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Planck constant

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(Redirected fromReduced Planck constant)
Physical constant in quantum mechanics
For the law governing black-body radiation, seePlanck's law.
Planck constant
Common symbols
h{\displaystyle h}
SI unitJ·s (joule-second)
InSI base unitskg⋅m2⋅s−1
DimensionML2T−1{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}
Value6.62607015×10−34 J⋅Hz−1
4.135667696...×10−15 eV⋅Hz−1
Reduced Planck constant
Common symbols
ℏ{\displaystyle \hbar }
SI unitJ·s (joule-second)
InSI base unitskg⋅m2⋅s−1
Derivations from
other quantities
DimensionML2T−1{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}
Value1.054571817...×10−34 J⋅s
6.582119569...×10−16 eV⋅s

ThePlanck constant, orPlanck's constant, denoted byh{\displaystyle h}, is a fundamentalphysical constant of foundational importance inquantum mechanics: aphoton's energy is equal to itsfrequency multiplied by the Planck constant, and a particle'smomentum is equal to thewavenumber of the associatedmatter wave (the reciprocal of itswavelength) multiplied by the Planck constant.

The constant was postulated byMax Planck in 1900 as aproportionality constant needed to explain experimentalblack-body radiation.[1] Planck later referred to the constant as the "quantum ofaction".[2] In 1905,Albert Einstein associated the "quantum" or minimal element of the energy to the electromagnetic wave itself. Max Planck received the 1918Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

Inmetrology, the Planck constant is used, together with other constants, to define thekilogram, theSI unit of mass.[3] TheSI units are defined such that it has the exact valueh{\displaystyle h} =6.62607015×10−34 J⋅Hz−1‍[4] when the Planck constant is expressed in SI units.

The closely relatedreduced Planck constant, denotedℏ{\textstyle \hbar } (h-bar), equal to the Planck constant divided by2π:ℏ=h2π{\textstyle \hbar ={\frac {h}{2\pi }}}, is commonly used in quantum physics equations. It relates the energy of a photon to itsangular frequency, and the linear momentum of a particle to theangular wavenumber of its associated matter wave. Ash{\displaystyle h} has an exact defined value, the value ofℏ{\textstyle \hbar } can be calculated to arbitrary precision:ℏ{\displaystyle \hbar } =1.054571817...×10−34 J⋅s.[5] As aproportionality constant in relationships involving angular quantities, the unit ofℏ{\textstyle \hbar } may be given as J·s/rad, with the same numerical value, as theradian is the naturaldimensionless unit ofangle.

History

[edit]

Origin of the constant

[edit]
Main article:Planck's law
Plaque at theHumboldt University of Berlin: "In this edifice taught Max Planck, the discoverer of the elementary quantum of actionh, from 1889 to 1928."
Intensity of light emitted from ablack body. Each curve represents behavior at different body temperatures. The Planck constanth is used to explain the shape of these curves.

The Planck constant was formulated as part of Max Planck's successful effort to produce a mathematical expression that accurately predicted the observed spectral distribution ofblack-body radiation.[6] This expression is known as Planck's law.

In the last years of the 19th century, Max Planck was investigating the problem of black-body radiation posed byKirchhoff some 40 years earlier. Everyphysical body spontaneously and continuously emitselectromagnetic 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.[6]: 141  Also around this time, but unknown to Planck,Lord Rayleigh had derived theoretically a formula, later known as theRayleigh–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 ofharmonic oscillators, one for each possible frequency. He examined how theentropy 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,[1] 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{\displaystyle h}, which is thought to be forHilfsgröße (auxiliary quantity),[7] and subsequently became known as the Planck constant. The expression formulated by Planck showed that the spectral radiance per unit frequency of a body forfrequencyν atabsolute temperatureT is given byBν(ν,T)dν=2hν3c21ehνkBT−1dν,{\displaystyle B_{\nu }(\nu ,T)d\nu ={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}d\nu ,}wherekB{\displaystyle k_{\text{B}}} is theBoltzmann constant,h{\displaystyle h} is the Planck constant, andc{\displaystyle c} is thespeed of light in the medium, whether material or vacuum.[8][9][10]

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.[1] To save his theory, Planck resorted to using the then-controversial theory ofstatistical mechanics,[1] which he described as "an act of desperation".[11] One of his new boundary conditions was

to interpretUN ['the vibrational energy ofN 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 elementε;

— Planck, "On the Law of Distribution of Energy in the Normal Spectrum"[1]

With this new condition, Planck had imposed the quantization of the energy of the oscillators, in his own words, "a purely formal assumption ... actually I did not think much about it",[12] 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 sometimes termed thePlanck–Einstein relation:E=hf.{\displaystyle E=hf.}

Planck was able to calculate the value ofh{\displaystyle h} from experimental data on black-body radiation: his result,6.55×10−34 J⋅s, is within 1.2% of the currently defined value.[1] He also made the first determination of theBoltzmann constantkB{\displaystyle k_{\text{B}}} from the same data and theory.[13]

The observed Planck curves at different temperatures, and the divergence of the theoretical Rayleigh–Jeans (black) curve from the observed Planck curve at 5000 K.

Development and application

[edit]

The black-body problem was revisited in 1905, whenLord Rayleigh andJames Jeans (together) andAlbert Einstein independently proved that classical electromagnetism couldnever account for the observed spectrum. These proofs are commonly known as the "ultraviolet catastrophe", a name coined byPaul Ehrenfest in 1911. They contributed greatly (along with Einstein's work on thephotoelectric effect) in convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The firstSolvay Conference in 1911 was devoted to "the theory of radiation and quanta".[14]

Photoelectric effect

[edit]
Main article: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 byAlexandre Edmond Becquerel in 1839, although credit is usually reserved forHeinrich Hertz,[15] who published the first thorough investigation in 1887. Another particularly thorough investigation was published byPhilipp Lenard (Lénárd Fülöp) in 1902.[16] Einstein's 1905 paper[17] discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921,[15] after his predictions had been confirmed by the experimental work ofRobert Andrews Millikan.[18] 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.[19][20]

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 itsintensity. The light from a theatre spotlight is moreintense 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 certainkinetic energy, which can be measured. This kinetic energy (for each photoelectron) isindependent of the intensity of the light,[16] but depends linearly on the frequency;[18] and if the frequency is too low (corresponding to a photon energy that is less than thework 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).[21][22] 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.[16]

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 namedphotons, was to be the same as Planck's "energy element", giving the modern version of the Planck–Einstein relation:E=hf.{\displaystyle E=hf.}

Einstein's postulate was later proven experimentally: the constant of proportionality between the frequency of incident lightf{\displaystyle f} and the kinetic energy of photoelectronsE{\displaystyle E} was shown to be equal to the Planck constanth{\displaystyle h}.[18]

Atomic structure

[edit]
Main article:Bohr model
A schematization of the Bohr model of the hydrogen atom. The transition shown from then = 3 level to then = 2 level gives rise to visible light of wavelength 656 nm (red), as the model predicts.

In 1912John William Nicholson developed[23] an atomic model and found the angular momentum of the electrons in the model were related byh/2π.[24][25] Nicholson's nuclear quantum atomic model influenced the development ofNiels Bohr 's atomic model[26][27][25] and Bohr quoted him in his 1913 paper of the Bohr model of the atom.[28] Bohr's model went beyond Planck's abstract harmonic oscillator concept: an electron in a Bohr atom could only have certain defined energiesEn{\displaystyle E_{n}}, defined byEn=−hcR∞n2,{\displaystyle E_{n}=-{\frac {hcR_{\infty }}{n^{2}}},}wherec{\displaystyle c} is the speed of light in vacuum,R∞{\displaystyle R_{\infty }} is an experimentally determined constant (theRydberg constant) andn∈{1,2,3,...}{\displaystyle n\in \{1,2,3,...\}}. This approach also allowed Bohr to account for theRydberg formula, an empirical description of the atomic spectrum of hydrogen, and to account for the value of the Rydberg constantR∞{\displaystyle R_{\infty }} in terms of other fundamental constants.In discussing angular momentum of the electrons in his model Bohr introduced the quantityh2π{\textstyle {\frac {h}{2\pi }}}, known as thereduced Planck constant as the quantum ofangular momentum.[28]

Uncertainty principle

[edit]
Main article:Uncertainty principle

The Planck constant also occurs in statements ofWerner Heisenberg's uncertainty principle. Given numerous particles prepared in the same state, theuncertainty in their position,Δx{\displaystyle \Delta x}, and the uncertainty in their momentum,Δpx{\displaystyle \Delta p_{x}}, obeyΔxΔpx≥ℏ2,{\displaystyle \Delta x\,\Delta p_{x}\geq {\frac {\hbar }{2}},}where the uncertainty is given as thestandard deviation of the measured value from itsexpected value. There are several other such pairs of physically measurableconjugate 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 thecommutator relationship between theposition operatorx^{\displaystyle {\hat {x}}} and themomentum operatorp^{\displaystyle {\hat {p}}}:[p^i,x^j]=−iℏδij,{\displaystyle [{\hat {p}}_{i},{\hat {x}}_{j}]=-i\hbar \delta _{ij},}whereδij{\displaystyle \delta _{ij}} is theKronecker delta.

Photon energy

[edit]

ThePlanck relation connects the particularphoton energyE with its associated wave frequencyf:E=hf.{\displaystyle E=hf.}This energy is extremely small in terms of ordinarily perceived everyday objects.

Since the frequencyf,wavelengthλ, andspeed of lightc are related byf=cλ{\displaystyle f={\frac {c}{\lambda }}}, the relation can also be expressed asE=hcλ.{\displaystyle E={\frac {hc}{\lambda }}.}

de Broglie wavelength

[edit]

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, includingelectrodynamics. Thede Broglie wavelengthλ of the particle is given byλ=hp,{\displaystyle \lambda ={\frac {h}{p}},}wherep denotes the linearmomentum of a particle, such as a photon, or any otherelementary particle.

Theenergy of a photon with angular frequencyω = 2πf is given byE=ℏω,{\displaystyle E=\hbar \omega ,}while its linear momentum relates top=ℏk,{\displaystyle p=\hbar k,}wherek is anangular wavenumber.

These two relations are the temporal and spatial parts of the special relativistic expression using4-vectors.Pμ=(Ec,p→)=ℏKμ=ℏ(ωc,k→).{\displaystyle P^{\mu }=\left({\frac {E}{c}},{\vec {p}}\right)=\hbar K^{\mu }=\hbar \left({\frac {\omega }{c}},{\vec {k}}\right).}

Statistical mechanics

[edit]

Classicalstatistical mechanics requires the existence ofh (but does not define its value).[29] Eventually, following Planck's discovery, it was speculated that physicalaction could not have an arbitrary value, but instead was restricted to integer multiples of a very small quantity, the "[elementary]quantum of action", called thePlanck constant.[30] This was a significant concept of the "old quantum theory" developed by physicists includingBohr,Sommerfeld, andIshiwara, in which particle trajectories exist but arehidden, but quantum laws constrain them based on their action. This view has been replaced by modern quantum theory, in which fixed trajectories of motion do not even exist; rather, the particle is represented by a wavefunction spread out in space and time.[31]: 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

[edit]

The Planck constant has the samedimensions asaction and asangular momentum (both with unit J·s = kg·m2·s−1). The Planck constant is fixed ath{\displaystyle h} =6.62607015×10−34 J⋅Hz−1‍[4] as part of the definition of theSI units.[32] Alternatively, if theradian were considered abase unit, thenh{\displaystyle h} would have the dimension of action (unit J·s), whileℏ{\displaystyle \hbar } would have the dimension of angular momentum (unit J·s·rad−1), instead.[33]

This value is used to define the SI unit of mass, thekilogram: "the kilogram [...] is defined by taking the fixed numerical value ofh to be6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where themetre and thesecond are defined in terms ofspeed of lightc and duration ofhyperfine transition of theground state of an unperturbedcaesium-133 atomΔνCs."[32] Technologies of massmetrology such as theKibble balance measure the kilogram by fixing the Planck constant.

Ash{\displaystyle h} has an exact defined value, the value of the reduced Planck constantℏ{\displaystyle \hbar } can be calculated to arbitrary precision without any limiting uncertainty:

ℏ=h2π=6.626070152π×10−34J⋅s=1.054571817...×10−34J⋅s{\displaystyle \hbar ={\frac {h}{2\pi }}={\frac {6.626\,070\,15}{2\pi }}\times 10^{-34}\,\mathrm {J{\cdot }s} =1.054\,571\,817...\times 10^{-34}\,\mathrm {J{\cdot }s} }

As aproportionality constant in relationships involving angular quantities, the unit ofℏ{\textstyle \hbar } may be given as J·s/rad, with the same numerical value, as the radian is the naturaldimensionless unit ofangle. This is analogous to the use ofhertz (Hz) for ordinaryfrequency andradians per second (rad/s) forangular frequency, both dimensionally equal to s−1.

Significance of the value

[edit]

The Planck constant is one of the smallest constants used in physics. This reflects the fact that on ascale 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 theproduct of energy and time for a physical event approaches the Planck constant,quantum effects dominate.[34]

Equivalently, the order of the Planck constant reflects the fact that everyday objects and systems are made of alarge number ofmicroscopic particles. For example, ingreen light (with awavelength of 555 nanometres or afrequency of540 THz) eachphoton has anenergyE =hf =3.58×10−19 J. This 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 individualatoms ormolecules. An amount oflight more typical in everyday experience (though much larger than the smallest amount perceivable by thehuman eye) is the energy of onemole of photons, which can be computed by multiplying the photon energy by theAvogadro number,6.02214076×1023,[35] with the result of216 kJ, about equal to thefood energy in a small freshapple.[36]

Reduced Planck constant

[edit]

Many equations in quantum physics are customarily written using thereduced Planck constant,[37]: 104  also known as theDirac constant, equal toh2π{\textstyle {\frac {h}{2\pi }}} and denotedℏ{\textstyle \hbar } (pronouncedh-bar[38]: 336 ).

History

[edit]

The combinationh2π{\textstyle {\frac {h}{2\pi }}} appeared inNiels Bohr's 1913 paper,[39]: 15  where it was denoted byM0{\textstyle M_{0}}.[25]: 169 [a] For the next 15 years, the combination continued to appear in the literature, but normally without a separate symbol.[40]: 180 [b] Then, in 1926, in their seminal papers,Schrödinger andDirac again introduced special symbols for it:K{\textstyle K} in the case of Schrödinger,[52] andh{\textstyle h} in the case of Dirac.[53] Dirac continued to useh{\textstyle h} in this way until 1930,[54]: 291  when he introduced the symbolℏ{\textstyle \hbar } in his bookThe Principles of Quantum Mechanics.[54]: 291 [55]

See also

[edit]

Notes

[edit]
  1. ^Bohr denoted byM{\textstyle M} the angular momentum of the electron around the nucleus, and wrote the quantization condition asM=τM0{\textstyle M=\tau M_{0}}, whereτ{\textstyle \tau } is a positive integer (seeBohr model).
  2. ^Here are some papers that are mentioned in[40] and in whichh2π{\textstyle {\frac {h}{2\pi }}} appeared without a separate symbol:[41]: 428 [42]: 549 [43]: 508 [44]: 230 [45]: 458 [46][47]: 276 [48][49][50].[51]

References

[edit]

Citations

[edit]
  1. ^abcdefPlanck, Max (1901),"Ueber das Gesetz der Energieverteilung im Normalspectrum"(PDF),Annalen der Physik (in German),309 (3):553–63,Bibcode:1901AnP...309..553P,doi:10.1002/andp.19013090310,archived(PDF) from the original on 10 June 2012, retrieved15 December 2008. English translations:
  2. ^"Max Planck Nobel Lecture".Archived from the original on 14 July 2023. Retrieved14 July 2023.
  3. ^The International System of Units(PDF), V3.01 (9th ed.), International Bureau of Weights and Measures, August 2024, p. 131,ISBN 978-92-822-2272-0
  4. ^ab"2022 CODATA Value: Planck constant".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved18 May 2024.
  5. ^"2022 CODATA Value: reduced Planck constant".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved18 May 2024.
  6. ^abBitter, Francis; Medicus, Heinrich A. (1973).Fields and particles. New York: Elsevier. pp. 137–144.
  7. ^Boya, Luis J. (2004). "The Thermal Radiation Formula of Planck (1900)".arXiv:physics/0402064v1.
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  9. ^Chandrasekhar, S. (1960) [1950].Radiative Transfer (revised reprint ed.).Dover. p. 8.ISBN 978-0-486-60590-6.{{cite book}}:ISBN / Date incompatibility (help)
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  16. ^abcLenard, P. (1902)."Ueber die lichtelektrische Wirkung".Annalen der Physik.313 (5):149–198.Bibcode:1902AnP...313..149L.doi:10.1002/andp.19023130510.Archived from the original on 18 August 2019. Retrieved3 July 2019.
  17. ^Einstein, Albert (1905)."Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt"(PDF).Annalen der Physik.17 (6):132–48.Bibcode:1905AnP...322..132E.doi:10.1002/andp.19053220607.Archived(PDF) from the original on 9 July 2011. Retrieved3 December 2009.
  18. ^abcMillikan, R. A. (1916)."A Direct Photoelectric Determination of Planck'sh".Physical Review.7 (3):355–88.Bibcode:1916PhRv....7..355M.doi:10.1103/PhysRev.7.355.
  19. ^Isaacson, Walter (10 April 2007).Einstein: His Life and Universe. Simon and Schuster. pp. 309–314.ISBN 978-1-4165-3932-2.Archived from the original on 9 January 2020. Retrieved31 October 2021.
  20. ^"The Nobel Prize in Physics 1921". Nobel Foundation.Archived from the original on 3 July 2018. Retrieved23 April 2014.
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  22. ^Smith, Richard (1963)."Two-Photon Photoelectric Effect".Physical Review.130 (6): 2599.Bibcode:1963PhRv..130.2599S.doi:10.1103/PhysRev.130.2599.4.
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  54. ^abMehra, Jagdish;Rechenberg, Helmut (2000).The Historical Development of Quantum Theory. Vol. 6. New York: Springer.
  55. ^Dirac, Paul A. M. (1930).The Principles of Quantum Mechanics (1st ed.). Oxford, U.K.: Clarendon.

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