Planck constant

Value of h Units Ref.
6.62607015×10−34 Js [1][note 1]
4.135667696×10−15 eVs [note 2]
Values of ħ (h-bar) Units Ref.
1.054571817×10−34 Js [note 2]
6.582119569×10−16 eVs [note 2]
Values of hc Units Ref.
1.98644568×10−25 Jm [note 3]
1.23984193 eVμm [note 3]
Values of ħc (h-bar) Units Ref.
3.16152649×10−26 Jm [note 3]
0.1973269804 eVμm [note 4]
Plaque at the Humboldt University of Berlin: "In this house taught Max Planck, the discoverer of the elementary quantum of action , from 1889 to 1928."

The Planck constant, or Planck's constant, is the quantum of electromagnetic action that relates a photon's energy to its frequency. The Planck constant multiplied by a photon's frequency is equal to a photon's energy. The Planck constant is a fundamental physical constant denoted as , and of fundamental importance in quantum mechanics. In metrology it is used to define the kilogram in SI units.[2]

The Planck constant is defined to have the exact value 6.62607015×10−34 J⋅s in SI units.[3][4]

At the end of the 19th century, accurate measurements of the spectrum of black body radiation existed, but predictions of the frequency distribution of the radiation by then-existing theories diverged significantly at higher frequencies. In 1900, Max Planck derived empirically a formula for the observed spectrum. He assumed that a hypothetical electrically charged oscillator in a cavity that contained black-body radiation could only change its energy in a minimal increment, , that was proportional to the frequency of its associated electromagnetic wave.[5] He was able to calculate the proportionality constant, , from the experimental measurements, and that constant is named in his honor. In 1905, the value was associated by Albert Einstein with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle. It was eventually called a photon. Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

Since energy and mass are equivalent, the Planck constant also relates mass to frequency.

Origin of the constantEdit

Intensity of light emitted from a black body Each curve represents behavior at different body temperatures. Max Planck was the first 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 (black-body radiation).[6] 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.[6]: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,[5] 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,  , which subsequently became known as the Planck Constant. The expression formulated by Planck showed that the spectral radiance of a body for frequency ν at absolute temperature T is given by


where   is the Boltzmann constant,   is the Planck constant, and   is the speed of light in the medium, whether material or vacuum.[7][8][9]

The spectral radiance of a body,  , 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   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.[10]

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   are W·sr−1·m−2·Hz−1, while those of   are 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.[5] To save his theory, Planck resorted to using the then-controversial theory of statistical mechanics,[5] which he described as "an act of despair … I was ready to sacrifice any of my previous convictions about physics."[11] One of his new boundary conditions was


— Planck, On the Law of Distribution of Energy in the Normal Spectrum[5]

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,[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 now sometimes termed the "Planck–Einstein relation":


Planck was able to calculate the value of   from experimental data on black-body radiation: his result, 6.55×10−34 J⋅s, is within 1.2% of the currently accepted value.[5] He also made the first determination of the Boltzmann constant   from the same data and theory.[13]

The divergence of the theoretical Rayleigh-Jeans (black) curve from the observed Planck curves at different temperatures.

Development and applicationEdit


Photoelectric effectEdit

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,[15] who published the first thorough investigation in 1887. Another particularly thorough investigation was published by Philipp Lenard 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 of Robert 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 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 didn't seem to agree with the wave description of light.

The "photoelectrons" emitted as a result of the photoelectric effect have a certain kinetic energy, which can be measured. This kinetic energy (for each photoelectron) is independent of the intensity of the light,[16] but depends linearly on the frequency;[18] 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).[21] 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 postulate was later proven experimentally: the constant of proportionality between the frequency of incident light   and the kinetic energy of photoelectrons   was shown to be equal to the Planck constant  .[18]

Atomic structureEdit




where   is the speed of light in vacuum,   is an experimentally determined constant (the Rydberg constant) and  . Once the electron reached the lowest energy level ( ), 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   in terms of other fundamental constants.



Uncertainty principleEdit

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,  , and the uncertainty in their momentum,  , 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.



where   is the Kronecker delta.

Photon energyEdit

The Planck–Einstein relation connects the particular photon energy E with its associated wave frequency f:





The de Broglie wavelength λ of the particle is given by


where 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 ω = 2πf is given by


while its linear momentum relates to


where k is an angular wavenumber. In 1923, Louis de Broglie generalized the Planck–Einstein relation by postulating that the Planck constant represents the proportionality between the momentum and the quantum wavelength of not just the photon, but the quantum wavelength of any particle. This was confirmed by experiments soon afterward. This holds throughout the quantum theory, including electrodynamics.

Problems can arise when dealing with frequency or the Planck constant because the units of angular measure (cycle or radian) are omitted in SI.[23][24][25][26][27] In the language of quantity calculus,[28] the expression for the "value" of the Planck constant, or of a frequency, is the product of a "numerical value" and a "unit of measurement". When we use the symbol f (or ν) for the value of a frequency it implies the units cycles per second or hertz, but when we use the symbol ω for its value it implies the units radians per second; the numerical values of these two ways of expressing the value of a frequency have a ratio of 2π, but their values are equal. Omitting the units of angular measure "cycle" and "radian" can lead to an error of 2π. A similar state of affairs occurs for the Planck constant. We use the symbol h when we express the value of the Planck constant in J⋅s/cycle, and we use the symbol ħ when we express its value in J⋅s/rad. Since both represent the value of the Planck constant, but in different units, we have h = ħ. Their "values" are equal but, as discussed below, their "numerical values" have a ratio of 2π. In this Wikipedia article the word "value" as used in the tables means "numerical value", and the equations involving the Planck constant and/or frequency actually involve their numerical values using the appropriate implied units. The distinction between "value" and "numerical value" as it applies to frequency and the Planck constant is explained in more detail in this pdf file Link.

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


Classical statistical mechanics requires the existence of h (but does not define its value).[29] Eventually, following upon Planck's discovery, it was recognized that physical action cannot take on an arbitrary value. Instead, it must be some integer multiple of a very small quantity, the "quantum of action", now called the reduced Planck constant or the natural unit of action. This is the so-called "old quantum theory" developed by Bohr and Sommerfeld, in which particle trajectories exist but are hidden, but quantum laws constrain them based on their action. This view has been largely replaced by fully modern quantum theory, in which definite trajectories of motion do not even exist, rather, the particle is represented by a wavefunction spread out in space and in time. Thus there is no value of the action as classically defined. Related to this is the concept of energy quantization which existed in old quantum theory and also exists in altered form in modern quantum physics. Classical physics cannot explain either quantization of energy or the lack of classical particle motion.



The Planck constant has dimensions of physical action; i.e., energy multiplied by time, or momentum multiplied by distance, or angular momentum. In SI units, the Planck constant is expressed in joule-seconds (J⋅s or Nms or kg⋅m2⋅s−1). Implicit in the dimensions of the Planck constant is the fact that the SI unit of frequency, the Hertz, represents one complete cycle, 360 degrees or 2π radians, per second. An angular frequency in radians per second is often more natural in mathematics and physics and many formulas use a reduced Planck constant (pronounced h-bar)

The above values are recommended by 2018 CODATA.

In atomic units,


Understanding the 'fixing' of the value of hEdit

Since 2019, the numerical value of the Planck constant has been fixed, with finite significant figures. Under the present definition of the kilogram, which states that "The kilogram [...] is defined by taking the fixed numerical value of h to be 6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the metre and the second are defined in terms of speed of light c and duration of hyperfine transition of the ground state of an unperturbed cesium-133 atom ΔνCs."[31] This implies that mass metrology is now aimed to find the value of one kilogram, and thus it is kilogram which is compensating. Every experiment aiming to measure the kilogram (such as the Kibble balance and the X-ray crystal density method), will essentially refine the value of a kilogram.

As an illustration of this, suppose the decision of making h to be exact was taken in 2010, when its measured value was 6.62606957×10−34 J⋅s, thus the present definition of kilogram was also enforced. In future, the value of one kilogram must have become refined to 6.62607015/6.626069571.0000001 times the mass of the International Prototype of the Kilogram (IPK), neglecting the metre and second units' share, for sake of simplicity.

Significance of the valueEdit

The Planck constant is related to the quantization of light and matter. It can be seen as a subatomic-scale constant. In a unit system adapted to subatomic scales, the electronvolt is the appropriate unit of energy and the petahertz the appropriate unit of frequency. Atomic unit systems are based (in part) on the Planck constant. The physical meaning of the Planck's constant could suggest some basic features of our physical world.

The Planck constant is one of the smallest constants used in physics. This reflects the fact that on a scale adapted to humans, where energies are typical of the order of kilojoules and times are typical of the order of seconds or minutes, the Planck constant (the quantum of action) is very small. One can regard the Planck constant to be only relevant to the microscopic scale instead of the macroscopic scale in our everyday experience.

Equivalently, the order of the Planck constant reflects the fact that everyday objects and systems are made of a large number of microscopic particles. For example, green light with a wavelength of 555 nanometres (a wavelength that can be perceived by the human eye to be green) has a frequency of 540 THz (540×1012 Hz). Each photon has an energy E = hf = 3.58×10−19 J. That is a very small amount of energy in terms of everyday experience, but everyday experience is not concerned with individual photons any more than with individual atoms or molecules. An amount of light 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, NA = 6.02214076×1023 mol−1, with the result of 216 kJ/mol, about the food energy in three apples.



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

Josephson constantEdit



The Josephson constant may be measured by comparing the potential difference generated by an array of Josephson junctions with a potential difference which is known in SI volts. The measurement of the potential difference in SI units is done by allowing an electrostatic force to cancel out a measurable gravitational force, in a Kibble balance. Assuming the validity of the theoretical treatment of the Josephson effect, KJ is related to the Planck constant by


Kibble balanceEdit



Magnetic resonanceEdit









Faraday constantEdit



X-ray crystal densityEdit



Particle acceleratorEdit


See alsoEdit


  1. ^ Set on 20 November 2018, by the CGPM to this exact value. This value took effect on 20 May 2019.
  2. ^ a b c The value is exact but not expressible as a finite decimal; approximated to 9 decimal places only.
  3. ^ a b c The value is exact but not expressible as a finite decimal; approximated to 8 decimal places only.
  4. ^ The value is exact but not expressible as a finite decimal; approximated to 10 decimal places only.



  1. ^ "Resolutions of the 26th CGPM" (PDF). BIPM. 2018-11-16. Retrieved 2018-11-20.
  3. ^ "2018 CODATA Value: Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
  4. ^ "Resolutions of the 26th CGPM" (PDF). BIPM. 2018-11-16. Archived from the original (PDF) on 2018-11-19. Retrieved 2018-11-20.
  5. ^ a b c d e f Planck, Max (1901), "Ueber das Gesetz der Energieverteilung im Normalspectrum" (PDF), Ann. Phys., 309 (3): 553–63, Bibcode:1901AnP...309..553P, doi:10.1002/andp.19013090310. English translation: "On the Law of Distribution of Energy in the Normal Spectrum Archived 2008-04-18 at the Wayback Machine"."Archived copy" (PDF). Archived from the original (PDF) on 2011-10-06. Retrieved 2011-10-13.CS1 maint: archived copy as title (link)
  6. ^ a b Bitter, Francis; Medicus, Heinrich A. (1973). Fields and particles. New York: Elsevier. pp. 137–144.
  7. ^ Planck, M. (1914). The Theory of Heat Radiation. Masius, M. (transl.) (2nd ed.). P. Blakiston's Son. pp. 6, 168. OL 7154661M.
  8. ^ Chandrasekhar, S. (1960) [1950]. Radiative Transfer (Revised reprint ed.). Dover. p. 8. ISBN 978-0-486-60590-6.
  9. ^ Rybicki, G. B.; Lightman, A. P. (1979). Radiative Processes in Astrophysics. Wiley. p. 22. ISBN 978-0-471-82759-7.
  10. ^ Shao, Gaofeng; et al. (2019). "Improved oxidation resistance of high emissivity coatings on fibrous ceramic for reusable space systems". Corrosion Science. 146: 233–246. arXiv:1902.03943. doi:10.1016/j.corsci.2018.11.006.
  12. ^ Kragh, Helge (1999), Quantum Generations: A History of Physics in the Twentieth Century, Princeton University Press, p. 62, ISBN 978-0-691-09552-3
  14. ^ Previous Solvay Conferences on Physics, International Solvay Institutes, archived from the original on 16 December 2008, retrieved 12 December 2008
  19. ^ Isaacson, Walter (2007-04-10), Einstein: His Life and Universe, ISBN 978-1-4165-3932-2, pp. 309–314.
  22. ^ Bohr, Niels (1913), "On the Constitution of Atoms and Molecules", Phil. Mag., 6th Series, 26 (153): 1–25, Bibcode:1913PMag...26..476B, doi:10.1080/14786441308634993
  23. ^ Mohr, J. C.; Phillips, W. D. (2015). "Dimensionless Units in the SI". Metrologia. 52 (1): 40–47. arXiv:1409.2794. Bibcode:2015Metro..52...40M. doi:10.1088/0026-1394/52/1/40.
  24. ^ Mills, I. M. (2016). "On the units radian and cycle for the quantity plane angle". Metrologia. 53 (3): 991–997. Bibcode:2016Metro..53..991M. doi:10.1088/0026-1394/53/3/991.
  25. ^ "SI units need reform to avoid confusion". Editorial. Nature. 548 (7666): 135. 7 August 2011. doi:10.1038/548135b. PMID 28796224.
  26. ^ P. R. Bunker; I. M. Mills; Per Jensen (2019). "The Planck constant and its units". J Quant Spectrosc Radiat Transfer. 237: 106594. doi:10.1016/j.jqsrt.2019.106594.
  27. ^ P. R. Bunker; Per Jensen (2020). "The Planck constant of action  A". J Quant Spectrosc Radiat Transfer. 243: 106835. doi:10.1016/j.jqsrt.2020.106835.
  28. ^ Maxwell J.C. (1873) A Treatise on Electricity and Magnetism, Oxford University Press
  30. ^ Einstein, Albert (2003), "Physics and Reality" (PDF), Daedalus, 132 (4): 24, doi:10.1162/001152603771338742, archived from the original (PDF) on 2012-04-15, The question is first: How can one assign a discrete succession of energy value Hσ to a system specified in the sense of classical mechanics (the energy function is a given function of the coordinates qr and the corresponding momenta pr)? The Planck constant h relates the frequency Hσ/h to the energy values Hσ. It is therefore sufficient to give to the system a succession of discrete frequency values.
  31. ^ 9th edition, SI BROCHURE. "BIPM" (PDF). BIPM.
  32. ^ Materese, Robin (2018-05-14). "Kilogram: The Kibble Balance". NIST. Retrieved 2018-11-13.


External linksEdit