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{{Short description|Physical constant in quantum mechanics}}{{For|the law governing black-body radiation|Planck’s law}}

factoids

second{{sup>−1}} | dimension = mathsf{M} mathsf{L}^{2} mathsf{T}^{-1}| derivations = hh_eV/Hz|ref=no}}}}

factoids

second (unit){{sup>−1}} | dimension = mathsf{M} mathsf{L}^{2} mathsf{T}^{-1}| derivations = {hide}ublist

| hbar {{={edih} h/(2 pi)

}}hbarhbar_eVs|ref=no}}}}The Planck constant, or Planck’s constant, denoted by {{nowrap|h,}}WEB, 20 May 2019, Planck constant,physics.nist.gov/cgi-bin/cuu/Value?h, live,web.archive.org/web/20220527011505/https://physics.nist.gov/cgi-bin/cuu/Value?h, 2022-05-27, 2023-09-03, The NIST Reference on Constants, Units, and Uncertainty, National Institute of Standards and Technology, NIST, is a fundamental physical constant of foundational importance in quantum mechanics: a photon’s energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a matter wave equals the Planck constant divided by the associated particle momentum.The constant was postulated by Max Planck in 1900 as a proportionality constant needed to explain experimental black-body radiation. Planck later referred to the constant as the “quantum of action”.WEB, Max Planck Nobel Lecture,www.nobelprize.org/prizes/physics/1918/planck/lecture/, live,web.archive.org/web/20230714164215/https://www.nobelprize.org/prizes/physics/1918/planck/lecture/, 2023-07-14, 2023-07-14, In 1905, 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, the Planck constant is used, together with other constants, to define the kilogram, the SI unit of mass.{{SIbrochure9th|page=131}} 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|h {{=}} {{physconst|h|after=.}}}}WEB, 2018-11-16, Resolutions of the 26th CGPM,www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf, dead,web.archive.org/web/20181119214326/https://www.bipm.org/utils/common/pdf/CGPM-2018/26th-CGPM-Resolutions.pdf, 2018-11-19, 2018-11-20, BIPM, en-GB, It is often used with units of electronvolt (eV), which corresponds to the SI unit per elementary charge.

History

Origin of the constant

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 black bodyblack bodyPlanck’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 (black-body radiation).BOOK, Bitter, Francis, Fields and particles, Medicus, Heinrich A., 1973, Elsevier, New York, 137–144, Francis Bitter, 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 Kirchhoff some 40 years earlier. Every physical body spontaneously and continuously emits electromagnetic radiation. There was no expression or explanation for the overall shape of the observed emission spectrum. At the time, Wien’s law fit the data for short wavelengths and high temperatures, but failed for long wavelengths.{{rp|141}} Also around this time, but unknown to Planck, Lord Rayleigh had derived theoretically a formula, now known as the 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 harmonic oscillators, 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,{{citation | first = Max | last = Planck | author-link = Max Planck | title = Ueber das Gesetz der Energieverteilung im Normalspectrum | url =www.physik.uni-augsburg.de/annalen/history/historic-papers/1901_309_553-563.pdf | journal = 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 =www.physik.uni-augsburg.de/annalen/history/historic-papers/1901_309_553-563.pdf" title="web.archive.org/web/20120610124128www.physik.uni-augsburg.de/annalen/history/historic-papers/1901_309_553-563.pdf">web.archive.org/web/20120610124128www.physik.uni-augsburg.de/annalen/history/historic-papers/1901_309_553-563.pdf | url-status = live }}. English translation: WEB, On the Law of Distribution of Energy in the Normal Spectrum,dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html, dead,dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html," title="web.archive.org/web/20080418002757dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html,">web.archive.org/web/20080418002757dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html, 2008-04-18, ”. WEB,theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf, On the Law of Distribution of Energy in the Normal Spectrum, 2011-10-13, dead,theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf," title="web.archive.org/web/20111006162543theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf,">web.archive.org/web/20111006162543theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf, 2011-10-06, 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, h , which is thought to be for Hilfsgrösse (auxiliary variable),ARXIV, physics/0402064v1, Luis J., Boya, The Thermal Radiation Formula of Planck (1900), 2004, and subsequently became known as the Planck constant. The expression formulated by Planck showed that the spectral radiance of a body for frequency {{math|ν}} at absolute temperature {{math|T}} is given by where k_text{B} is the Boltzmann constant, h is the Planck constant, and c is the speed of light in the medium, whether material or vacuum.BOOK, Planck, M.,archive.org/details/theoryofheatradi00planrich, The Theory of Heat Radiation, P. Blakiston’s Son, Masius, M. (transl.), 1914, 2nd, 6, 168, 7154661M, Max Planck, BOOK, Chandrasekhar, S.,archive.org/details/radiativetransfe0000chan, Radiative Transfer, Dover Publications, Dover, 1960, 978-0-486-60590-6, Revised reprint, 8, Subrahmanyan Chandrasekhar, registration, 1950, BOOK, Rybicki, G. B.,books.google.com/books?id=LtdEjNABMlsC, Radiative Processes in Astrophysics, Lightman, A. P., John Wiley & Sons, Wiley, 1979, 978-0-471-82759-7, 22, Alan Lightman, 2020-05-20,web.archive.org/web/20200727111701/https://books.google.com/books?id=LtdEjNABMlsC, 2020-07-27, live, The spectral radiance of a body, B_{nu} , 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 lambda instead of per unit frequency. In this case, it is given by showing how radiated energy emitted at shorter wavelengths increases more rapidly with temperature than energy emitted at longer wavelengths.JOURNAL, Shao, Gaofeng, etal, 2019, Improved oxidation resistance of high emissivity coatings on fibrous ceramic for reusable space systems, Corrosion Science, 146, 233–246, 1902.03943, 10.1016/j.corsci.2018.11.006, 118927116, 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 SI units of B_{nu} are {{nowrap|W·sr−1·m−2·Hz−1}}, while those of B_{lambda} are {{nowrap|W·sr−1·m−3}}.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. To save his theory, Planck resorted to using the then-controversial theory of statistical mechanics, which he described as “an act of desperation”.{{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/20090108204645physicsworld.com/cws/article/print/373 |publisher=PhysicsWorld.com |archive-date=2009-01-08 |author-link=Helge Kragh}} One of his new boundary conditions was{{blockquote|text=to interpret UN [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}}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,{{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/20211206180414books.google.com/books?id=ELrFDIldlawC |url-status=live |publisher=Princeton University Press |isbn=978-0-691-09552-3 |archive-date=2021-12-06}} 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 “Planck–Einstein relation”: Planck was able to calculate the value of h 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. He also made the first determination of the Boltzmann constant k_text{B} from the same data and theory.{{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/20110715190331nobelprize.org/nobel_prizes/physics/laureates/1918/planck-lecture.html |url-status=live |archive-date=15 July 2011 |author-link=Max Planck}}(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 Lord Rayleigh and James Jeans (on the one hand) and 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 “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 first Solvay Conference in 1911 was devoted to “the theory of radiation and quanta”.{{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/20081216120021www.solvayinstitutes.be/Conseils%20Solvay/PreviousPhysics.html |url-status=dead |publisher=International Solvay Institutes |archive-date=16 December 2008}}

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,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/20110904232203www.nobelprize.org/nobel_prizes/physics/laureates/1921/press.html |url-status=live |archive-date=4 September 2011 |author-link=Svante Arrhenius}} who published the first thorough investigation in 1887. Another particularly thorough investigation was published by Philipp Lenard (Lénárd Fülöp) in 1902.{{Citation |last=Lenard |first=P. |title=Ueber die lichtelektrische Wirkung |url=https://zenodo.org/record/1424009 |journal=Annalen der Physik |volume=313 |issue=5 |pages=149–98 |year=1902 |access-date=2019-07-03 |archive-url=https://web.archive.org/web/20190818204108zenodo.org/record/1424009 |url-status=live |bibcode=1902AnP...313..149L |doi=10.1002/andp.19023130510 |archive-date=2019-08-18 |author-link=Philipp Lenard}} Einstein’s 1905 paper{{Citation |last=Einstein |first=Albert |title=Ãœber 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 |volume=17 |issue=6 |pages=132–48 |year=1905 |access-date=2009-12-03 |archive-url=https://web.archive.org/web/20110709180735www.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}} discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921, after his predictions had been confirmed by the experimental work of Robert Andrews Millikan.{{Citation |last=Millikan |first=R. A. |title=A Direct Photoelectric Determination of Planck’s h |journal=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}} 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.{{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/20200109170326books.google.com/books?id=cdxWNE7NY6QC |url-status=live |isbn=978-1-4165-3932-2 |archive-date=2020-01-09}}, pp. 309–314.WEB, The Nobel Prize in Physics 1921,www.nobelprize.org/nobel_prizes/physics/laureates/1921/, live,web.archive.org/web/20180703190346/https://www.nobelprize.org/nobel_prizes/physics/laureates/1921/, 2018-07-03, 2014-04-23, Nobelprize.org, 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 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 kinetic energy, which can be measured. This kinetic energy (for each photoelectron) is independent of the intensity of the light, but depends linearly on the frequency; 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).*JOURNAL, Smith, Richard, 1962, Two Photon Photoelectric Effect, Physical Review, 128, 5, 2225, 1962PhRv..128.2225S, 10.1103/PhysRev.128.2225,

• JOURNAL, 10.1103/PhysRev.130.2599.4, Two-Photon Photoelectric Effect, 1963, Smith, Richard, Physical Review, 130, 6, 2599, 1963PhRv..130.2599S, free, 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.

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 photons, was to be the same as Planck’s “energy element”, giving the modern version of the Planck–Einstein relation: Einstein’s postulate was later proven experimentally: the constant of proportionality between the frequency of incident light f and the kinetic energy of photoelectrons E was shown to be equal to the Planck constant h.

Atomic structure

(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}}.JOURNAL, Heilbron, John L., 2013, The path to the quantum atom, Nature (journal), Nature, 498, 7452, 27–30, 10.1038/498027a, 23739408, 4355108, JOURNAL, Nicholson, J. W., 1911, The spectrum of Nebulium, Monthly Notices of the Royal Astronomical Society, 72, 49, 1911MNRAS..72...49N, 10.1093/mnras/72.1.49, free, *JOURNAL, Nicholson, J. W., 1911, The Constitution of the Solar Corona I, Monthly Notices of the Royal Astronomical Society, 72, 139, 1911MNRAS..72..139N, 10.1093/mnras/72.2.139, free,

• JOURNAL, 10.1093/mnras/72.8.677, The Constitution of the Solar Corona II, 1912, Nicholson, J. W., Monthly Notices of the Royal Astronomical Society, 72, 8, 677–693, free,
• JOURNAL, 10.1093/mnras/72.9.729, The Constitution of the Solar Corona III, 1912, Nicholson, J. W., Monthly Notices of the Royal Astronomical Society, 72, 9, 729–740, free, JOURNAL, Nicholson, J. W., 1912, On the new nebular line at λ4353, Monthly Notices of the Royal Astronomical Society, 72, 8, 693, 1912MNRAS..72..693N, 10.1093/mnras/72.8.693, free, JOURNAL, McCormmach, Russell, 1966, The Atomic Theory of John William Nicholson, Archive for History of Exact Sciences, 3, 2, 160–184, 10.1007/BF00357268, 41133258, 120797894, Niels Bohr quoted him in his 1913 paper of the Bohr model of the atom.JOURNAL, Bohr, N., 1913, On the constitution of atoms and molecules,zenodo.org/record/2493915, live, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 6th series, 26, 151, 1–25, 1913PMag...26..476B, 10.1080/14786441308634955,web.archive.org/web/20230307022713/https://zenodo.org/record/2493915, 2023-03-07, 2023-07-23, The influence of the work of Nicholson’s nuclear quantum atomic model on Bohr’s model has been written about by many historians.JOURNAL, Hirosige, Tetu, Nisio, Sigeko, 1964, Formation of Bohr’s theory of atomic constitution, Japanese Studies in History of Science, 3, 6–28, 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).

Niels Bohr introduced the third quantized model of the atom in 1913, in an attempt to overcome a major shortcoming of 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. In classical electrodynamics, a charge moving in a circle should radiate electromagnetic radiation. If that charge were to be an electron orbiting a nucleus, the radiation would cause it to lose energy and spiral down into the nucleus. Bohr solved this paradox with explicit reference to Planck’s work: an electron in a Bohr atom could only have certain defined energies E_n where c is the speed of light in vacuum, R_{infty} is an experimentally determined constant (the Rydberg constant) and n in {1, 2, 3, ... }. Once the electron reached the lowest energy level (n = 1), it could not get any closer to the nucleus (lower energy). This approach also allowed Bohr to account for the Rydberg formula, an empirical description of the atomic spectrum of hydrogen, and to account for the value of the Rydberg constant R_{infty} in terms of other fundamental constants.Bohr also introduced the quantity hbar=frac{h}{2pi}, now known as the reduced Planck constant or Dirac constant, as the quantum of angular momentum. At first, Bohr thought that this was the angular momentum of each electron in an atom: this proved incorrect and, despite developments by Sommerfeld and others, an accurate description of the electron angular momentum proved beyond the Bohr model. The correct quantization rules for electrons – in which the energy reduces to the Bohr model equation in the case of the hydrogen atom – were given by Heisenberg’s matrix mechanics in 1925 and the Schrödinger wave equation in 1926: the reduced Planck constant remains the fundamental quantum of angular momentum. In modern terms, if J is the total angular momentum of a system with rotational invariance, and J_z the angular momentum measured along any given direction, these quantities can only take on the values begin{align}J^2 = j(j+1) hbar^2,qquad & j = 0, tfrac{1}{2}, 1, tfrac{3}{2}, ldots, J_z = m hbar, qquadqquadquad & m = -j, -j+1, ldots, j.end{align}

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, Delta x, and the uncertainty in their momentum, Delta p_{x}, obey where the uncertainty is given as the standard deviation of the measured value from its 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 relationship between the position operator hat{x} and the momentum operator hat{p}: where delta_{ij} is the Kronecker delta.

Photon energy

The Planck relation connects the particular photon energy {{math|E}} with its associated wave frequency {{math|f}}: 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 f= frac{c}{lambda} , the relation can also be expressed as

de Broglie wavelength

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 afterward. This holds throughout the quantum theory, including electrodynamics. The de Broglie wavelength {{math|λ}} of the particle is given by where {{math|p}} denotes the linear momentum of a particle, such as a photon, or any other elementary particle.The energy of a photon with angular frequency {{math|1=ω = 2πf}} is given by while its linear momentum relates to where {{math|k}} is an angular wavenumber.These two relations are the temporal and spatial parts of the special relativistic expression using 4-vectors.

Statistical mechanics

Classical statistical mechanics requires the existence of {{math|h}} (but does not define its value).{{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/20211206180408books.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 }} Eventually, following upon Planck’s discovery, it was speculated that physical action could not take on an arbitrary value, but instead was restricted to integer multiples of a very small quantity, the “[elementary] quantum of action”, now called the Planck constant.BOOK, ter Haar, D.,archive.org/details/oldquantumtheory0000haar, The Old Quantum Theory, Pergamon Press, 1967, 978-0-08-012101-7, 133, registration, This was a significant conceptual part of the so-called “old quantum theory” developed by physicists including Bohr, Sommerfeld, and 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.{{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/20120415132339www.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 {{math|Hσ}} to a system specified in the sense of classical mechanics (the energy function is a given function of the coordinates {{math |qr}} and the corresponding momenta {{math |pr}})? The Planck constant {{math |h}} relates the frequency {{math |Hσ/h}} to the energy values {{math |Hσ}}. It is therefore sufficient to give to the system a succession of discrete frequency values.}}{{rp|373}}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 quantization of energy.

Dimension and value

The Planck constant has the same dimensions as action and as angular momentum. In SI units, the Planck constant is expressed with the unit joule per hertz (Jâ‹…Hz{{sup|−1}}) or joule-second (Jâ‹…s).h = mathrm{6.626 070 15times 10^{-34} {J cdot s}}hbar = {h over 2pi} = mathrm{1.054 571 817 times 10^{-34} {J cdot s}} = mathrm{ 6.582 119 569... times 10^{-16} {eV cdot s}}physics.nist.gov/cgi-bin/cuu/Value?hbarevThe above values have been adopted as fixed in the 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: “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 Jâ‹…s, which is equal to kgâ‹…m2â‹…s−1, where the metre and the second are defined in terms of speed of light {{Mvar|c}} and duration of hyperfine transition of the ground state of an unperturbed caesium-133 atom {{Math|ΔνCs}}.“{{SIbrochure9th}} Technologies of mass metrology such as the 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 product of energy and time for a physical event approaches the Planck constant, quantum effects dominate.WEB, The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action,www.feynmanlectures.caltech.edu/II_19.html, 2023-11-03, www.feynmanlectures.caltech.edu, 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 light (with a wavelength of 555 nanometres or a frequency of {{val|540|u=THz}}) 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 of photons; its energy can be computed by multiplying the photon energy by the 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

In many applications, the Planck constant h naturally appears in combination with 2pi as {{nowrap|h/(2pi),}} which can be traced to the fact that in these applications it is natural to use the angular frequency (in radians per second) rather than plain frequency (in cycles per second or hertz). For this reason, it is often useful to absorb that factor of {{math|2Ï€}} into the Planck constant by introducing the reduced Planck constantWEB, 20 May 2019, reduced Planck constant,physics.nist.gov/cgi-bin/cuu/Value?hbar, live,web.archive.org/web/20230408225435/https://physics.nist.gov/cgi-bin/cuu/Value?hbar, 2023-04-08, 2023-09-03, The NIST Reference on Constants, Units, and Uncertainty, National Institute of Standards and Technology, NIST, BOOK, Lyth, David H., David H. Lyth, Liddle, Andrew R., Andrew R. Liddle, The Primordial Density Perturbation: Cosmology, Inflation and the Origin of Structure, 11 June 2009, Cambridge University Press, 978-1-139-64374-0,books.google.com/books?id=IPwgAwAAQBAJ, en, {{rp|page=482}} (or reduced Planck’s constantBOOK, Huang, Kerson, Kerson Huang, Quantum Field Theory: From Operators to Path Integrals, 26 April 2010, John Wiley & Sons, 978-3-527-40846-7,books.google.com/books?id=5ZWZRpvZzUEC, en, {{rp|page=5}} BOOK, Schmitz, Kenneth S., Physical Chemistry: Concepts and Theory, 11 November 2016, Elsevier, 978-0-12-800600-9,books.google.com/books?id=Y1ydBgAAQBAJ, en, {{rp|page=788}}), equal to the Planck constant divided by 2 pi and denoted by hbar (pronounced h-barBOOK, Chabay, Ruth W., Ruth Chabay,books.google.com/books?id=zMWHEAAAQBAJ, Matter and Interactions, Sherwood, Bruce A., 20 November 2017, John Wiley & Sons, 978-1-119-45575-2, en, {{rp|page=336}}). Many of the most important equations, relations, definitions, and results of quantum mechanics are customarily written using the reduced Planck constant hbar rather than the Planck constant h, including the Schrödinger equation, momentum operator, canonical commutation relation, Heisenberg’s uncertainty principle, and Planck units.BOOK, Schwarz, Patricia M., Schwarz, John H., John Henry Schwarz, Special Relativity: From Einstein to Strings, 25 March 2004, Cambridge University Press, 978-1-139-44950-2,books.google.com/books?id=BmhQxWwk0K4C, en, {{rp|page=104}}Because the fundamental equations look simpler when written using hbar as opposed to {{nowrap|h,}} it is usually hbar rather than h that gives the most reliable results when used in order-of-magnitude estimates.BOOK, Lévy-Leblond, Jean-Marc, Jean-Marc Lévy-Leblond, The meanings of Planck’s constant,ssqg.ans.pw.edu.pl/en/wp-content/uploads/2021/04/JMLL_Meanings-of-h.pdf,web.archive.org/web/20231014082224/https://ssqg.ans.pw.edu.pl/en/wp-content/uploads/2021/04/JMLL_Meanings-of-h.pdf, 2023-10-14, Beltrametti, E., Rimini, A., Robotti, Nadia, Nadia Robotti, One Hundred Years of H: Pavia, 14-16 September 2000, 2002, Italian Physical Society, 978-88-7438-003-9,books.google.com/books?id=tqbvAAAAMAAJ, en, {{rp|pages=8–9}} {{efn|As examples, the preceding reference shows what happens when one uses dimensional analysis to obtain estimates for the ionization energy and the size of a hydrogen atom. If we use the Gaussian units, then the relevant parameters that determine the ionization energy E_{text{i}} are the mass of the electron {{nowrap|m_{text{e}},}} the electron charge {{nowrap|e,}} and either the Planck constant h or the reduced Planck constant hbar (since h and hbar have the same dimensions, they will enter the dimensional analysis in the same way). One obtains that E_{text{i}} must be proportional to m_{text{e}} e^{4}/h^{2} if we used {{nowrap|h,}} and to m_{text{e}} e^{4}/hbar^{2} if we used {{nowrap|hbar.}} In an order-of-magnitude estimate, we take that the constant of proportionality is 1. Now, the actual correct answer is {{nowrap|E_{text{i}}=m_{text{e}} e^{4}/(2hbar^{2});}}BOOK, Shu, Frank, The Physical Universe: An Introduction to Astronomy, 1982, University Science Books, 978-0-935702-05-7,books.google.com/books?id=v_6PbAfapSAC, en, {{rp|page=45}} therefore, if we choose to use hbar as one of our parameters, our estimate will off by a factor of 2, whereas if we choose to use {{nowrap|h,}} it will be off by a factor of {{nowrap|4pi^{2}/2 approx 20.}} Similarly for the estimate of the size of a hydrogen atom: depending on whether we use h or hbar as one of the parameters, we get either h^{2}/(m_{text{e}} e^{2}) or {{nowrap|hbar^{2}/(m_{text{e}} e^{2}).}} The latter happens to be exactly correct,WEB, Bohr Radius -- from Eric Weisstein’s World of Physics,scienceworld.wolfram.com/physics/BohrRadius.html, scienceworld.wolfram.com, 14 October 2023, en,web.archive.org/web/20231014072210/https://scienceworld.wolfram.com/physics/BohrRadius.html, 2023-10-14, whereas the estimate using h is off by a factor of {{nowrap|4pi^{2} approx 40.}}}}

Names

The reduced Planck constant is known by many other names: the rationalized Planck constantENCYCLOPEDIA, Planck constant, A Dictionary of Physics, OUP Oxford, Oxford, UK, {{google books, y, YPmFDwAAQBAJ, |date=2017 |editor-last=Rennie |editor-first=Richard |isbn=978-0198821472 |editor2-last=Law |editor2-first=Jonathan|edition= 7th|series=Oxford Quick Reference}}{{rp|page=726}} BOOK, The International Encyclopedia of Physical Chemistry and Chemical Physics, 1960, Pergamon Press,books.google.com/books?id=ufTvAAAAMAAJ, en, {{rp|page=10}} BOOK, Vértes, Attila, Nagy, Sándor, Klencsár, Zoltán, Lovas, Rezso György, Rösch, Frank, Handbook of Nuclear Chemistry, 10 December 2010, Springer Science & Business Media, 978-1-4419-0719-6,books.google.com/books?id=NQyF6KaUScQC, en, {{rp|page=-}} (or rationalized Planck’s constantBOOK, Flügge, Siegfried, Siegfried Flügge, Handbuch der Physik: Atome I-II, 1957, Springer,books.google.com/books?id=A-APAQAAMAAJ, Bethe, Hans A., Hans Bethe, Salpeter, Edwin E., Edwin E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, en, {{rp|page=334}} BOOK, Lang, Kenneth, Astrophysical Formulae: A Compendium for the Physicist and Astrophysicist, 11 November 2013, Springer Science & Business Media, 978-3-662-11188-8,books.google.com/books?id=LyzrCAAAQBAJ, en, {{rp|page=ix}} BOOK, Galgani, L., Carati, A., Pozzi, B., The Problem of the Rate of Thermalization, and the Relations between Classical and Quantum Mechanics, December 2002, 111–122, 10.1142/9789812776273_0011, Fabrizio, Mauro, Morro, Angelo, Mathematical Models and Methods for Smart Materials, Cortona, Italy, 25 – 29 June 2001, 978-981-238-235-1, {{rp|page=112}} ), the Dirac constantBOOK, Fox, Mark,books.google.com/books?id=pVtlDwAAQBAJ, A Student’s Guide to Atomic Physics, 14 June 2018, Cambridge University Press, 978-1-316-99309-5, en, {{rp|page=275}} {{rp|page=726}} BOOK, Kleiss, Ronald, Quantum Field Theory: A Diagrammatic Approach, 10 June 2021, Cambridge University Press, 978-1-108-78750-5,books.google.com/books?id=G_UvEAAAQBAJ, en, {{rp|page=xv}} (or Dirac’s constantBOOK, Zohuri, Bahman, Thermal Effects of High Power Laser Energy on Materials, 5 January 2021, Springer Nature, 978-3-030-63064-5,books.google.com/books?id=P3gSEAAAQBAJ, en, {{rp|page=148}} BOOK, Balian, Roger, Roger Balian, From Microphysics to Macrophysics: Methods and Applications of Statistical Physics. Volume II, 26 June 2007, Springer Science & Business Media, 978-3-540-45480-9,books.google.com/books?id=cUY_AAAAQBAJ, en, {{rp|page=604}} BOOK, Chen, C. Julian, Physics of Solar Energy, 15 August 2011, John Wiley & Sons, 978-1-118-04459-9,books.google.com/books?id=dFKz6GwSVNEC, en, {{rp|page=313}}), the Dirac hENCYCLOPEDIA, Britannica, Dirac h,www.britannica.com/science/Dirac-h, 2023-09-27,web.archive.org/web/20230217112008/https://www.britannica.com/science/Dirac-h, 2023-02-17, live, BOOK, Shoenberg, D., David Shoenberg, Magnetic Oscillations in Metals, 3 September 2009, Cambridge University Press, 978-1-316-58317-3,books.google.com/books?id=HM_LCgAAQBAJ, en, {{rp|page=xviii}} (or Dirac’s hBOOK, Powell, John L., Crasemann, Bernd, Quantum Mechanics, 5 May 2015, Courier Dover Publications, 978-0-486-80478-1,books.google.com/books?id=GHifCQAAQBAJ, en, {{rp|page=17}} ), the Dirac hbarBOOK, Dresden, Max, Max Dresden, H.A. Kramers Between Tradition and Revolution, 6 December 2012, Springer Science & Business Media, 978-1-4612-4622-0,books.google.com/books?id=1ggDCAAAQBAJ, en, {{rp|page=187}} (or Dirac’s hbarBOOK, Johnson, R. E., Introduction to Atomic and Molecular Collisions, 6 December 2012, Springer Science & Business Media, 978-1-4684-8448-9,books.google.com/books?id=cKPTBwAAQBAJ, en, {{rp|page=273}} BOOK, Garcia, Alejandro, Henley, Ernest M., Ernest M. Henley, Subatomic Physics, 13 July 2007, World Scientific Publishing Company, 978-981-310-167-8,books.google.com/books?id=j847DQAAQBAJ, en, 3rd, {{rp|page=14}} ), and h-bar.BOOK, Holbrow, Charles H., Charles H. Holbrow,books.google.com/books?id=CYE_AAAAQBAJ, Modern Introductory Physics, Lloyd, James N., Amato, Joseph C., Galvez, Enrique, Parks, M. Elizabeth, 14 September 2010, Springer Science & Business Media, 978-0-387-79080-0, New York, en, {{rp|page=558}}BOOK, Polyanin, Andrei D., Andrei Polyanin,books.google.com/books?id=ejzScufwDRUC, A Concise Handbook of Mathematics, Physics, and Engineering Sciences, Chernoutsan, Alexei, 18 October 2010, CRC Press, 978-1-4398-0640-1, en, {{rp|page=561}} It is also common to refer to this hbar as “Planck’s constant“BOOK, Dowling, Jonathan P., Jonathan Dowling, Schrödinger’s Web: Race to Build the Quantum Internet, 24 August 2020, CRC Press, 978-1-000-08017-9,books.google.com/books?id=unP2DwAAQBAJ, en, {{rp|page=55}} {{efn|Notable examples of such usage include Landau and LifshitzBOOK, Landau, L. D., Lev Landau,books.google.com/books?id=neBbAwAAQBAJ, Quantum Mechanics: Non-Relativistic Theory, Lifshitz, E. M., Evgeny Lifshitz, 22 October 2013, Elsevier, 978-1-4831-4912-7, en, {{rp|page=20}} and Giffiths,BOOK, Griffiths, David J., David J. Griffiths,books.google.com/books?id=zU3bDwAAQBAJ, Introduction to Quantum Mechanics, Schroeter, Darrell F., 20 November 2019, Cambridge University Press, 978-1-108-10314-5, en, {{rp|page=3}} but there are many others, e.g.ENCYCLOPEDIA, Planck’s constant, The Great Soviet Encyclopedia, The Gale Group,encyclopedia2.thefreedictionary.com/Planck%27s+constant, 1970-1979, 3rd, BOOK, Itzykson, Claude, Claude Itzykson, Zuber, Jean-Bernard, Jean-Bernard Zuber, Quantum Field Theory, 20 September 2012, Courier Corporation, 978-0-486-13469-7,books.google.com/books?id=CxYCMNrUnTEC, en, {{rp|page=449}} BOOK, Kaku, Michio, Michio Kaku, Quantum Field Theory: A Modern Introduction, 1993, Oxford University Press, 978-0-19-507652-3,books.google.com/books?id=6J7vAAAAMAAJ, en, {{rp|page=284}} BOOK, Bogoli︠u︡bov, NikolaÄ­ Nikolaevich, Nikolay Bogolyubov, Shirkov, DmitriÄ­ Vasilʹevich, Dmitry Shirkov, Quantum Fields, 1982, Benjamin/Cummings Publishing Company, Advanced Book Program/World Science Division, 978-0-8053-0983-6,books.google.com/books?id=wczvAAAAMAAJ, en, {{rp|page=3}} BOOK, Aitchison, Ian J. R., Ian Aitchison, Hey, Anthony J. G., Tony Hey, Gauge Theories in Particle Physics: A Practical Introduction: From Relativistic Quantum Mechanics to QED, Fourth Edition, 17 December 2012, CRC Press, 978-1-4665-1299-3,books.google.com/books?id=-v6sPfuyUt8C, en, {{rp|page=365}} BOOK, de Wit, B., Bernard de Wit, Smith, J., Field Theory in Particle Physics, Volume 1, 2 December 2012, Elsevier, 978-0-444-59622-2,books.google.com/books?id=cCAGmj6XEfkC, en, {{rp|page=14}} BOOK, Brown, Lowell S., Lowell S. Brown, Quantum Field Theory, 1992, Cambridge University Press, 978-0-521-46946-3,books.google.com/books?id=mKsRAVX1910C, en, {{rp|page=18}} BOOK, Buchbinder, Iosif L., Shapiro, Ilya, Introduction to Quantum Field Theory with Applications to Quantum Gravity, March 2021, Oxford University Press, 978-0-19-883831-9,books.google.com/books?id=tL4cEAAAQBAJ, en, {{rp|page=4}} BOOK, Jaffe, Arthur, Arthur Jaffe, 9. Where does quantum field theory fit into the big picture?, Cao, Tian Yu, Conceptual Foundations of Quantum Field Theory, 25 March 2004, Cambridge University Press, 978-0-521-60272-3,books.google.com/books?id=d0wS0EJHZ3MC, en, {{rp|page=138}} BOOK, Cabibbo, Nicola, Nicola Cabibbo, Maiani, Luciano, Luciano Maiani, Benhar, Omar, An Introduction to Gauge Theories, 28 July 2017, CRC Press, 978-1-4987-3452-3,books.google.com/books?id=6WxGDwAAQBAJ, en, {{rp|page=251}} BOOK, Casalbuoni, Roberto, Introduction To Quantum Field Theory, 6 April 2017, World Scientific Publishing Company, 978-981-314-668-6,books.google.com/books?id=MFEyDwAAQBAJ, en, Second, {{rp|page=1}} BOOK, Das, Ashok, Ashok Das, Lectures On Quantum Field Theory, 2nd, 24 July 2020, World Scientific, 978-981-12-2088-3,books.google.com/books?id=gXr1DwAAQBAJ, en, {{rp|page=622}} BOOK, Desai, Bipin R., Quantum Mechanics with Basic Field Theory, 2010, Cambridge University Press, 978-0-521-87760-2,books.google.com/books?id=cScBCJ6wLpYC, en, {{rp|page=xx}} BOOK, Donoghue, John, Sorbo, Lorenzo, A Prelude to Quantum Field Theory, 8 March 2022, Princeton University Press, 978-0-691-22348-3,books.google.com/books?id=Wv5FEAAAQBAJ, en, {{rp|page=20}} BOOK, Folland, Gerald B., Gerald Folland, Quantum Field Theory: A Tourist Guide for Mathematicians, 3 February 2021, American Mathematical Soc., 978-1-4704-6483-7,books.google.com/books?id=nPEbEAAAQBAJ, en, {{rp|page=4}} BOOK, Fradkin, Eduardo, Quantum Field Theory: An Integrated Approach, 23 March 2021, Princeton University Press, 978-0-691-14908-0,books.google.com/books?id=ruEIEAAAQBAJ, en, {{rp|page=36}} BOOK, Gelis, François, Quantum Field Theory, 11 July 2019, Cambridge University Press, 978-1-108-48090-1,books.google.com/books?id=BzGdDwAAQBAJ, en, {{rp|page=41}} BOOK, Greiner, Walter, Walter Greiner, Reinhardt, Joachim, Quantum Electrodynamics, 9 March 2013, Springer Science & Business Media, 978-3-662-05246-4,books.google.com/books?id=6y3rCAAAQBAJ, en, {{rp|page=199}} BOOK, Liboff, Richard L., Richard Liboff,books.google.com/books?id=xMIlAgQSRUsC, Introductory Quantum Mechanics, 2003, Pearson Education, 978-81-317-0441-7, 4th, San Francisco, en, {{rp|page=846}} JOURNAL, Barut, A. O., Asım Orhan Barut, 1 August 1978, The Creation of a Photon: A Heuristic Calculation of Planck’s Constant ħ or the Fine Structure Constant α, Zeitschrift für Naturforschung A, 33, 8, 993–994, 10.1515/zna-1978-0819, 1978ZNatA..33..993B, 45829793, free, JOURNAL, Kocia, Lucas, Love, Peter, 12 July 2018, Measurement contextuality and Planck’s constant, New Journal of Physics, 20, 7, 073020, 10.1088/1367-2630/aacef2, 1711.08066, 2018NJPh...20g3020K, 73623448, JOURNAL, Humpherys, David, 28 November 2022, The Implicit Structure of Planck’s Constant, European Journal of Applied Physics, 4, 6, 22–25, 10.24018/ejphysics.2022.4.6.227, 254359279, free, {{rp|page=25}} BOOK, Bais, F. Alexander, Philosophy of Information, Farmer, J. Doyne, J. Doyne Farmer, 2008, North-Holland, 978-0-444-51726-5, Adriaans, Pieter, Handbook of the Philosophy of Science, 8, Amsterdam, The Physics of Information, van Benthem, Johan, Johan van Benthem (logician),www.sciencedirect.com/science/article/abs/pii/B9780444517265500200, 0708.2837, {{rp|pages=653}}}} while retaining the relationship hbar, {{=}} h/(2 pi).

Symbols

By far the most common symbol for the reduced Planck constant is hbar . However, there are some sources that denote it by h instead, in which case they usually refer to it as the “Dirac h“BOOK, Hirota, E., Sakakima, H., Inomata, K., Giant Magneto-Resistance Devices, 9 March 2013, Springer Science & Business Media, 978-3-662-04777-4,books.google.com/books?id=mvDrCAAAQBAJ, en, {{rp|page=43}} JOURNAL, Gardner, John H., An Invariance Theory, Encyclia, 1988, 65, 139, {{rp|page=151}} (or “Dirac’s {{nowrap|h“}}BOOK, Levine, Raphael D., Raphael David Levine, Molecular Reaction Dynamics, 4 June 2009, Cambridge University Press, 978-1-139-44287-9,books.google.com/books?id=FVqyS31OM7sC, en, {{rp|page=21}}).

History

The combination h/(2pi) first made its appearance{{efn|Some sourcesJOURNAL, Heilbron, John L., June 2013, The path to the quantum atom, Nature, 498, 7452, 27–30, 10.1038/498027a, 23739408, 4355108, JOURNAL, McCormmach, Russell, 1 January 1966, The atomic theory of John William Nicholson, Archive for History of Exact Sciences, 3, 2, 160–184, 10.1007/BF00357268, 41133258, 120797894, {{rp|169}} BOOK, Mehra, Jagdish, Jagdish_Mehra, Rechenberg, Helmut, Helmut Rechenberg, The Historical Development of Quantum Theory, 1, 3 August 1982, Springer New York, 978-0-387-90642-3,books.google.com/books?id=oLWNzwEACAAJ, en, {{rp|page=180}} claim that John William Nicholson discovered the quantization of angular momentum in units of h/(2pi) in his 1912 paper,JOURNAL, Nicholson, J. W., 14 June 1912, The Constitution of the Solar Corona. II., Monthly Notices of the Royal Astronomical Society, Oxford University Press, 72, 8, 677–693, 10.1093/mnras/72.8.677, 0035-8711, free, 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.“{{rp|page=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 h 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,{{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.{{rp|692}} The literal combination h/(2pi) does not appear in that paper. A biographical memoir of NicholsonJOURNAL, Wilson, W., 10.1098/rsbm.1956.0014, John William Nicholson 1881-1955, Biographical Memoirs of Fellows of the Royal Society, 2, 209–214, 1956, 769485, states that Nicholson only “later” realized that the discrete changes in angular momentum are integral multiples of {{nowrap|h/(2pi),}} 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 Niels Bohr’s 1913 paper,JOURNAL, Bohr, N., July 1913, I. On the constitution of atoms and molecules,zenodo.org/record/2493915, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 26, 151, 1–25, 10.1080/14786441308634955, {{rp|page=15}} where it was denoted by {{nowrap|M_{0}.}}{{efn|Bohr denoted by M the angular momentum of the electron around the nucleus, and wrote the quantization condition as {{nowrap|M=tau M_{0},}} where tau is a positive integer. (See the Bohr model.)}} 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 and in which h/(2pi) appeared without a separate symbol: JOURNAL, Sommerfeld, A., Arnold Sommerfeld, Zur Theorie der Balmerschen Serie, Sitzungsberichte der mathematisch-physikalischen Klasse der K. B. Akademie der Wissenschaften zu München, 1915, 33, 198, 425–458,static-content.springer.com/esm/art%3A10.1140%2Fepjh%2Fe2013-40053-8/MediaObjects/13129_2013_121_MOESM1_ESM.pdf, 10.1140/epjh/e2013-40053-8, {{rp|page=428}} JOURNAL, Schwarzschild, K., Karl Schwarzschild, Zur Quantenhypothese, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1916, 548–568, {{rp|page=549}} JOURNAL, Ehrenfest, P., Paul Ehrenfest, XLVIII. Adiabatic invariants and the theory of quanta, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, June 1917, 33, 198, 500–513, 10.1080/14786440608635664, {{rp|page=508}} JOURNAL, Landé, A., Alfred Landé, Das Serienspektrum des Heliums, Physikalische Zeitschrift, June 1919, 20, 228–234, {{rp|page=230}} JOURNAL, Bohr, N., Niels Bohr, Ãœber die Serienspektra der Elemente, Zeitschrift für Physik, October 1920, 2, 5, 423–469, 10.1007/BF01329978, {{rp|page=458}} JOURNAL, Stern, Otto, Otto Stern, Ein Weg zur experimentellen Prüfung der Richtungsquantelung im Magnetfeld, Zeitschrift für Physik, December 1921, 7, 1, 249–253, 10.1007/BF01332793,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, JOURNAL, Heisenberg, Werner, Werner Heisenberg, Zur Quantentheorie der Linienstruktur und der anomalen Zeemaneflekte, Zeitschrift für Physik, December 1922, 8, 1, 273–297, 10.1007/BF01329602, {{rp|page=276}} JOURNAL, Kramers, H. A., Hans Kramers, Pauli, W., Wolfgang Pauli, Zur Theorie der Bandenspektren, Zeitschrift für Physik, December 1923, 13, 1, 351–367, 10.1007/BF01328226,books.google.com/books?id=9PpMAQAAIAAJ&dq=%22negativen+Halogenionen+ein+resultierendes%22&pg=RA1-PA351, JOURNAL, Born, M., Max Born, Jordan, P., Pascual Jordan, Zur Quantenmechanik, Zeitschrift für Physik, December 1925, 34, 1, 858–888, 10.1007/BF01328531, JOURNAL, Dirac, P. A. M., Paul Dirac, The fundamental equations of quantum mechanics, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, December 1925, 109, 752, 642–653, 10.1098/rspa.1925.0150, free, .JOURNAL, Born, M., Max Born, Heisenberg, W., Werner Heisenberg, Jordan, P., Pascual Jordan, Zur Quantenmechanik. II., Zeitschrift für Physik, August 1926, 35, 8–9, 557–615, 10.1007/BF01379806, }} Then, in 1926, in their seminal papers, Schrödinger and Dirac again introduced special symbols for it: K in the case of Schrödinger,JOURNAL, Schrödinger, E., Erwin Schrödinger, Quantisierung als Eigenwertproblem, Annalen der Physik, 1926, 384, 4, 361–376, 10.1002/andp.19263840404, free, and h in the case of Dirac.JOURNAL, Dirac, P. A. M., Paul Dirac, On the theory of quantum mechanics, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, October 1926, 112, 762, 661–677, 10.1098/rspa.1926.0133, free, Dirac continued to use h in this way until 1930,BOOK, Mehra, Jagdish, Jagdish_Mehra, Rechenberg, Helmut, Helmut Rechenberg, The Historical Development of Quantum Theory, 6, 2000, Springer, New York, en, {{rp|page=291}} when he introduced the symbol hbar in his book The Principles of Quantum Mechanics.{{rp|page=291}} BOOK, Dirac, P. A. M., Paul Dirac, The Principles of Quantum Mechanics, 1930, Clarendon, Oxford, U.K., 1st,

See also

• CODATA 2018
• International System of Units
• Introduction to quantum mechanics
• List of scientists whose names are used in physical constants
• Planck units
• Wave–particle duality

Notes

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References

Citations

{{Reflist}}

Sources

• {{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 |author-link=John D. Barrow |url-access=registration}}

External links

• “The role of the Planck constant in physics” – presentation at 26th CGPM meeting at Versailles, France, November 2018 when voting took place.
• “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.

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