration}}</ref> This was a significant conceptual part of the so-called "[[Physics:Old quantum theory|old quantum theory]]" developed by physicists including [[Biography:Niels Bohr|Bohr]], [[Biography:Arnold Sommerfeld|Sommerfeld]], and [[Biography:Jun Ishiwara|Ishiwara]], in which particle trajectories exist but are hidden, but quantum laws constrain them based on their action. This view has been 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. 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 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 |first=Albert |title=Physics and Reality |url=http://www.kostic.niu.edu/Physics_and_RealityAlbert_Einstein.pdf |journal=Daedalus |volume=132 |issue=4 |page=24 |year=2003 |archive-url=https://web.archive.org/web/20120415132339/http://www.kostic.niu.edu/Physics_and_Reality-Albert_Einstein.pdf |url-status=dead |doi=10.1162/001152603771338742 |s2cid=57559543 |archive-date=2012-04-15 |quote=The question is first: How can one assign a discrete succession of energy values&nbsp;{{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&nbsp;{{math  |''q<sub>r</sub>''}} and the corresponding momenta&nbsp;{{math |''p<sub>r</sub>''}})? The Planck constant {{math |''h''}} relates the frequency&nbsp;{{math |''H<sub>σ</sub>''/''h''}} to the energy values&nbsp;{{math |''H<sub>σ</sub>''}}. It is therefore sufficient to give to the system a succession of discrete frequency values.}}</ref>

== Dimension and value ==
The Planck constant has the same dimensions as [[Physics:Action|action]] and as [[Physics:Angular momentum|angular momentum]]. In [[International System of Units|SI units]], the Planck constant is expressed with the unit [[Joule|joule]] per [[Hertz|hertz]] (J⋅Hz{{sup|−1}}) or joule-second (J⋅s).

<math display="block">h = \mathrm{6.626\ 070\ 15\times 10^{-34} {J \cdot s}}</math>
<math display="block">\hbar = {h \over 2\pi} = \mathrm{1.054\ 571\ 817...\times 10^{-34} \ {J \cdot s}} = \mathrm{ 6.582\ 119\ 569... \times 10^{-16}\ {eV \cdot s}}.</math>

The above values have been adopted as fixed in the [[2019 redefinition of the SI base units|2019 redefinition of the SI base units]].

Since 2019, the numerical value of the Planck constant has been fixed, with a finite decimal representation.  This fixed value is used to define the Si unit of mass, the [[Kilogram|kilogram]]: "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|metre]] and the [[Second|second]] are defined in terms of [[Physics:Speed of light|speed of light]] {{Mvar|c}} and duration of [[Physics:Hyperfine structure|hyperfine transition]] of the [[Physics:Ground state|ground state]] of an unperturbed [[Physics:Caesium-133|caesium-133]] atom {{Math|Δ''ν''<sub>Cs</sub>}}."<ref>{{SIbrochure9th}}</ref> Technologies of mass [[Metrology|metrology]] such as the [[Engineering:Kibble balance|Kibble balance]] measure refine the value of kilogram applying fixed value of the Planck constant.

=== Significance of the value ===
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 typical of the order of kilojoules and times are typical of the order of seconds or minutes, the Planck constant is very small. When the [[Physics:Action|product of energy and time]] for a physical event approaches the Planck constant, quantum effects dominate.<ref name=FeynmanII>{{Cite web |title=The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action |url=https://www.feynmanlectures.caltech.edu/II_19.html |access-date=2023-11-03 |website=www.feynmanlectures.caltech.edu}}</ref>

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, in [[Green|green]] light (with a [[Wavelength|wavelength]] of 555&nbsp;[[Nanometre|nanometre]]s or a frequency of {{val|540|u=THz}}) each [[Physics:Photon|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 [[Physics:Avogadro constant|Avogadro constant]], {{physconst|NA|symbol=yes}}, with the result of {{val|216|u=kJ}}, about the food energy in three apples.{{citation needed|date=October 2023}}

== {{anchor|Reduced Planck constant|Reduced Planck constant ℏ|Dirac constant}} Reduced Planck constant ℏ==
<!--'Reduced Planck constant' and 'Dirac constant' redirect here-->
In many applications, the Planck constant <math display="inline">h</math>  naturally appears in combination with <math display="inline">2\pi</math>  as {{nowrap|<math display="inline">h/(2\pi)</math>,}} which can be traced to the fact that in these applications it is natural to use the [[Physics:Angular frequency|angular frequency]] (in [[Radian|radian]]s per second) rather than plain [[Physics:Frequency|frequency]] (in [[Physics:Cycle per second|cycle]]s per second or [[Hertz|hertz]]). For this reason, it is often useful to absorb that factor of {{math|2''π''}} into the Planck constant by introducing the '''reduced Planck constant'''<!--
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--><ref name="NIST reduced h">{{Cite web |date=20 May 2019 |title=reduced Planck constant |url=https://physics.nist.gov/cgi-bin/cuu/Value?hbar |url-status=live |archive-url=https://web.archive.org/web/20230408225435/https://physics.nist.gov/cgi-bin/cuu/Value?hbar |archive-date=2023-04-08 |access-date=2023-09-03 |website=The NIST Reference on Constants, Units, and Uncertainty |publisher=[[Organization:National Institute of Standards and Technology|NIST]]}}</ref><!--
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--><ref>{{cite book |last1=Lyth |first1=David H. | last2=Liddle |first2=Andrew R. |author-link2=Andrew R. Liddle|title=The Primordial Density Perturbation: Cosmology, Inflation and the Origin of Structure |date=11 June 2009 |publisher=Cambridge University Press |isbn=978-1-139-64374-0 |url=https://books.google.com/books?id=IPwgAwAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/The_Primordial_Density_Perturbation/IPwgAwAAQBAJ?hl=en&gbpv=1&dq=%22reduced+Planck+constant%22&pg=PA482&printsec=frontcover 482]}} <!--
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-->(or '''reduced Planck's constant'''<!--
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--><ref>{{cite book |last1=Huang |first1=Kerson | title=Quantum Field Theory: From Operators to Path Integrals |date=26 April 2010 |publisher=John Wiley & Sons |isbn=978-3-527-40846-7 |url=https://books.google.com/books?id=5ZWZRpvZzUEC |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Field_Theory/5ZWZRpvZzUEC?hl=en&gbpv=1&dq=%22are+operators+satisfying%22+%22is+the+reduced+Planck%E2%80%99s+constant,+and%22&pg=PA5&printsec=frontcover 5]}} <!--
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--><ref>{{cite book |last1=Schmitz |first1=Kenneth S. |title=Physical Chemistry: Concepts and Theory |date=11 November 2016 |publisher=Elsevier |isbn=978-0-12-800600-9 |url=https://books.google.com/books?id=Y1ydBgAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Physical_Chemistry/Y1ydBgAAQBAJ?hl=en&gbpv=1&dq=%22reduced+Planck%27s+constant%22&pg=PA788&printsec=frontcover 788]}}<!--
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-->), equal to the Planck constant divided by <math display="inline">2 \pi</math><ref name="NIST reduced h" /> and denoted by <math display="inline">\hbar</math> (pronounced ''h-bar''<!--
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--><ref name="Chabay and Sherwood">{{Cite book |last1=Chabay |first1=Ruth W. |url=https://books.google.com/books?id=zMWHEAAAQBAJ |title=Matter and Interactions |last2=Sherwood |first2=Bruce A. |date=20 November 2017 |publisher=John Wiley & Sons |isbn=978-1-119-45575-2 |language=en}}</ref>{{rp|page=[https://books.google.com/books?id=zMWHEAAAQBAJ&dq=%22pronounced+h-bar%22&pg=PA336 336]}}). <!--
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Many of the most important equations, relations, definitions, and results of quantum mechanics are customarily written using the reduced Planck constant <math display="inline">\hbar</math>  rather than the Planck constant <math display="inline">h</math>, including the [[Physics:Schrödinger equation|Schrödinger equation]], [[Physics:Momentum operator|momentum operator]], [[Physics:Canonical commutation relation|canonical commutation relation]], Heisenberg's uncertainty principle, and [[Planck units]].<!--
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--><ref>{{cite book |last1=Schwarz |first1=Patricia M. |last2=Schwarz |first2=John H. |title=Special Relativity: From Einstein to Strings |date=25 March 2004 |publisher=Cambridge University Press |isbn=978-1-139-44950-2 |url=https://books.google.com/books?id=BmhQxWwk0K4C |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Special_Relativity/BmhQxWwk0K4C?hl=en&gbpv=1&dq=%22The%20more%20commonly%20used%20constant%2C%20however%2C%20is%22&pg=PA105&printsec=frontcover 104]}}

Because the fundamental equations look simpler when written using <math display="inline">\hbar</math> as opposed to {{nowrap|<math display="inline">h</math>,}} it is usually <math display="inline">\hbar</math> rather than <math display="inline">h</math> that gives the most reliable results when used in [[Fermi problem|order-of-magnitude estimates]].<!--
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--><ref>{{cite book |last1=Lévy-Leblond|first1=Jean-Marc|chapter=The meanings of Planck's constant|chapter-url=https://ssqg.ans.pw.edu.pl/en/wp-content/uploads/2021/04/JMLL_Meanings-of-h.pdf|archive-url=https://web.archive.org/web/20231014082224/https://ssqg.ans.pw.edu.pl/en/wp-content/uploads/2021/04/JMLL_Meanings-of-h.pdf|archive-date=2023-10-14|editor-last1=Beltrametti |editor-first1=E. |editor-last2=Rimini |editor-first2=A. |editor-last3=Robotti |editor-first3=Nadia |title=One Hundred Years of H: Pavia, 14-16 September 2000 |date=2002 |publisher=Italian Physical Society |isbn=978-88-7438-003-9 |url=https://books.google.com/books?id=tqbvAAAAMAAJ |language=en}}</ref>{{rp|pages=8–9}} <!--
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--> {{efn|As examples, the preceding reference shows what happens when one uses [[Dimensional analysis|dimensional analysis]] to obtain estimates for the ionization energy and the size of a hydrogen atom. If we use the [[Physics:Gaussian units|Gaussian units]], then the relevant parameters that determine the ionization energy <math display="inline">E_{\text{i}}</math> are the mass of the electron {{nowrap|<math display="inline">m_{\text{e}}</math>,}} the electron charge {{nowrap|<math display="inline">e</math>,}} and either the Planck constant <math display="inline">h</math> or the reduced Planck constant <math display="inline">\hbar</math> (since <math display="inline">h</math> and <math display="inline">\hbar</math> have the same dimensions, they will enter the dimensional analysis in the same way). One obtains that <math display="inline">E_{\text{i}}</math> must be proportional to <math display="inline">m_{\text{e}} e^{4}/h^{2}</math> if we used {{nowrap|<math display="inline">h</math>,}} and to <math display="inline">m_{\text{e}} e^{4}/\hbar^{2}</math> is we used {{nowrap|<math display="inline">\hbar</math>.}} In an order-of-magnitude estimate, we take that the constant of proportionality is 1. Now, the actual correct answer is {{nowrap|<math display="inline">E_{\text{i}}=m_{\text{e}} e^{4}/(2\hbar^{2})</math>;}}<!--
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--><ref>{{cite book |last1=Shu |first1=Frank |title=The Physical Universe: An Introduction to Astronomy |date=1982 |publisher=University Science Books |isbn=978-0-935702-05-7 |url=https://books.google.com/books?id=v_6PbAfapSAC |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/The_Physical_Universe/v_6PbAfapSAC?hl=en&gbpv=1&bsq=%22values%20are%20given%20by%22 45]}} <!--
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-->therefore, if we choose to use <math display="inline">\hbar</math> as one of our parameters, our estimate will off by a factor of 2, whereas if we choose to use {{nowrap|<math display="inline">h</math>,}} it will be off by a factor of {{nowrap|<math display="inline">4\pi^{2}/2 \approx 20</math>.}}  Similarly for the estimate of the size of a hydrogen atom: depending on whether we use <math display="inline">h</math> or <math display="inline">\hbar</math> as one of the parameters, we get either <math display="inline">h^{2}/(m_{\text{e}} e^{2})</math> or {{nowrap|<math display="inline">\hbar^{2}/(m_{\text{e}} e^{2})</math>.}} The latter happens to be exactly correct,<ref>{{cite web |title=Bohr Radius -- from Eric Weisstein's World of Physics |url=https://scienceworld.wolfram.com/physics/BohrRadius.html |website=scienceworld.wolfram.com |access-date=14 October 2023 |language=en|archive-url=https://web.archive.org/web/20231014072210/https://scienceworld.wolfram.com/physics/BohrRadius.html|archive-date=2023-10-14}}</ref> whereas the estimate using <math display="inline">h</math> is off by a factor of {{nowrap|<math display="inline">4\pi^{2} \approx 40</math>.}}}}

=== Names ===

The reduced Planck constant is known by many other names: the '''rationalized Planck constant'''<!--
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--><ref name="Oxford Planck constant">{{Cite encyclopedia |title=Planck constant |encyclopedia=A Dictionary of Physics |publisher=OUP Oxford |location=Oxford, UK |url={{google books|plainurl=y|id=YPmFDwAAQBAJ}} |date=2017 |editor-last=Rennie |editor-first=Richard |isbn=978-0198821472 |editor2-last=Law |editor2-first=Jonathan|edition= 7th|series=Oxford Quick Reference}}</ref>{{rp|page=[https://www.google.com/books/edition/A_Dictionary_of_Science/40RdDgAAQBAJ?hl=en&gbpv=1&dq=%22physics)%20the%20rationalized%20Planck%20constant%20(or%20Dirac%20constant)%22&pg=PT1470&printsec=frontcover 726]}} <!--
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--><ref>{{cite book |title=The International Encyclopedia of Physical Chemistry and Chemical Physics |date=1960 |publisher=Pergamon Press |url=https://books.google.com/books?id=ufTvAAAAMAAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/The_International_Encyclopedia_of_Physic/ufTvAAAAMAAJ?hl=en&gbpv=1&bsq=%22rationalized%20Planck%22&printsec=frontcover 10]}} <!--
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--><ref>{{cite book |last1=Vértes |first1=Attila |last2=Nagy |first2=Sándor |last3=Klencsár |first3=Zoltán |last4=Lovas |first4=Rezso György |last5=Rösch |first5=Frank |title=Handbook of Nuclear Chemistry|date=10 December 2010 |publisher=Springer Science & Business Media |isbn=978-1-4419-0719-6 |url=https://books.google.com/books?id=NQyF6KaUScQC |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Handbook_of_Nuclear_Chemistry/NQyF6KaUScQC?hl=en&gbpv=1&dq=%22called+rationalized+Planck+constant%22&pg=PT616&printsec=frontcover -]}} <!--
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-->(or '''rationalized Planck's constant'''<!--
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--><ref name="Bethe and Salpeter">{{cite book |editor-last1=Flügge |editor-first1=Siegfried |title=Handbuch der Physik: Atome I-II |date=1957 |publisher=Springer |url=https://books.google.com/books?id=A-APAQAAMAAJ |last1=Bethe |first1=Hans A. |last2=Salpeter |first2=Edwin E. |author-link2=Edwin E. Salpeter|chapter=Quantum Mechanics of One- and Two-Electron Atoms |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Handbuch_der_Physik_Atome_I_II/A-APAQAAMAAJ?hl=en&gbpv=1&bsq=rationalized 334]}} <!--
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--><ref>{{cite book |last1=Lang |first1=Kenneth |title=Astrophysical Formulae: A Compendium for the Physicist and Astrophysicist |date=11 November 2013 |publisher=Springer Science & Business Media |isbn=978-3-662-11188-8 |url=https://books.google.com/books?id=LyzrCAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Astrophysical_Formulae/LyzrCAAAQBAJ?hl=en&gbpv=1&dq=%22Astrophysical+Formulae%22+%22rationalized+Planck%27s%22&pg=PR9&printsec=frontcover ix]}} <!--
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--><ref>{{cite book |last1=Galgani |first1=L. |last2=Carati |first2=A. |last3=Pozzi |first3=B. |chapter=The Problem of the Rate of Thermalization, and the Relations between Classical and Quantum Mechanics |date=December 2002 |pages=111–122 |doi=10.1142/9789812776273_0011|editor-last1=Fabrizio|editor-first1=Mauro |editor-last2=Morro|editor-first2=Angelo|title=Mathematical Models and Methods for Smart Materials, Cortona, Italy, 25 – 29 June 2001 |isbn=978-981-238-235-1 }}</ref>{{rp|page=[https://www.google.com/books/edition/Mathematical_Models_and_Methods_for_Smar/wDRpDQAAQBAJ?hl=en&gbpv=1&dq=%22is+the+(rationalized)+Planck%27s+constant%22&pg=PA112&printsec=frontcover 112]}} <!--
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-->), the '''Dirac constant'''<!--
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--><ref name="Fox Student Guide Atomic">{{Cite book |last=Fox |first=Mark |url=https://books.google.com/books?id=pVtlDwAAQBAJ |title=A Student's Guide to Atomic Physics |date=14 June 2018 |publisher=Cambridge University Press |isbn=978-1-316-99309-5 |language=en}}</ref>{{rp|page=[https://books.google.com/books?id=pVtlDwAAQBAJ&dq=%22Dirac+constant%22&pg=PA275 275]}} <!--
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--><ref name="Oxford Planck constant" />{{rp|page=[https://www.google.com/books/edition/A_Dictionary_of_Science/40RdDgAAQBAJ?hl=en&gbpv=1&dq=%22physics)%20the%20rationalized%20Planck%20constant%20(or%20Dirac%20constant)%22&pg=PT1470&printsec=frontcover 726]}} <!--
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--><ref>{{cite book |last1=Kleiss |first1=Ronald |title=Quantum Field Theory: A Diagrammatic Approach |date=10 June 2021 |publisher=Cambridge University Press |isbn=978-1-108-78750-5 |url=https://books.google.com/books?id=G_UvEAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Field_Theory/G_UvEAAAQBAJ?hl=en&gbpv=1&dq=%22We+shall+use+the+reduced+Planck+constant%22&pg=PR15&printsec=frontcover xv]}} <!--
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-->(or '''Dirac's constant'''<!--
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--><ref>{{cite book |last1=Zohuri |first1=Bahman |title=Thermal Effects of High Power Laser Energy on Materials |date=5 January 2021 |publisher=Springer Nature |isbn=978-3-030-63064-5 |url=https://books.google.com/books?id=P3gSEAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Thermal_Effects_of_High_Power_Laser_Ener/P3gSEAAAQBAJ?hl=en&gbpv=1&dq=%22Dirac%27s+constant%22&pg=PA148&printsec=frontcover 148]}} <!--
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--><ref>{{cite book |last1=Balian |first1=Roger |title=From Microphysics to Macrophysics: Methods and Applications of Statistical Physics. Volume II |date=26 June 2007 |publisher=Springer Science & Business Media |isbn=978-3-540-45480-9 |url=https://books.google.com/books?id=cUY_AAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/From_Microphysics_to_Macrophysics/cUY_AAAAQBAJ?hl=en&gbpv=1&dq=%22Dirac%27s+constant%22&pg=PA604&printsec=frontcover 604]}} <!--
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--><ref>{{cite book |last1=Chen |first1=C. Julian |title=Physics of Solar Energy |date=15 August 2011 |publisher=John Wiley & Sons |isbn=978-1-118-04459-9 |url=https://books.google.com/books?id=dFKz6GwSVNEC |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Physics_of_Solar_Energy/dFKz6GwSVNEC?hl=en&gbpv=1&dq=%22Dirac%27s+constant%22&pg=PA313&printsec=frontcover 313]}}<!--
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-->), the '''Dirac <math display="inline">h</math>'''<!--
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--><ref>{{cite encyclopedia |encyclopedia= Britannica |title=Dirac h |url= https://www.britannica.com/science/Dirac-h|access-date=2023-09-27  |archive-url=https://web.archive.org/web/20230217112008/https://www.britannica.com/science/Dirac-h |archive-date=2023-02-17 |url-status=live }}</ref><!--
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--><ref>{{cite book |last1=Shoenberg |first1=D. |title=Magnetic Oscillations in Metals |date=3 September 2009 |publisher=Cambridge University Press |isbn=978-1-316-58317-3 |url=https://books.google.com/books?id=HM_LCgAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Magnetic_Oscillations_in_Metals/HM_LCgAAQBAJ?hl=en&gbpv=1&dq=%22Dirac+h%22&pg=PR18&printsec=frontcover xviii]}} <!--
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-->(or '''Dirac's <math display="inline">h</math>'''<!--
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--><ref>{{cite book |last1=Powell |first1=John L. |last2=Crasemann |first2=Bernd |title=Quantum Mechanics |date=5 May 2015 |publisher=Courier Dover Publications |isbn=978-0-486-80478-1 |url=https://books.google.com/books?id=GHifCQAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Mechanics/GHifCQAAQBAJ?hl=en&gbpv=1&dq=%22Dirac+h%22&pg=PA17&printsec=frontcover 17]}} <!--
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-->), the '''Dirac <math display="inline">\hbar</math>'''<!--
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--><ref>{{cite book |last1=Dresden |first1=Max |title=H.A. Kramers Between Tradition and Revolution |date=6 December 2012 |publisher=Springer Science & Business Media |isbn=978-1-4612-4622-0 |url=https://books.google.com/books?id=1ggDCAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/H_A_Kramers_Between_Tradition_and_Revolu/1ggDCAAAQBAJ?hl=en&gbpv=1&dq=%22Dirac+h%22&pg=PA187&printsec=frontcover 187]}} <!--
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-->(or '''Dirac's <math display="inline">\hbar</math>'''<!--
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--><ref>{{cite book |last1=Johnson |first1=R. E. |title=Introduction to Atomic and Molecular Collisions |date=6 December 2012 |publisher=Springer Science & Business Media |isbn=978-1-4684-8448-9 |url=https://books.google.com/books?id=cKPTBwAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Introduction_to_Atomic_and_Molecular_Col/cKPTBwAAQBAJ?hl=en&gbpv=1&dq=%22Dirac+h%22&pg=PA273&printsec=frontcover 273]}} <!--
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--><ref>{{cite book |last1=Garcia |first1=Alejandro |last2=Henley |first2=Ernest M. |title=Subatomic Physics |date=13 July 2007 |publisher=World Scientific Publishing Company |isbn=978-981-310-167-8 |url=https://books.google.com/books?id=j847DQAAQBAJ |language=en |edition=3rd }}</ref>{{rp|page=[https://www.google.com/books/edition/Subatomic_Physics_3rd_Edition/j847DQAAQBAJ?hl=en&gbpv=1&dq=%22Dirac+h%22&pg=PA14&printsec=frontcover 14]}} <!--
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--><!--

<ref>{{cite book |last1=Kragh |first1=Helge |title=Dirac: A Scientific Biography |date=30 March 1990 |publisher=Cambridge University Press |isbn=978-0-521-38089-8 |url=https://www.google.com/books/edition/Dirac/zXm1Bso1VREC |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Dirac/zXm1Bso1VREC?hl=en&gbpv=1&bsq=%22Dirac%27s%20h%22 23]}}

--><!--
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-->), and '''h-bar'''.<!--
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--><ref name="Holbrow et al Modern Introductory Physics">{{Cite book |last1=Holbrow |first1=Charles H. | url=https://books.google.com/books?id=CYE_AAAAQBAJ |title=Modern Introductory Physics |last2=Lloyd |first2=James N. |last3=Amato |first3=Joseph C. |last4=Galvez |first4=Enrique |last5=Parks |first5=M. Elizabeth |date=14 September 2010 |publisher=Springer Science & Business Media |isbn=978-0-387-79080-0 |location=New York |language=en}}</ref>{{rp|page=[https://books.google.com/books?id=CYE_AAAAQBAJ&dq=%22and+is+called+h-bar%22&pg=PA558 558]}}<!--
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--><ref>{{Cite book |last1=Polyanin |first1=Andrei D. |url=https://books.google.com/books?id=ejzScufwDRUC |title=A Concise Handbook of Mathematics, Physics, and Engineering Sciences |last2=Chernoutsan |first2=Alexei |date=18 October 2010 |publisher=CRC Press |isbn=978-1-4398-0640-1 |language=en}}</ref>{{rp|page=[https://books.google.com/books?id=ejzScufwDRUC&dq=%22called+h-bar%22&pg=PA561 561]}}<!--
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--> It is also common to refer to this <math display="inline">\hbar</math> as “Planck's constant”<!--
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--><ref>{{cite book |last1=Dowling |first1=Jonathan P.|title=Schrödinger's Web: Race to Build the Quantum Internet |date=24 August 2020 |publisher=CRC Press |isbn=978-1-000-08017-9 |url=https://books.google.com/books?id=unP2DwAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Schr%C3%B6dinger_s_Web/unP2DwAAQBAJ?hl=en&gbpv=1&dq=People+often+refer+to+%C4%A7+%3D+h/(2%CF%80)+also+as+Plancks+constant&pg=PA55&printsec=frontcover 55]}} <!--
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-->{{efn|Notable examples of such usage include  [[Physics:Course of Theoretical Physics#Volume 3|Landau and Lifshitz]]<!--
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--><ref name="Landau and Lifshit Quantum Mechanics">{{Cite book |last1=Landau |first1=L. D. | url=https://books.google.com/books?id=neBbAwAAQBAJ |title=Quantum Mechanics: Non-Relativistic Theory |last2=Lifshitz |first2=E. M. |author-link2= Evgeny Lifshitz |date=22 October 2013 |publisher=Elsevier |isbn=978-1-4831-4912-7 |language=en}}</ref>{{rp|page=[https://books.google.com/books?id=neBbAwAAQBAJ&dq=%22The+constant+of+proportionality+is+called+Planck%27s%22&pg=PA20 20]}} <!--
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-->and Giffiths,<!--
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--><ref name="Griffiths Quantum">{{Cite book |last1=Griffiths |first1=David J.|url=https://books.google.com/books?id=zU3bDwAAQBAJ |title=Introduction to Quantum Mechanics |last2=Schroeter |first2=Darrell F. |date=20 November 2019 |publisher=Cambridge University Press |isbn=978-1-108-10314-5 |language=en}}</ref>{{rp|page=[https://books.google.com/books?id=zU3bDwAAQBAJ&dq=%22or+rather,+his+original+constant%22&pg=PA3 3]}} <!--
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-->but there are many others, e.g.<!--
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--><ref name="Soviet Planck constant">{{Cite encyclopedia |title=Planck's constant |encyclopedia=The Great Soviet Encyclopedia |publisher=The Gale Group |url=https://encyclopedia2.thefreedictionary.com/Planck%27s+constant |edition=1970-1979, 3rd}}</ref><!--
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--><ref>{{cite book |last1=Itzykson |first1=Claude | last2=Zuber |first2=Jean-Bernard | author-link2=Jean-Bernard Zuber| title=Quantum Field Theory |date=20 September 2012 |publisher=Courier Corporation |isbn=978-0-486-13469-7 |url=https://books.google.com/books?id=CxYCMNrUnTEC |language=en}}</ref>{{rp|page=449}} <!--
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--><ref>{{cite book |last1=Kaku |first1=Michio | title=Quantum Field Theory: A Modern Introduction |date=1993 |publisher=Oxford University Press |isbn=978-0-19-507652-3 |url=https://books.google.com/books?id=6J7vAAAAMAAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Field_Theory/6J7vAAAAMAAJ?hl=en&gbpv=1&bsq=%22corresponds+to+a+power+expansion+in%22&dq=%22corresponds+to+a+power+expansion+in%22&printsec=frontcover 284]}} <!--
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--><ref>{{cite book |last1=Bogoli︠u︡bov |first1=Nikolaĭ Nikolaevich | last2=Shirkov |first2=Dmitriĭ Vasilʹevich | author-link2=Dmitry Shirkov |title=Quantum Fields |date=1982 |publisher=Benjamin/Cummings Publishing Company, Advanced Book Program/World Science Division |isbn=978-0-8053-0983-6 |url=https://books.google.com/books?id=wczvAAAAMAAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Fields/wczvAAAAMAAJ?hl=en&gbpv=1&bsq=%22Throughout+the+book+the+so-called+rational+system+of+units+is+used%22&dq=%22Throughout+the+book+the+so-called+rational+system+of+units+is+used%22&printsec=frontcover 3]}} <!--
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--><ref>{{cite book |last1=Aitchison |first1=Ian J. R.|last2=Hey |first2=Anthony J. G. |author-link2=Tony Hey|title=Gauge Theories in Particle Physics: A Practical Introduction: From Relativistic Quantum Mechanics to QED, Fourth Edition |date=17 December 2012 |publisher=CRC Press |isbn=978-1-4665-1299-3 |url=https://books.google.com/books?id=-v6sPfuyUt8C |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Gauge_Theories_in_Particle_Physics_A_Pra/-v6sPfuyUt8C?hl=en&gbpv=1&dq=%22The+numerical+value+of+Planck%27s+constant+is%22&pg=PA365&printsec=frontcover 365]}} <!--
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--><ref>{{cite book |last1=de Wit |first1=B. | last2=Smith |first2=J. |title=Field Theory in Particle Physics, Volume 1 |date=2 December 2012 |publisher=Elsevier |isbn=978-0-444-59622-2 |url=https://books.google.com/books?id=cCAGmj6XEfkC |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Field_Theory_in_Particle_Physics_Volume/cCAGmj6XEfkC?hl=en&gbpv=1&dq=%22to+choose+the+velocity+of+light+in+vacuum+and+Planck%27s+constant+to+be%22&pg=PA14&printsec=frontcover 14]}} <!--
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--><ref>{{cite book |last1=Brown |first1=Lowell S. | title=Quantum Field Theory |date=1992 |publisher=Cambridge University Press |isbn=978-0-521-46946-3 |url=https://books.google.com/books?id=mKsRAVX1910C |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Field_Theory/mKsRAVX1910C?hl=en&gbpv=1&dq=%22+Temporarily+reverting+to+ordinary+units+that%22&pg=PA18&printsec=frontcover 18]}} <!--
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--> <ref>{{cite book |last1=Buchbinder |first1=Iosif L. |last2=Shapiro |first2=Ilya |title=Introduction to Quantum Field Theory with Applications to Quantum Gravity |date=March 2021 |publisher=Oxford University Press |isbn=978-0-19-883831-9 |url=https://books.google.com/books?id=tL4cEAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Introduction_to_Quantum_Field_Theory_wit/tL4cEAAAQBAJ?hl=en&gbpv=1&dq=%22is+the+Planck+constant.+According+to+the+standard+interpretation%22&pg=PA4&printsec=frontcover 4]}} <!--
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--><ref>{{cite book |author-last=Jaffe|author-first=Arthur|chapter=9. Where does quantum field theory fit into the big picture?|editor-last1=Cao |editor-first1=Tian Yu |title=Conceptual Foundations of Quantum Field Theory |date=25 March 2004 |publisher=Cambridge University Press |isbn=978-0-521-60272-3 |url=https://books.google.com/books?id=d0wS0EJHZ3MC |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Conceptual_Foundations_of_Quantum_Field/d0wS0EJHZ3MC?hl=en&gbpv=1&dq=%22We+are+using+units+in+which+the+velocity+of+light+c%22&pg=PA138&printsec=frontcover 138]}} <!--
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--><ref>{{cite book |last1=Cabibbo |first1=Nicola |last2=Maiani |first2=Luciano |author-link2=Luciano Maiani|last3=Benhar |first3=Omar |title=An Introduction to Gauge Theories |date=28 July 2017 |publisher=CRC Press |isbn=978-1-4987-3452-3 |url=https://books.google.com/books?id=6WxGDwAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/An_Introduction_to_Gauge_Theories/6WxGDwAAQBAJ?hl=en&gbpv=1&dq=%22of+the+potential+can+be+calculated+by+an+expansion+in+powers+of+Planck%E2%80%99s%22&pg=PA251&printsec=frontcover 251]}} <!--
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--><ref>{{cite book |last1=Casalbuoni |first1=Roberto |title=Introduction To Quantum Field Theory |date=6 April 2017 |publisher=World Scientific Publishing Company |isbn=978-981-314-668-6 |url=https://books.google.com/books?id=MFEyDwAAQBAJ |language=en |edition=Second }}</ref>{{rp|page=[https://www.google.com/books/edition/Introduction_To_Quantum_Field_Theory_Sec/MFEyDwAAQBAJ?hl=en&gbpv=1&dq=%22of+light,+c,+and+the+Planck+constant%22+%22appear+almost+everywhere.%22&pg=PA1&printsec=frontcover 1]}} <!--
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--><ref>{{cite book |last1=Das |first1=Ashok |title=Lectures On Quantum Field Theory |edition=2nd|date=24 July 2020 |publisher=World Scientific |isbn=978-981-12-2088-3 |url=https://books.google.com/books?id=gXr1DwAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Lectures_On_Quantum_Field_Theory_Second/gXr1DwAAQBAJ?hl=en&gbpv=1&dq=%22represents+the+Planck%E2%80%99s+constant+and+we+see+that+it+can+be%22&pg=PA622&printsec=frontcover 622]}} <!--
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--><ref>{{cite book |last1=Desai |first1=Bipin R. |title=Quantum Mechanics with Basic Field Theory |date=2010 |publisher=Cambridge University Press |isbn=978-0-521-87760-2 |url=https://books.google.com/books?id=cScBCJ6wLpYC |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Mechanics_with_Basic_Field_Theor/cScBCJ6wLpYC?hl=en&gbpv=1&dq=Planck%E2%80%99s%20constant&pg=PR20&printsec=frontcover xx]}} <!--
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--><ref>{{cite book |last1=Donoghue |first1=John |last2=Sorbo |first2=Lorenzo |title=A Prelude to Quantum Field Theory |date=8 March 2022 |publisher=Princeton University Press |isbn=978-0-691-22348-3 |url=https://books.google.com/books?id=Wv5FEAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/A_Prelude_to_Quantum_Field_Theory/Wv5FEAAAQBAJ?hl=en&gbpv=1&dq=%22Because+the+speed+of+light+c+and+the+Planck+constant%22&pg=PA20&printsec=frontcover 20]}} <!--
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--><ref>{{cite book |last1=Folland |first1=Gerald B. |title=Quantum Field Theory: A Tourist Guide for Mathematicians |date=3 February 2021 |publisher=American Mathematical Soc. |isbn=978-1-4704-6483-7 |url=https://books.google.com/books?id=nPEbEAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Field_Theory_A_Tourist_Guide_for/nPEbEAAAQBAJ?hl=en&gbpv=1&dq=%22In+quantum+mechanics+this+factor+is+Planck%E2%80%99s+constant%22&pg=PA4&printsec=frontcover 4]}} <!--
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--><ref>{{cite book |last1=Fradkin |first1=Eduardo |title=Quantum Field Theory: An Integrated Approach |date=23 March 2021 |publisher=Princeton University Press |isbn=978-0-691-14908-0 |url=https://books.google.com/books?id=ruEIEAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Field_Theory/ruEIEAAAQBAJ?hl=en&gbpv=1&dq=%22that+if+T+is+formally+identified+with+the+Planck+constant%22&pg=PA36&printsec=frontcover 36]}} <!--
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--><ref>{{cite book |last1=Gelis |first1=François |title=Quantum Field Theory |date=11 July 2019 |publisher=Cambridge University Press |isbn=978-1-108-48090-1 |url=https://books.google.com/books?id=BzGdDwAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Field_Theory/BzGdDwAAQBAJ?hl=en&gbpv=1&dq=%22It+turns+out+that+the+number+of+loops+also+counts+the+order+in+the+Planck+constant%22&pg=PA41&printsec=frontcover 41]}} <!--
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--><ref>{{cite book |last1=Greiner |first1=Walter|last2=Reinhardt |first2=Joachim |title=Quantum Electrodynamics |date=9 March 2013 |publisher=Springer Science & Business Media |isbn=978-3-662-05246-4 |url=https://books.google.com/books?id=6y3rCAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Quantum_Electrodynamics/6y3rCAAAQBAJ?hl=en&gbpv=1&dq=%22not+depend+on+Planck%E2%80%99s+constant%22+%22Writing+out+the+constants+%22&pg=PA199&printsec=frontcover 199]}} <!--
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--><ref>{{Cite book |last=Liboff |first=Richard L. |url=https://books.google.com/books?id=xMIlAgQSRUsC |title=Introductory Quantum Mechanics |date=2003 |publisher=Pearson Education |isbn=978-81-317-0441-7 |edition=4th |location=San Francisco |language=en}}</ref>{{rp|page=846}} <!--
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--><ref>{{Cite journal |last=Barut |first=A. O.| date=1 August 1978 |title=The Creation of a Photon: A Heuristic Calculation of Planck's Constant ħ or the Fine Structure Constant α |journal=Zeitschrift für Naturforschung A |volume=33 |issue=8 |pages=993–994 |doi=10.1515/zna-1978-0819|bibcode=1978ZNatA..33..993B |s2cid=45829793 |doi-access=free }}</ref><!--
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--><ref>{{Cite journal |last1=Kocia |first1=Lucas |last2=Love |first2=Peter |date=12 July 2018 |title=Measurement contextuality and Planck's constant |journal=New Journal of Physics |volume=20 |issue=7 |pages=073020 |doi=10.1088/1367-2630/aacef2|arxiv=1711.08066 |bibcode=2018NJPh...20g3020K |s2cid=73623448 }}</ref><!--
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--><ref>{{Cite journal |last=Humpherys |first=David |date=28 November 2022 |title=The Implicit Structure of Planck's Constant |journal=European Journal of Applied Physics |volume=4 |issue=6 |pages=22–25 |doi=10.24018/ejphysics.2022.4.6.227|s2cid=254359279 |doi-access=free }}</ref>{{rp|page=25}} <!--
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--><ref>{{Cite book |last1=Bais |first1=F. Alexander |title=Philosophy of Information |last2=Farmer |first2=J. Doyne | date=2008 |publisher=North-Holland |isbn=978-0-444-51726-5 |editor-last=Adriaans |editor-first=Pieter |series=Handbook of the Philosophy of Science |volume=8 |publication-place=Amsterdam |chapter=The Physics of Information |editor-last2=van Benthem |editor-first2=Johan |chapter-url=https://www.sciencedirect.com/science/article/abs/pii/B9780444517265500200 |arxiv=0708.2837}}</ref>{{rp|pages=653}}}} <!--
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-->while retaining the relationship <math display="inline">\hbar\, {{=}} h/(2 \pi)</math>.

=== Symbols ===

By far the most common symbol for the reduced Planck constant is <math display="inline">\hbar</math> . However, there are some sources that denote it by <math display="inline">h</math> instead, in which case they usually refer to it as the “Dirac <math display="inline">h</math>”<!--
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--><ref>{{cite book |last1=Hirota |first1=E. |last2=Sakakima |first2=H. |last3=Inomata |first3=K. |title=Giant Magneto-Resistance Devices |date=9 March 2013 |publisher=Springer Science & Business Media |isbn=978-3-662-04777-4 |url=https://books.google.com/books?id=mvDrCAAAQBAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Giant_Magneto_Resistance_Devices/mvDrCAAAQBAJ?hl=en&gbpv=1&dq=%22Dirac+h%22&pg=PA43&printsec=frontcover 43]}} <!--
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--><ref>{{cite journal |last1=Gardner |first1=John H. |title=An Invariance Theory |journal=Encyclia |date=1988 |volume=65 |page=139}}</ref>{{rp|page=[https://www.google.com/search?q=%22The+h+is+a+finite+constant+which+keeps+the+exponent+dimensionless.++This+constant+proves+to+be+Planck%27s+constant+divided+by+2%D0%BB+%2C+so+the+h+employed+here+is+the+Dirac+h+.%22&sca_esv=569891204&tbm=bks&ei=N9YZZa6AHbyr5NoPv4KikA0&ved=0ahUKEwju97_n19WBAxW8FVkFHT-BCNIQ4dUDCAg&uact=5&oq=%22The+h+is+a+finite+constant+which+keeps+the+exponent+dimensionless.++This+constant+proves+to+be+Planck%27s+constant+divided+by+2%D0%BB+%2C+so+the+h+employed+here+is+the+Dirac+h+.%22&gs_lp=Eg1nd3Mtd2l6LWJvb2tzIqsBIlRoZSBoIGlzIGEgZmluaXRlIGNvbnN0YW50IHdoaWNoIGtlZXBzIHRoZSBleHBvbmVudCBkaW1lbnNpb25sZXNzLiAgVGhpcyBjb25zdGFudCBwcm92ZXMgdG8gYmUgUGxhbmNrJ3MgY29uc3RhbnQgZGl2aWRlZCBieSAy0LsgLCBzbyB0aGUgaCBlbXBsb3llZCBoZXJlIGlzIHRoZSBEaXJhYyBoIC4iSABQAFgAcAB4AJABAJgBAKABAKoBALgBA8gBAPgBAvgBAQ&sclient=gws-wiz-books 151]}} <!--
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-->(or “Dirac's {{nowrap|<math display="inline">h</math>”}}<!--
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--><ref>{{cite book |last1=Levine |first1=Raphael D. | title=Molecular Reaction Dynamics |date=4 June 2009 |publisher=Cambridge University Press |isbn=978-1-139-44287-9 |url=https://books.google.com/books?id=FVqyS31OM7sC |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/Molecular_Reaction_Dynamics/FVqyS31OM7sC?hl=en&gbpv=1&dq=%22Dirac+h%22&pg=PA21&printsec=frontcover 21]}}<!--
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=== History ===

The combination <math display="inline">h/(2\pi)</math> first made its appearance{{efn|Some sources<!--
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--><ref name="Heilbron2013">{{Cite journal |last=Heilbron |first=John L. |date=June 2013 |title=The path to the quantum atom |journal=Nature |volume=498 |issue=7452 |pages=27–30 |doi=10.1038/498027a |pmid=23739408 |s2cid=4355108}}</ref><!--
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--><ref name="McCormmach1966">{{Cite journal |last=McCormmach |first=Russell |date=1 January 1966 |title=The atomic theory of John William Nicholson |journal=Archive for History of Exact Sciences |volume=3 |issue=2 |pages=160–184 |doi=10.1007/BF00357268 |jstor=41133258 |s2cid=120797894}}</ref>{{rp|169}} <!--
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--> <ref name="Mehra and Rechenberg v1p1">{{cite book |last1=Mehra |first1=Jagdish |last2=Rechenberg |first2=Helmut|author-link2=Helmut Rechenberg |title=The Historical Development of Quantum Theory |volume=1|date=3 August 1982 |publisher=Springer New York |isbn=978-0-387-90642-3 |url=https://books.google.com/books?id=oLWNzwEACAAJ |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/The_Quantum_Theory_of_Planck_Einstein_Bo/MvxAAQAAIAAJ?hl=en&gbpv=1&bsq=%22electron+ring+in+protofluorine%22&dq=%22electron+ring+in+protofluorine%22&printsec=frontcover 180]}} <!--
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-->claim that John William Nicholson discovered the quantization of angular momentum in units of <math display="inline">h/(2\pi)</math> in his 1912 paper,<!--
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--><ref name="Nicholson1912">{{Cite journal |last=Nicholson |first=J. W. |date=14 June 1912 |title=The Constitution of the Solar Corona. II. |journal=Monthly Notices of the Royal Astronomical Society |publisher=Oxford University Press |volume=72 |issue=8 |pages=677–693 |doi=10.1093/mnras/72.8.677 |issn=0035-8711|doi-access=free }}</ref> <!--
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-->so prior to Bohr. True, Bohr does credit Nicholson for emphasizing “the possible importance of the angular momentum in the discussion of atomic systems in relation to Planck's theory.”<ref name="bohr1"/>{{rp|page=[https://www.google.com/books/edition/London_Edinburgh_and_Dublin_Philosophica/g1k3AQAAMAAJ?hl=en&gbpv=1&bsq=%22theory+is+emphasized+by+Nicholson%22&dq=%22theory+is+emphasized+by+Nicholson%22&printsec=frontcover 15]}} However, in his paper, Nicholson deals exclusively with the quantization of energy, not angular momentum—with the exception of one paragraph in which he says, ''if, therefore, the constant <math display="inline">h</math> of Planck has, as Sommerfeld has suggested, an atomic significance, it may mean that the angular momentum of an atom can only rise or fall by discrete amounts when electrons leave or return. It is readily seen that this view presents less difficulty to the mind than the more usual interpretation, which is believed to involve an atomic constitution of energy itself,''<ref name="Nicholson1912"/>{{rp|679}} and with the exception of the following text in the summary: ''in the present paper, the suggested theory of the coronal spectrum has been put upon a definite basis which is in accord with the recent theories of emission of energy by bodies. It is indicated that the key to the physical side of these theories lies in the fact that an expulsion or retention of an electron by any atom probably involves a discontinuous change in the angular momentum of the atom, which is dependent on the number of electrons already present.''<ref name="Nicholson1912"/>{{rp|692}} The literal combination <math display="inline">h/(2\pi)</math> does not appear in that paper. A biographical memoir of Nicholson<ref name="frs">{{Cite journal | last1 = Wilson | first1 = W. | doi = 10.1098/rsbm.1956.0014 | title = John William Nicholson 1881-1955 | journal = Biographical Memoirs of Fellows of the Royal Society | volume = 2 | pages = 209–214 | year = 1956 | jstor = 769485| doi-access =  }}</ref> states that Nicholson only “later” realized that the discrete changes in angular momentum are integral multiples of {{nowrap|<math display="inline">h/(2\pi)</math>,}} but unfortunately the memoir does not say if this realization occurred before or after Bohr published his paper, or whether Nicholson ever published it.}} in [[Biography:Niels Bohr|Niels Bohr]]'s 1913 paper,<!--
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--><ref name="bohr1">{{Cite journal |last=Bohr |first=N. |date=July 1913 |title=I. On the constitution of atoms and molecules |url=https://zenodo.org/record/2493915 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=26 |issue=151 |pages=1–25 |doi=10.1080/14786441308634955}}</ref>{{rp|page=[https://www.google.com/books/edition/London_Edinburgh_and_Dublin_Philosophica/g1k3AQAAMAAJ?hl=en&gbpv=1&bsq=%22If+we+therefore+assume+that+the+orbit+of+the+electron+in%22&dq=%22If+we+therefore+assume+that+the+orbit+of+the+electron+in%22&printsec=frontcover 15]}} <!--
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--> where it was denoted by {{nowrap|<math display="inline">M_{0}</math>.}}{{efn|Bohr denoted by <math display="inline">M</math> the angular momentum of the electron around the nucleus, and wrote the quantization condition as {{nowrap|<math display="inline">M=\tau M_{0}</math>,}} where <math display="inline">\tau</math> is a positive integer. (See the [[Physics:Bohr model|Bohr model]].)}}<!--
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--> For the next 15 years, the combination continued to appear in the literature, but normally without a separate symbol.{{efn|Here are some papers that are mentioned in<ref name="Mehra and Rechenberg v1p1"/>  and in which <math display="inline">h/(2\pi)</math> appeared without a separate symbol: <!--
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--><ref>{{cite journal |last1=Sommerfeld |first1=A. |title=Zur Theorie der Balmerschen Serie |journal=Sitzungsberichte der mathematisch-physikalischen Klasse der K. B. Akademie der Wissenschaften zu München |date=1915 |volume=33 |issue=198 |pages=425–458|url=https://static-content.springer.com/esm/art%3A10.1140%2Fepjh%2Fe2013-40053-8/MediaObjects/13129_2013_121_MOESM1_ESM.pdf|doi=10.1140/epjh/e2013-40053-8}}</ref>{{rp|page=428}} <!--
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--><ref>{{cite journal |last1=Schwarzschild |first1=K. |title=Zur Quantenhypothese |journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin|date=1916 |pages=548–568}}</ref>{{rp|page=[https://www.biodiversitylibrary.org/item/93032#page/599/mode/1up 549]}} <!--
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--><ref>{{cite journal |last1=Ehrenfest |first1=P. |title=XLVIII. Adiabatic invariants and the theory of quanta |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |date=June 1917 |volume=33 |issue=198 |pages=500–513 |doi=10.1080/14786440608635664}}</ref>{{rp|page=[https://www.google.com/books/edition/The_London_Edinburgh_and_Dublin_Philosop/B6lJAAAAYAAJ?hl=en&gbpv=1&dq=%22we+must+demand+that+p+can+take+no+other+values+than%22&pg=PA508&printsec=frontcover 508]}} <!--
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--><ref>{{cite journal |last1=Landé |first1=A. |title=Das Serienspektrum des Heliums |journal=Physikalische Zeitschrift |date=June 1919 |volume=20 |pages= 228–234}}</ref>{{rp|page=[https://www.google.com/books/edition/Selected_Scientific_Papers_of_Alfred_Lan/2ZPqCAAAQBAJ?hl=en&gbpv=1&dq=%22festgelegt+,+w%C3%A4hrend+das+Valenzelektron+mit%22&pg=PA102&printsec=frontcover 230]}} <!--
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--><ref>{{cite journal |last1=Bohr |first1=N.|title=Über die Serienspektra der Elemente |journal=Zeitschrift für Physik |date=October 1920 |volume=2 |issue=5 |pages=423–469 |doi=10.1007/BF01329978}}</ref>{{rp|page=458}} <!--
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--><ref>{{cite journal |last1=Stern |first1=Otto |title=Ein Weg zur experimentellen Prüfung der Richtungsquantelung im Magnetfeld |journal=Zeitschrift für Physik |date=December 1921 |volume=7 |issue=1 |pages=249–253 |doi=10.1007/BF01332793|url=https://books.google.com/books?id=hlZKAAAAMAAJ&dq=%22Ein+Weg+zur+experimentellen+Pr%C3%BCfung+der+Richtungsquantelung%22+%22Bringen+wir+also+ein+Gas+aus+Atomen%22&pg=PA249}}</ref><!--
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--><ref>{{cite journal |last1=Heisenberg |first1=Werner|title=Zur Quantentheorie der Linienstruktur und der anomalen Zeemaneflekte |journal=Zeitschrift für Physik |date=December 1922 |volume=8 |issue=1 |pages=273–297 |doi=10.1007/BF01329602}}</ref>{{rp|page=[https://www.google.com/books/edition/Original_Scientific_Papers_Wissenschaftl/KF2oBgAAQBAJ?hl=en&gbpv=1&dq=%22gegeben+ist,+hat+das+ganze+Atom+den+Drehimpuls%22&pg=PA137&printsec=frontcover 276]}} <!--
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--><ref>{{cite journal |last1=Kramers |first1=H. A.|last2=Pauli |first2=W.|author-link2=Wolfgang Pauli |title=Zur Theorie der Bandenspektren |journal=Zeitschrift für Physik |date=December 1923 |volume=13 |issue=1 |pages=351–367 |doi=10.1007/BF01328226|url=https://books.google.com/books?id=9PpMAQAAIAAJ&dq=%22negativen+Halogenionen+ein+resultierendes%22&pg=RA1-PA351}}</ref><!--
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--><ref>{{cite journal |last1=Born |first1=M. | last2=Jordan |first2=P.|author-link2=Pascual Jordan |title=Zur Quantenmechanik |journal=Zeitschrift für Physik |date=December 1925 |volume=34 |issue=1 |pages=858–888 |doi=10.1007/BF01328531}}</ref><!--
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--><ref>{{cite journal |last1=Dirac |first1=P. A. M.| title=The fundamental equations of quantum mechanics |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |date=December 1925 |volume=109 |issue=752 |pages=642–653 |doi=10.1098/rspa.1925.0150|doi-access=free }}</ref><!--
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-->.<ref>{{cite journal |last1=Born |first1=M.| last2=Heisenberg |first2=W.|author-link2=Werner Heisenberg |last3=Jordan |first3=P.|author-link3=Pascual Jordan |title=Zur Quantenmechanik. II. |journal=Zeitschrift für Physik |date=August 1926 |volume=35 |issue=8–9 |pages=557–615 |doi=10.1007/BF01379806}}</ref>}} <!--
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-->Then, in 1926, in their seminal papers, [[Biography:Erwin Schrödinger|Schrödinger]] and [[Biography:Paul Dirac|Dirac]] again introduced special symbols for it: <math display="inline">K</math> in the case of Schrödinger,<!--
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--><ref name="Schrodinger 1926 Erste">{{cite journal |last1=Schrödinger |first1=E.|title=Quantisierung als Eigenwertproblem |journal=Annalen der Physik |date=1926 |volume=384 |issue=4 |pages=361–376 |doi=10.1002/andp.19263840404|doi-access=free }}</ref> and <math display="inline">h</math> in the case of Dirac.<!--
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--><ref>{{cite journal |last1=Dirac |first1=P. A. M.|title=On the theory of quantum mechanics |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |date=October 1926 |volume=112 |issue=762 |pages=661–677 |doi=10.1098/rspa.1926.0133|doi-access=free }}</ref> <!--
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-->Dirac continued to use <math display="inline">h</math> in this way until 1930,<!--
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--><ref name="Mehra and Rechenberg v6">{{cite book |last1=Mehra |first1=Jagdish |last2=Rechenberg |first2=Helmut|author-link2=Helmut Rechenberg |title=The Historical Development of Quantum Theory |volume=6|date=2000 |publisher=Springer|location= New York |language=en}}</ref>{{rp|page=[https://www.google.com/books/edition/The_Historical_Development_of_Quantum_Th/warvAAAAMAAJ?hl=en&gbpv=1&bsq=%22Until+1930,+Dirac+always+wrote%22&dq=%22Until+1930,+Dirac+always+wrote%22&printsec=frontcover 291]}} <!--
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--> when he introduced the symbol <math display="inline">\hbar</math> in his book ''[[Physics:The Principles of Quantum Mechanics|The Principles of Quantum Mechanics]].''<ref name="Mehra and Rechenberg v6"/>{{rp|page=[https://www.google.com/books/edition/The_Historical_Development_of_Quantum_Th/warvAAAAMAAJ?hl=en&gbpv=1&bsq=%22Until+1930,+Dirac+always+wrote%22&dq=%22Until+1930,+Dirac+always+wrote%22&printsec=frontcover 291]}} <!--
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--><ref>{{cite book |last1=Dirac |first1=P. A. M.|title=The Principles of Quantum Mechanics |date=1930 |publisher=Clarendon |location=Oxford, U.K. |edition=1st}}</ref><!--
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== See also ==
* CODATA 2018
* [[International System of Units]]
* [[Physics:Introduction to quantum mechanics|Introduction to quantum mechanics]]
* List of scientists whose names are used in physical constants
* [[Planck units]]
* [[Physics:Wave–particle duality|Wave–particle duality]]

== Notes ==
{{notelist}}

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

=== Sources ===
{{refbegin}}
* {{Citation |last=Barrow |first=John D. |title=The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe |url=https://archive.org/details/constantsofnatur0000barr |year=2002 |publisher=Pantheon Books |isbn=978-0-375-42221-8 |url-access=registration}}
{{refend}}

== External links ==
* [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.]
* [https://www.youtube.com/watch?v=0nV9f5uqul0&t=84s “The Planck constant and its units” – presentation at the 35th Symposium on Chemical Physics at the University of Waterloo, Waterloo, Ontario, Canada, November 3 2019.]


[[Category:Fundamental constants]]

{{Sourceattribution|Planck constant}}