Planck constant

The Planck constant (denoted h, also called Planck's constant) is a physical constant that is the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant. The Planck constant is of fundamental importance in quantum mechanics, and in metrology it is the basis for the definition of the kilogram.

At the end of the 19th century, physicists were unable to explain why the observed spectrum of black body radiation, which by then had been accurately measured, diverged significantly at higher frequencies from that predicted by existing theories. In 1900, Max Planck empirically derived a formula for the observed spectrum. He assumed that a hypothetical electrically charged oscillator in a cavity that contained black body radiation could only change its energy in a minimal increment, E, that was proportional to the frequency of its associated electromagnetic wave.[7] He was able to calculate the proportionality constant, h, from the experimental measurements, and that constant is named in his honor. In 1905, the value E was associated by Albert Einstein with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle, as opposed to an electromagnetic wave. It was eventually called a photon. Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

Since energy and mass are equivalent, the Planck constant also relates mass to frequency. By 2017, the Planck constant had been measured with sufficient accuracy in terms of the SI base units, that it was central to the project of replacing the International Prototype of the Kilogram, a metal cylinder that had defined the kilogram since 1889.[8] The new definition was unanimously approved at the General Conference on Weights and Measures (CGPM) on 16 November 2018 as part of the 2019 redefinition of SI base units.[9] For this new definition of the kilogram, the Planck constant, as defined by the ISO standard, was set to 6.62607015×10−34 J⋅s exactly. The kilogram was the last SI base unit to be re-defined by a fundamental physical property to replace a physical artifact.[10][11][12]

Value of h (2019) Units Ref.
6.62607015×10−34 Js [1][2][3][4]
Values of h (2014) Units Ref.
6.626070150(81)×10−34 Js [5]
4.135667662(25)×10−15 eVs [6]
2π EPtP
Values of ħ (h-bar) Units Ref.
1.054571800(13)×10−34 Js [6]
6.582119514(40)×10−16 eVs [6]
1 EPtP
Values of hc Units Ref.
1.98644568×10−25 Jm
1.23984193 eVμm
2π EPP
Values of ħc (h-bar) Units Ref.
3.16152649×10−26 Jm
0.19732697 eVμm
Max Planck Wirkungsquantums 20050815
Plaque at the Humboldt University of Berlin: "Max Planck, discoverer of the elementary quantum of action h, taught in this building from 1889 to 1928."

Origin of the definition

Wiens law
Intensity of light emitted from a black body at any given wavelength. Each curve represents behaviour at a different body temperature. Max Planck was the first to explain the shape of these curves.

In the last years of the 19th century, Max Planck was investigating the problem of black-body radiation first posed by Kirchhoff some 40 years earlier.

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

Approaching this problem, Planck hypothesized that the equations of motion for light describe a set of harmonic 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 black-body spectrum.[7] To create Planck's law, which correctly predicts blackbody emissions by fitting the observed curves, he multiplied the classical expression by a factor that involves a constant, h, in both the numerator and the denominator, which subsequently became known as the Planck Constant.

The spectral radiance of a body, Bν, describes the amount of energy it emits at different radiation frequencies. It is the power emitted per unit area of the body, per unit solid angle of emission, per unit frequency.

Planck showed that the spectral radiance of a body for frequency ν at absolute temperature T is given by

where kB is the Boltzmann constant, h is the Planck constant, and c is the speed of light in the medium, whether material or vacuum.[13][14][15]

The spectral radiance can also be expressed per unit wavelength λ 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.[16]

The law may also be expressed in other terms, such as the number of photons emitted at a certain wavelength, or the energy density in a volume of radiation. The SI units of Bν are W·sr−1·m−2·Hz−1, while those of Bλ 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.[7] To save his theory, Planck resorted to using the then-controversial theory of statistical mechanics,[7] which he described as "an act of despair … I was ready to sacrifice any of my previous convictions about physics."[17] One of his new boundary conditions was

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;

— Planck, On the Law of Distribution of Energy in the Normal Spectrum[7]

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,[18] but one which would revolutionize physics. Applying this new approach to Wien's displacement law showed that the "energy element" must be proportional to the frequency of the oscillator, the first version of what is now sometimes termed the "Planck–Einstein relation":

Planck was able to calculate the value of h from experimental data on black-body radiation: his result, 6.55×10−34 J⋅s, is within 1.2% of the currently accepted value.[7] He also made the first determination of the Boltzmann constant kB from the same data and theory.[19]

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

Development and application

The black-body problem was revisited in 1905, when Rayleigh and Jeans (on the one hand) and Einstein (on the other hand) independently proved that classical electromagnetism could never account for the observed spectrum. These proofs are commonly known as the "ultraviolet catastrophe", a name coined by Paul Ehrenfest in 1911. They contributed greatly (along with Einstein's work on the photoelectric effect) in convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The very first Solvay Conference in 1911 was devoted to "the theory of radiation and quanta".[20]

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,[21] who published the first thorough investigation in 1887. Another particularly thorough investigation was published by Philipp Lenard in 1902.[22] Einstein's 1905 paper[23] discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921,[21] when his predictions had been confirmed by the experimental work of Robert Andrews Millikan.[24] 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.[25][26]

Prior to Einstein's paper, electromagnetic radiation such as visible light was considered to behave as a wave: hence the use of the terms "frequency" and "wavelength" to characterise different types of radiation. The energy transferred by a wave in a given time is called its intensity. The light from a theatre spotlight is more intense than the light from a domestic lightbulb; that is to say that the spotlight gives out more energy per unit time and per unit space (and hence consumes more electricity) than the ordinary bulb, even though the colour of the light might be very similar. Other waves, such as sound or the waves crashing against a seafront, also have their own intensity. However, the energy account of the photoelectric effect didn't seem to agree with the wave description of light.

The "photoelectrons" emitted as a result of the photoelectric effect have a certain kinetic energy, which can be measured. This kinetic energy (for each photoelectron) is independent of the intensity of the light,[22] but depends linearly on the frequency;[24] 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).[27] 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.[22]

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).[24]

Atomic structure

Bohr atom model
A schematization of the Bohr model of the hydrogen atom. The transition shown from the n = 3 level to the n = 2 level gives rise to visible light of wavelength 656 nm (red), as the model predicts.

Niels Bohr introduced the first quantized model of the atom in 1913, in an attempt to overcome a major shortcoming of Rutherford's classical model.[28] 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 En

where c0 is the speed of light in vacuum, R is an experimentally determined constant (the Rydberg constant) and n is any integer (n = 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 in terms of other fundamental constants.

Bohr also introduced the quantity , now known as the reduced Planck constant, as the quantum of angular momentum. At first, Bohr thought that this was the angular momentum of each electron in an atom: this proved incorrect and, despite developments by Sommerfeld and others, an accurate description of the electron angular momentum proved beyond the Bohr model. The correct quantization rules for electrons – in which the energy reduces to the Bohr model equation in the case of the hydrogen atom – were given by Heisenberg's matrix mechanics in 1925 and the Schrödinger wave equation in 1926: the reduced Planck constant remains the fundamental quantum of angular momentum. In modern terms, if J is the total angular momentum of a system with rotational invariance, and Jz the angular momentum measured along any given direction, these quantities can only take on the values

Uncertainty principle

The Planck constant also occurs in statements of Werner Heisenberg's uncertainty principle. Given a large number of particles prepared in the same state, the uncertainty in their position, Δx, and the uncertainty in their momentum (in the same direction), Δp, obey

where the uncertainty is given as the standard deviation of the measured value from its expected value. There are a number of other such pairs of physically measurable values which obey a similar rule. One example is time vs. energy. The either-or nature of uncertainty forces measurement attempts to choose between trade offs, and given that they are quanta, the trade offs often take the form of either-or (as in Fourier analysis), rather than the compromises and gray areas of time series analysis.

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 and the momentum operator :

where δij is the Kronecker delta.

Photon energy

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

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

Since the frequency f, wavelength λ, and speed of light c are related by , the relation can also be expressed as

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.

In applications where it is natural to use the angular frequency (i.e. where the frequency is expressed in terms of radians per second instead of cycles per second or hertz) it is often useful to absorb a factor of 2π into the Planck constant. The resulting constant is called the reduced Planck constant. It is equal to the Planck constant divided by 2π, and is denoted ħ (pronounced "h-bar"):

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 afterwards. This holds throughout quantum theory, including electrodynamics.

Problems can arise when dealing with frequency or the Planck constant because the units of angular measure (cycle or radian) are omitted in SI.[29][30][31] In the language of quantity calculus,[32] 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 component parts of the special relativistic expression using 4-vectors.

Classical statistical mechanics requires the existence of h (but does not define its value).[33] 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 a classical particle motion.

In many cases, such as for monochromatic light or for atoms, quantization of energy also implies that only certain energy levels are allowed, and values in between are forbidden.[34]


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)

On 16 November 2018, the International Bureau of Weights and Measures (BIPM) voted to redefine the kilogram by fixing the value of the Planck constant, thereby defining the kilogram in terms of the second and the speed of light. Starting 20 May 2019, the new value is exactly


In July 2017, the NIST measured the Planck constant using its Kibble balance instrument with an uncertainty of only 13 parts per billion, obtaining a value of 6.626069934(89)×10−34 J⋅s.[35] This measurement, along with others, allowed the redefinition of SI base units.[36] The two digits inside the parentheses denote the standard uncertainty in the last two digits of the value.

As of the 2014 CODATA release, the best measured value of the Planck constant was:[6]


The value of the reduced Planck constant (or Dirac constant) was:


The 2014 CODATA results were made available in June 2015[37] and represent the best-known, internationally accepted values for these constants, based on all data published as of 31 December 2014. New CODATA figures are normally produced every four years. However, in order to support the redefinition of the SI base units, CODATA made a special release that was published in October 2017.[38] It incorporates all data up to 1 July 2017 and determines the final numerical values of the Planck constant, h, Elementary charge, e, Boltzmann constant, k, and Avogadro constant, NA, that are to be used for the new SI definitions.

Significance of the value

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

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

Equivalently, the order of the Planck constant reflects the fact that everyday objects and systems are made of a large number of microscopic particles. For example, green light with a wavelength of 555 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, NA6.022140758(62)×1023 mol−1, with the result of 216 kJ/mol, about the food energy in three apples.

Dependent physical constants

There are several related constants for which more than 99% of the uncertainty in the 2014 CODATA values[39] is due to the uncertainty in the value of the Planck constant, as indicated by the square of the correlation coefficient (r2 > 0.99, r > 0.995). The Planck constant is (with one or two exceptions)[40] the fundamental physical constant which is known to the lowest level of precision, with a 1σ relative uncertainty ur of 1.2×10−8.

Rest mass of the electron

The normal textbook derivation of the Rydberg constant R defines it in terms of the electron mass me and a variety of other physical constants.


However, the Rydberg constant can be determined very accurately (5.9×10−12 from the atomic spectrum of hydrogen, whereas there is no direct method to measure the mass of a stationary electron in SI units. Hence the equation for the computation of becomes

where is the speed of light and is the fine-structure constant. The speed of light has an exactly defined value in SI units, and the fine-structure constant can be determined more accurately ( = 2.3×10−10) than the Planck constant. Thus, the uncertainty in the value of the electron rest mass is due entirely to the uncertainty in the value of the Planck constant .

Avogadro constant

The Avogadro constant NA is determined as the ratio of the mass of one mole of electrons to the mass of a single electron; the mass of one mole of electrons is the "relative atomic mass" of an electron , which can be measured in a Penning trap ( 2.9×10−11), multiplied by the molar mass constant Mu, which is defined as 0.001 M.

The dependence of the Avogadro constant on the Planck constant () also holds for the physical constants which are related to amount of substance, such as the atomic mass constant. The uncertainty in the value of the Planck constant limits the knowledge of the masses of atoms and subatomic particles when expressed in SI units. It is possible to measure the masses more precisely in atomic mass units, but not to convert them more precisely into kilograms.

Elementary charge

Sommerfeld originally defined the fine-structure constant α as:

where e is the elementary charge, ε0 is the electric constant (also called the permittivity of free space), and μ0 is the magnetic constant (also called the permeability of free space). The latter two constants have fixed values in the International System of Units. However, α can also be determined experimentally, notably by measuring the electron spin g-factor , then comparing the result with the value predicted by quantum electrodynamics.

At present, the most precise value for the elementary charge is obtained by rearranging the definition of to obtain the following definition of in terms of and :

Bohr magneton and nuclear magneton

The Bohr magneton and the nuclear magneton are units which are used to describe the magnetic properties of the electron and atomic nuclei respectively. The Bohr magneton is the magnetic moment which would be expected for an electron if it behaved as a spinning charge according to classical electrodynamics. It is defined in terms of the reduced Planck constant, the elementary charge and the electron mass, all of which depend on the Planck constant: the final dependence on h1/2 (r2 > 0.995) can be found by expanding the variables.

The nuclear magneton has a similar definition, but corrected for the fact that the proton is much more massive than the electron. The ratio of the electron relative atomic mass to the proton relative atomic mass can be determined experimentally to a high level of precision (ur = 9.5×10−11).


Method Value of h
(10−34 J⋅s)
Kibble (watt) balance 6.62606889(23) 3.4×10−8 [41][42][43]
X-ray crystal density 6.6260745(19) 2.9×10−7 [44]
Josephson constant 6.6260678(27) 4.1×10−7 [45][46]
Magnetic resonance 6.6260724(57) 8.6×10−7 [47][48]
Faraday constant 6.6260657(88) 1.3×10−6 [49]
CODATA 2010 6.62606957(29) 4.4×10−8 [50]
Kibble balance with superconducting magnet 6.62606979(30) 4.5×108 [51]
The nine recent determinations of the Planck constant cover five separate methods. Where there is more than one recent determination for a given method, the value of h given here is a weighted mean of the results, as calculated by CODATA.

In principle, the Planck constant could be determined by examining the spectrum of a black-body radiator or the kinetic energy of photoelectrons, and this is how its value was first calculated in the early twentieth century. In practice, these are no longer the most accurate methods. The CODATA value quoted here is based on three Kibble balance measurements of KJ2RK and one inter-laboratory determination of the molar volume of silicon,[52] but is mostly determined by a 2007 Kibble balance measurement made at the U.S. National Institute of Standards and Technology (NIST).[43] Five other measurements by three different methods were initially considered, but not included in the final refinement as they were too imprecise to affect the result.

There are both practical and theoretical difficulties in determining h. The practical difficulties can be illustrated by the fact that the two most accurate methods, the Kibble balance and the X-ray crystal density method, do not appear to agree with one another. The most likely reason is that the measurement uncertainty for one (or both) of the methods has been estimated too low – it is (or they are) not as precise as is currently believed – but for the time being there is no indication which method is at fault.

The theoretical difficulties arise from the fact that all of the methods except the X-ray crystal density method rely on the theoretical basis of the Josephson effect and the quantum Hall effect. If these theories are slightly inaccurate – though there is no evidence at present to suggest they are – the methods would not give accurate values for the Planck constant. More importantly, the values of the Planck constant obtained in this way cannot be used as tests of the theories without falling into a circular argument. There are other statistical ways of testing the theories, and the theories have yet to be refuted.[52]

Josephson constant

The Josephson constant KJ relates the potential difference U generated by the Josephson effect at a "Josephson junction" with the frequency ν of the microwave radiation. The theoretical treatment of Josephson effect suggests very strongly that KJ = 2e/h.

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. Assuming the validity of the theoretical treatment of the Josephson effect, KJ is related to the Planck constant by

Kibble balance

A Kibble balance (formerly known as a watt balance)[53] is an instrument for comparing two powers, one of which is measured in SI watts and the other of which is measured in conventional electrical units. From the definition of the conventional watt W90, this gives a measure of the product KJ2RK in SI units, where RK is the von Klitzing constant which appears in the quantum Hall effect. If the theoretical treatments of the Josephson effect and the quantum Hall effect are valid, and in particular assuming that RK = h/e2, the measurement of KJ2RK is a direct determination of the Planck constant.

Magnetic resonance

The gyromagnetic ratio γ is the constant of proportionality between the frequency ν of nuclear magnetic resonance (or electron paramagnetic resonance for electrons) and the applied magnetic field B: ν = γB. It is difficult to measure gyromagnetic ratios precisely because of the difficulties in precisely measuring B, but the value for protons in water at 25 °C is known to better than one part per million. The protons are said to be "shielded" from the applied magnetic field by the electrons in the water molecule, the same effect that gives rise to chemical shift in NMR spectroscopy, and this is indicated by a prime on the symbol for the gyromagnetic ratio, γp. The gyromagnetic ratio is related to the shielded proton magnetic moment μp, the spin number I (I = ​12 for protons) and the reduced Planck constant.

The ratio of the shielded proton magnetic moment μp to the electron magnetic moment μe can be measured separately and to high precision, as the imprecisely known value of the applied magnetic field cancels itself out in taking the ratio. The value of μe in Bohr magnetons is also known: it is half the electron g-factor ge. Hence

A further complication is that the measurement of γp involves the measurement of an electric current: this is invariably measured in conventional amperes rather than in SI amperes, so a conversion factor is required. The symbol Γp-90 is used for the measured gyromagnetic ratio using conventional electrical units. In addition, there are two methods of measuring the value, a "low-field" method and a "high-field" method, and the conversion factors are different in the two cases. Only the high-field value Γp-90(hi) is of interest in determining the Planck constant.

Substitution gives the expression for the Planck constant in terms of Γp-90(hi):

Faraday constant

The Faraday constant F is the charge of one mole of electrons, equal to the Avogadro constant NA multiplied by the elementary charge e. It can be determined by careful electrolysis experiments, measuring the amount of silver dissolved from an electrode in a given time and for a given electric current. In practice, it is measured in conventional electrical units, and so given the symbol F90. Substituting the definitions of NA and e, and converting from conventional electrical units to SI units, gives the relation to the Planck constant.

X-ray crystal density

The X-ray crystal density method is primarily a method for determining the Avogadro constant NA but as the Avogadro constant is related to the Planck constant it also determines a value for h. The principle behind the method is to determine NA as the ratio between the volume of the unit cell of a crystal, measured by X-ray crystallography, and the molar volume of the substance. Crystals of silicon are used, as they are available in high quality and purity by the technology developed for the semiconductor industry. The unit cell volume is calculated from the spacing between two crystal planes referred to as d220. The molar volume Vm(Si) requires a knowledge of the density of the crystal and the atomic weight of the silicon used. The Planck constant is given by

Particle accelerator

The experimental measurement of the Planck constant in the Large Hadron Collider laboratory was carried out in 2011. The study called PCC using a giant particle accelerator helped to better understand the relationships between the Planck constant and measuring distances in space.


As mentioned above, the numerical value of the Planck constant depends on the system of units used to describe it. Its value in SI units is known to 12 parts per billion but its value in atomic units is known exactly, because of the way the scale of atomic units is defined. The same is true of conventional electrical units, where the Planck constant (denoted h90 to distinguish it from its value in SI units) is given by

with KJ–90 and RK–90 being exactly defined constants. Atomic units and conventional electrical units are very useful in their respective fields, because the uncertainty in the final result does not depend on an uncertain conversion factor, only on the uncertainty of the measurement itself.

It is currently planned to redefine certain SI base units in terms of fundamental physical constants.[8] This has already been done for the metre, which since 1983 has been defined in terms of a fixed value of the speed of light. The most urgent unit on the list for redefinition is the kilogram, whose value has been fixed for all science (since 1889) by the mass of a small cylinder of platinumiridium alloy kept in a vault just outside Paris. While nobody knows if the mass of the International Prototype Kilogram has changed since 1889 – the value 1 kg of its mass expressed in kilograms is by definition unchanged and therein lies one of the problems – it is known that over such a timescale the many similar Pt–Ir alloy cylinders kept in national laboratories around the world have changed their relative masses by several tens of parts per billion, however carefully they are stored. A change of several tens of micrograms in one kilogram is equivalent to the current uncertainty in the value of the Planck constant in SI units.

The legal process to change the definition of the kilogram to one based on a fixed value of the Planck constant is already underway.[54] The 24th and 25th General Conferences on Weights and Measures (CGPM) in 2011 and 2014 approved of the redefinition in principle, but were not satisfied with the measurement uncertainty of the Planck constant. The limits they specified were reached in 2016,[8] and the redefinition is scheduled to occur on 16 November 2018, during the 26th CGPM.[55]

Kibble balances already measure mass in terms of the Planck constant: at present, standard kilogram prototypes are taken as fixed masses and the measurement is performed to determine the Planck constant but, once the Planck constant is fixed in SI units, the same experiment would be a measurement of the mass. The relative uncertainty in the measurement would remain the same.

Mass standards could also be constructed from silicon crystals or by other atom-counting methods. Such methods require a knowledge of the Avogadro constant, which fixes the proportionality between atomic mass and macroscopic mass but, with a defined value of the Planck constant, NA would be known to the same level of uncertainty (if not better) than current methods of comparing macroscopic mass.

See also


  1. ^ Set on 20 November 2018, by the CGPM to this exact value, with this value to take effect 20 May 2019, as explained in the other references.
  2. ^ "Resolutions of the 26th CGPM" (PDF). BIPM. 2018-11-16. Retrieved 2018-11-20.
  3. ^ Ghosh, Pallab (2018-11-16). "Kilogram gets a new definition". BBC News. Retrieved 2018-11-16.
  4. ^ Veritasium (2018-11-15), The kg is dead, long live the kg, retrieved 2018-11-16
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  • Barrow, John D. (2002), The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe, Pantheon Books, ISBN 978-0-375-42221-8

External links


Atomic mass constant

In physics and chemistry, the atomic mass constant, mu, is one twelfth of the mass of an unbound atom of carbon-12 at rest and in its ground state. It serves to define the atomic mass unit and is, by definition, equal to 1 u. It is inverse of Avogadro constant (1/NA) when expressed in grams (instead of SI unit kilogram). The CODATA recommended value is 1.66053906660(50)×10−27 kg.

In practice, the atomic mass constant is determined from the electron rest mass me and the electron relative atomic mass Ar(e) (that is, the mass of the electron on a scale where 12C = 12). The relative atomic mass of the electron can be measured in cyclotron experiments, while the rest mass of the electron can be derived from other physical constants.

where c is the speed of light, h is the Planck constant, α is the fine-structure constant, and R is the Rydberg constant.

The current (CODATA 2014) uncertainty in the value of the atomic mass constant – relative uncertainty 1.2×10−8 – is almost entirely due to the uncertainty in the value of the Planck constant in SI units. With the 2019 redefinition of SI base units, the relative uncertainty will improve to 4.7×10−10, which will be almost entirely due to the uncertainty in the fine-structure constant.

Bohr magneton

In atomic physics, the Bohr magneton (symbol μB) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by either its orbital or spin angular momentum.

The Bohr magneton is defined in SI units by

and in Gaussian CGS units by


e is the elementary charge,
ħ is the reduced Planck constant,
me is the electron rest mass and
c is the speed of light.

The electron magnetic moment, which is the electron's intrinsic spin magnetic moment, is approximately one Bohr magneton.

David Cho (director)

Cho Sung-kyu (born March 3, 1969), also known as David Cho, is a South Korean film producer, executive producer, director and screenwriter. Apart from producing and investing in numerous films as CEO of Sponge Entertainment, Cho wrote and directed Second Half (2010), The Heaven Is Only Open to the Single! (2012), The Winter of the Year Was Warm (2012), Santa Barbara (2014), and Planck Constant (2015).

H with stroke

Ħ (minuscule: ħ) is a letter of the Latin alphabet, derived from H with the addition of a bar. It is used in Maltese and in Tunisian Arabic transliteration (based on Maltese with additional letters) for a voiceless pharyngeal fricative consonant (corresponding to the letter heth of Semitic abjads). Lowercase ħ is used in the International Phonetic Alphabet for the same sound.

In quantum mechanics, an italic ℏ (U+210F) with a line, represents the reduced Planck constant. In this context, it is pronounced "h-bar".

The lowercase resembles the Cyrillic letter Tshe (ћ), or the astronomical symbol of Saturn (♄).

A white uppercase Ħ on a red square is the logo of Heritage Malta.


The hartree (symbol: Eh or Ha), also known as the Hartree energy, is the atomic unit of energy, named after the British physicist Douglas Hartree. It is defined as

2R∞hc, where R∞ is the Rydberg constant, h is the Planck constant and c is the speed of light.

The 2014 CODATA recommended value is Eh = 4.359 744 650(54)×10−18 J = 27.211 386 02(17) eV.The hartree energy is approximately the electric potential energy of the hydrogen atom in its ground state and, by the virial theorem, approximately twice its ionization energy; the relationships are not exact because of the finite mass of the nucleus of the hydrogen atom and relativistic corrections.

The hartree is usually used as a unit of energy in atomic physics and computational chemistry: for experimental measurements at the atomic scale, the electronvolt (eV) or the reciprocal centimetre (cm−1) are much more widely used.

Hydrogen line

The hydrogen line, 21-centimeter line or H I line refers to the electromagnetic radiation spectral line that is created by a change in the energy state of neutral hydrogen atoms. This electromagnetic radiation is at the precise frequency of 1420405751.7667±0.0009 Hz, which is equivalent to the vacuum wavelength of 21.1061140542 cm in free space. This wavelength falls within the microwave region of the electromagnetic spectrum, and it is observed frequently in radio astronomy, since those radio waves can penetrate the large clouds of interstellar cosmic dust that are opaque to visible light.

The microwaves of the hydrogen line come from the atomic transition of an electron between the two hyperfine levels of the hydrogen 1s ground state that have an energy difference of ≈ 5.87433 µeV. It is called the spin-flip transition. The frequency, ν, of the quanta that are emitted by this transition between two different energy levels is given by the Planck–Einstein relation E = hν. According to that relation, the photon energy of a 1,420,405,751.7667 Hz photon is ≈ 5.87433 µeV. The constant of proportionality, h, is known as the Planck constant.


The kilogram, also kilogramme and kilo, is the base unit of mass in the International System of Units (SI), having the unit symbol kg. It is defined in terms of the Planck constant.The kilogram was originally defined in 1795 as the mass of a litre (cubic decimetre) of water. This was a convenient definition, but hard to replicate precisely. In 1799, a platinum artefact replaced it as a standard mass sample. Later, the International Prototype Kilogram (IPK) remained the standard of the unit of mass for the metric system until 20 May 2019. In spite of best efforts to maintain it, the IPK diverged from its replicas by approximately 50 micrograms since their manufacture late in the 19th century. This led to efforts to develop measurement technology precise enough to allow replacing the kilogram artefact with a definition based directly on physical fundamental constants, which was adopted in 2019.The new definition is based on invariant constants of nature, in particular the Planck constant, which was changed to being defined rather than measured, thereby fixing the value of the kilogram in terms of the second and the metre, and eliminating the need for the IPK. The new definition was approved by the General Conference on Weights and Measures (CGPM) on 16 November 2018. The Planck constant relates a light particle's energy, and hence mass, to its frequency. The new definition only became possible when instruments were devised to measure the Planck constant with sufficient accuracy based on the IPK definition of the kilogram.

List of scientific constants named after people

This is a list of physical and mathematical constants named after people.Eponymous constants and their influence on scientific citations have been discussed in the literature.

  • Reduced Planck constant or Dirac constant (-bar, ħ) – Max Planck, Paul Dirac
Matter wave

Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter can exhibit wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave. The concept that matter behaves like a wave was proposed by Louis de Broglie (/dəˈbrɔɪ/) in 1924. It is also referred to as the de Broglie hypothesis. Matter waves are referred to as de Broglie waves.

The de Broglie wavelength is the wavelength, λ, associated with a massive particle and is related to its momentum, p, through the Planck constant, h:

Wave-like behavior of matter was first experimentally demonstrated by George Paget Thomson's thin metal diffraction experiment, and independently in the Davisson–Germer experiment both using electrons, and it has also been confirmed for other elementary particles, neutral atoms and even molecules.

Nuclear magneton

The nuclear magneton (symbol μN), is a physical constant of magnetic moment, defined in SI units by:

and in Gaussian CGS units by:


e is the elementary charge,
ħ is the reduced Planck constant,
mp is the proton rest mass, and
c is the speed of light

In SI units, its value is approximately:

μN = 5.050783699(31)×10−27 J/T

In Gaussian CGS units, its value can be given in convenient units as

μN = 0.10515446 e⋅fm

The nuclear magneton is the natural unit for expressing magnetic dipole moments of heavy particles such as nucleons and atomic nuclei.

Due to the fact that neutrons and protons consist of quarks and thus are no real Dirac particles, their magnetic moment differ from :

The magnetic dipole moment of the electron, which is much larger as a consequence of much larger charge-to-mass ratio, is usually expressed in units of the Bohr magneton. The Bohr magneton, which is calculated in the same fashion as the nuclear magneton, is larger than μN by a factor equal to the ratio of the proton to electron mass, or about a factor of 1836.

Physical constant

A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement.

There are many physical constants in science, some of the most widely recognized being the speed of light in vacuum c, the gravitational constant G, the Planck constant h, the electric constant ε0, and the elementary charge e. Physical constants can take many dimensional forms: the speed of light signifies a maximum speed for any object and its dimension is length divided by time; while the fine-structure constant α, which characterizes the strength of the electromagnetic interaction, is dimensionless.

The term fundamental physical constant is sometimes used to refer to universal but dimensioned physical constants such as those mentioned above. Increasingly, however, physicists reserve the use of the term fundamental physical constant for dimensionless physical constants, such as the fine-structure constant α.

Physical constant in the sense under discussion in this article should not be confused with other quantities called "constants" that are assumed to be constant in a given context without the implication that they are fundamental, such as the "time constant" characteristic of a given system, or material constants, such as the Madelung constant, electrical resistivity, and heat capacity.

The International Bureau of Weights and Measures decided to redefine several SI base units as from 20 May 2019 by fixing the SI value of several physical constants, including the Planck constant, h, the elementary charge, e, the Boltzmann constant, kB, and the Avogadro constant, NA. The new fixed values are based on the best measurements of the constants based on the earlier definitions, including the kilogram, to ensure minimal impact. As a consequence, the uncertainty in the value of many physical constants when expressed in SI units are substantially reduced.

Planck length

In physics, the Planck length, denoted ℓP, is a unit of length that is the distance light travels in one unit of Planck time. It is equal to 1.616255(18)×10−35 m. It is a base unit in the system of Planck units, developed by physicist Max Planck. The Planck length can be defined from three fundamental physical constants: the speed of light in a vacuum, the Planck constant, and the gravitational constant.

Planck mass

In physics, the Planck mass, denoted by mP, is the unit of mass in the system of natural units known as Planck units. It is approximately 0.02 milligrams. Unlike some other Planck units, such as Planck length, Planck mass is not a fundamental lower or upper bound; instead, Planck mass is a unit of mass defined using only what Max Planck considered fundamental and universal units. One Planck mass is roughly the mass of a flea egg. For comparison, this value is of the order of 1015 (a quadrillion) times larger than the highest energy available to contemporary particle accelerators.

It is defined as:

where c is the speed of light in a vacuum, G is the gravitational constant, and ħ is the reduced Planck constant.

Substituting values for the various components in this definition gives the approximate equivalent value of this unit in terms of other units of mass:

For the Planck mass , the Schwarzschild radius () and the Compton wavelength () are of the same order as the Planck length .

Particle physicists and cosmologists often use an alternative normalization with the reduced Planck mass, which is

The factor of simplifies a number of equations in general relativity.

Planck time

In quantum mechanics, the Planck time (tP) is the unit of time in the system of natural units known as Planck units. A Planck time unit is the time required for light to travel a distance of 1 Planck length in a vacuum, which is a time interval of approximately 5.39 × 10 −44 s. The unit is named after Max Planck, who was the first to propose it.

The Planck time is defined as:


ħ = ​h2π is the reduced Planck constant (sometimes h is used instead of ħ in the definition)
G = gravitational constant
c = speed of light in vacuum

Using the known values of the constants, the approximate equivalent value in terms of the SI unit, the second, is

where the two digits between parentheses denote the standard error of the approximated value.

Planck–Einstein relation

The Planck–Einstein relation is also referred to as the Einstein relation, Planck's energy–frequency relation, the Planck relation, and the Planck equation. Also the eponym Planck formula belongs on this list, but also often refers to Planck's law instead. These various eponyms are far from standard; they are used only sporadically, neither regularly nor very widely. They refer to a formula integral to quantum mechanics, which states that the energy of a photon, E, known as photon energy, is proportional to its frequency, ν:

The constant of proportionality, h, is known as the Planck constant. Several equivalent forms of the relation exist, including in terms of angular frequency, ω:

The relation accounts for quantized nature of light, and plays a key role in understanding phenomena such as the photoelectric effect, and Planck's law of black-body radiation. See also the Planck postulate.

Quantum calculus

Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus", where h ostensibly stands for Planck's constant while q stands for quantum. The two parameters are related by the formula

where is the reduced Planck constant.

Resonance (particle physics)

In particle physics, a resonance is the peak located around a certain energy found in differential cross sections of scattering experiments. These peaks are associated with subatomic particles, which include a variety of bosons, quarks and hadrons (such as nucleons, delta baryons or upsilon mesons) and their excitations. In common usage, "resonance" only describes particles with very short lifetimes, mostly high-energy hadrons existing for 10−23 seconds or less.

The width of the resonance (Γ) is related to the mean lifetime (τ) of the particle (or its excited state) by the relation

where h is the Planck constant and .

Thus, the lifetime of a particle is the direct inverse of the particle's resonance width. For example, the charged pion has the second-longest lifetime of any meson, at 2.6033×10−8 s. Therefore, its resonance width is very small, about 2.528×10−8 eV or about 6.11 MHz. Pions are generally not considered as "resonances". The charged rho meson has a very short lifetime, about 4.41×10−24 s. Correspondingly, its resonance width is very large, at 149.1 MeV or about 36 ZHz. This amounts to nearly one-fifth of the particle's rest mass.

Rotational temperature

The characteristic rotational temperature (θR or θrot) is commonly used in statistical thermodynamics to simplify the expression of the rotational partition function and the rotational contribution to molecular thermodynamic properties. It has units of temperature and is defined as

where is the rotational constant, I is a molecular moment of inertia, h is the Planck constant, c is the speed of light, ħ = h/2π is the reduced Planck constant and kB is the Boltzmann constant.

The physical meaning of θR is as an estimate of the temperature at which thermal energy (of the order of kBT) is comparable to the spacing between rotational energy levels (of the order of hcB). At about this temperature the population of excited rotational levels becomes important. Some typical values are given in the table. In each case the value refers to the most common isotopic species.

Stefan–Boltzmann constant

The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter σ (sigma), is the constant of proportionality in the Stefan–Boltzmann law: "the total intensity radiated over all wavelengths increases as the temperature increases", of a black body which is proportional to the fourth power of the thermodynamic temperature. The theory of thermal radiation lays down the theory of quantum mechanics, by using physics to relate to molecular, atomic and sub-atomic levels. Slovenian physicist Josef Stefan formulated the constant in 1879, and it was later derived in 1884 by Austrian physicist Ludwig Boltzmann. The equation can also be derived from Planck's law, by integrating over all wavelengths at a given temperature, which will represent a small flat black body box. "The amount of thermal radiation emitted increases rapidly and the principal frequency of the radiation becomes higher with increasing temperatures". The Stefan–Boltzmann constant can be used to measure the amount of heat that is emitted by a blackbody, which absorbs all of the radiant energy that hits it, and will emit all the radiant energy. Furthermore, the Stefan–Boltzmann constant allows for temperature (K) to be converted to units for intensity (W⋅m−2), which is power per unit area.

The value of the Stefan–Boltzmann constant is given in SI units by

σ = 5.670374419...×10−8 W⋅m−2⋅K−4.

In cgs units the Stefan–Boltzmann constant is:

σ5.6704×10−5 erg⋅cm−2⋅s−1⋅K−4.

In thermochemistry the Stefan–Boltzmann constant is often expressed in cal⋅cm−2⋅day−1⋅K−4:

σ11.7×10−8 cal cm−2⋅day−1⋅K−4.

In US customary units the Stefan–Boltzmann constant is:

σ1.714×10−9 BTU⋅hr−1⋅ft−2⋅°R−4.

The value of the Stefan–Boltzmann constant is derivable as well as experimentally determinable; see Stefan–Boltzmann law for details. It can be defined in terms of the Boltzmann constant as


The CODATA recommended value is calculated from the measured value of the gas constant:


Dimensional formula: M1T−3Θ−4

A related constant is the radiation constant (or radiation density constant) a which is given by:

Base Planck units
Derived Planck units
Physical constants

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