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The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. The constant gives the relationship between the energy of a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
and its
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, and by the mass-energy equivalence, the relationship between mass and frequency. Specifically, a photon's energy is equal to its frequency multiplied by the Planck constant. The constant is generally denoted by $h$. The reduced Planck constant, or Dirac constant, equal to the constant divided by $2 \pi$, is denoted by $\hbar$. In metrology it is used, together with other constants, to define the kilogram, the SI unit of mass. The SI units are defined in such a way that, when the Planck constant is expressed in SI units, it has the exact value The constant was first postulated by Max Planck in 1900 as part of a solution to the ultraviolet catastrophe. At the end of the 19th century, accurate measurements of the spectrum of black body radiation existed, but the distribution of those measurements at higher frequencies diverged significantly from what was predicted by then-existing theories. 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 can only change its
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
in quantized steps, and that the energies of those steps are proportional to the frequency of the oscillator's associated
electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ...
. He was able to calculate the proportionality constant from experimental measurements, and that constant is named in his honor. In 1905,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
determined 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, and was eventually called a
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
. Max Planck received the 1918
Nobel Prize in Physics ) , image = Nobel Prize.png , alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
"in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

# Origin of the constant

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). 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 In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
. 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. Also around this time, but unknown to Planck,
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. A ...
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 Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
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,. English translation: ". which gave a simple empirical formula for long wavelengths. Planck tried to find a mathematical expression that could reproduce Wien's law (for short wavelengths) and the empirical formula (for long wavelengths). This expression included a constant, $h$, which is thought to be for Hilfsgrösse (auxiliary variable), and subsequently became known as the Planck constant. The expression formulated by Planck showed that the spectral radiance of a body for
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
at
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...
is given by :$B_\nu\left(\nu, T\right) = \frac \frac,$ where $k_\text$ is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, $h$ is the Planck constant, and $c$ is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
in the medium, whether material or vacuum. The spectral radiance of a body, $B_$, 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 In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
$\lambda$ instead of per unit frequency. In this case, it is given by :$B_\lambda\left(\lambda, T\right) =\frac\frac,$ showing how radiated energy emitted at shorter wavelengths increases more rapidly with temperature than energy emitted at longer wavelengths. Planck's law may also be expressed in other terms, such as the number of photons emitted at a certain wavelength, or the energy density in a volume of radiation. The SI units of $B_$ are , while those of $B_$ are . Planck soon realized that his solution was not unique. There were several different solutions, each of which gave a different value for the entropy of the oscillators. To save his theory, Planck resorted to using the then-controversial theory of statistical mechanics, which he described as "an act of despair … I was ready to sacrifice any of my previous convictions about physics." One of his new boundary conditions was 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, 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 The Planck relationFrench & Taylor (1978), pp. 24, 55.Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11. (referred to as Planck's energy–frequency relation,Schwinger (2001), p. 203. the Planck relation, Planck equation, and Planck formula, ...
": :$E = hf.$ Planck was able to calculate the value of $h$ from experimental data on black-body radiation: his result, , is within 1.2% of the currently accepted value. He also made the first determination of the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
$k_\text$ from the same data and theory.

# Development and application

The black-body problem was revisited in 1905, when
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. A ...
and James Jeans (on the one hand) and
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
(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 Paul Ehrenfest (18 January 1880 – 25 September 1933) was an Austrian theoretical physicist, who made major contributions to the field of statistical mechanics and its relations with quantum mechanics, including the theory of phase transition ...
in 1911. They contributed greatly (along with Einstein's work on the photoelectric effect) in convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The first Solvay Conference in 1911 was devoted to "the theory of radiation and quanta".

## 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 Alexandre-Edmond Becquerel (24 March 1820 – 11 May 1891), known as Edmond Becquerel, was a French physicist who studied the solar spectrum, magnetism, electricity and optics. He is credited with the discovery of the photovoltaic effect, the ...
in 1839, although credit is usually reserved for
Heinrich Hertz Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's equations of electromagnetism. The uni ...
,See, e.g., who published the first thorough investigation in 1887. Another particularly thorough investigation was published by Philipp Lenard (Lénárd Fülöp) in 1902. Einstein's 1905 paper discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921, after his predictions had been confirmed by the experimental work of
Robert Andrews Millikan Robert Andrews Millikan (March 22, 1868 – December 19, 1953) was an American experimental physicist honored with the Nobel Prize for Physics in 1923 for the measurement of the elementary electric charge and for his work on the photoelectric ...
. 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. 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 In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
, which can be measured. This kinetic energy (for each photoelectron) is ''independent'' of the intensity of the light, but depends linearly on the frequency; and if the frequency is too low (corresponding to a photon energy that is less than the
work function In solid-state physics, the work function (sometimes spelt workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately ...
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). Assuming the frequency is high enough to cause the photoelectric effect, a rise in intensity of the light source causes more photoelectrons to be emitted with the same kinetic energy, rather than the same number of photoelectrons to be emitted with higher kinetic energy. Einstein's explanation for these observations was that light itself is quantized; that the energy of light is not transferred continuously as in a classical wave, but only in small "packets" or quanta. The size of these "packets" of energy, which would later be named
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s, was to be the same as Planck's "energy element", giving the modern version of the Planck–Einstein relation: :$E = hf .$ Einstein's postulate was later proven experimentally: the constant of proportionality between the frequency of incident light $f$ and the kinetic energy of photoelectrons $E$ was shown to be equal to the Planck constant $h$.

## Atomic structure

It was John William Nicholson in 1912 who introduced h-bar into the theory of the atom which was the first quantum and nuclear atom and the first to quantize angular momentum as ''h''/2. Niels Bohr quoted him in his 1913 paper of the Bohr model of the atom. The influence of the work of Nicholson’s nuclear quantum atomic model on Bohr’s model has been written about by many historians. Niels Bohr introduced the third quantized model of the atom in 1913, in an attempt to overcome a major shortcoming of Rutherford's classical model. The first quantized model of the atom was introduced in 1910 by
Arthur Erich Haas Arthur Erich Haas (April 30, 1884 in Brno – February 20, 1941 in Chicago) was an Austrian physicist, noted for a 1910 paper he submitted in support of his habilitation as '' Privatdocent'' at the University of Vienna that outlined a treatm ...
and was discussed at the 1911 Solvay conference. In classical electrodynamics, a charge moving in a circle should radiate electromagnetic radiation. If that charge were to be an electron orbiting a
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: * Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
, the radiation would cause it to lose energy and spiral down into the nucleus. Bohr solved this paradox with explicit reference to Planck's work: an electron in a Bohr atom could only have certain defined energies $E_n$ :$E_n = -\frac ,$ where $c$ is the speed of light in vacuum, $R_$ is an experimentally determined constant (the Rydberg constant) and $n \in \$. Once the electron reached the lowest energy level ($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 $\hbar=\frac$, now known as the reduced Planck constant or Dirac constant, as the quantum of
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. At first, Bohr thought that this was the angular momentum of each electron in an atom: this proved incorrect and, despite developments by
Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
and others, an accurate description of the electron angular momentum proved beyond the Bohr model. The correct quantization rules for electrons – in which the energy reduces to the Bohr model equation in the case of the hydrogen atom – were given by Heisenberg's matrix mechanics in 1925 and the Schrödinger wave equation in 1926: the reduced Planck constant remains the fundamental quantum of angular momentum. In modern terms, if $J$ is the total angular momentum of a system with rotational invariance, and $J_z$ the angular momentum measured along any given direction, these quantities can only take on the values :$\begin J^2 = j\left(j+1\right) \hbar^2,\qquad & j = 0, \tfrac, 1, \tfrac, \ldots, \\ J_z = m \hbar, \qquad\qquad\quad & m = -j, -j+1, \ldots, j. \end$

## Uncertainty principle

The Planck constant also occurs in statements of Werner Heisenberg's uncertainty principle. Given numerous particles prepared in the same state, the
uncertainty Uncertainty refers to epistemic situations involving imperfect or unknown information. It applies to predictions of future events, to physical measurements that are already made, or to the unknown. Uncertainty arises in partially observable ...
in their position, $\Delta x$, and the uncertainty in their momentum, $\Delta p_$, obey :$\Delta x\, \Delta p_ \ge \frac ,$ where the uncertainty is given as the standard deviation of the measured value from its expected value. There are several other such pairs of physically measurable conjugate variables which obey a similar rule. One example is time vs. energy. The inverse relationship between the uncertainty of the two conjugate variables forces a tradeoff in quantum experiments, as measuring one quantity more precisely results in the other quantity becoming imprecise. In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental cornerstones to the entire theory lies in the commutator relationship between the position operator $\hat$ and the momentum operator $\hat$: : where $\delta_$ is the Kronecker delta.

# Photon energy

The Planck relation connects the particular photon energy with its associated wave frequency : :$E = hf.$ This energy is extremely small in terms of ordinarily perceived everyday objects. Since the frequency ,
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
, and
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
are related by $f= \frac$, the relation can also be expressed as :$E = \frac .$

## de Broglie wavelength

In 1923, Louis de Broglie generalized the Planck–Einstein relation by postulating that the Planck constant represents the proportionality between the momentum and the quantum wavelength of not just the photon, but the quantum wavelength of any particle. This was confirmed by experiments soon afterward. This holds throughout the quantum theory, including electrodynamics. The de Broglie wavelength of the particle is given by :$\lambda = \frac,$ where denotes the linear momentum of a particle, such as a photon, or any other
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions ( quarks, leptons, ...
. The energy of a photon with angular frequency is given by :$E = \hbar \omega ,$ while its linear momentum relates to :$p = \hbar k ,$ where is an angular wavenumber. These two relations are the temporal and spatial parts of the special relativistic expression using 4-vectors. :$P^\mu = \left\left(\frac, \vec\right\right) = \hbar K^\mu = \hbar\left\left(\frac, \vec\right\right).$

## Statistical mechanics

Classical statistical mechanics requires the existence of (but does not define its value). Eventually, following upon Planck's discovery, it was speculated that physical action could not take on an arbitrary value, but instead was restricted to integer multiples of a very small quantity, the " lementary quantum of action", now called the ''Planck constant''. This was a significant conceptual part of the so-called " old quantum theory" developed by physicists including
Bohr Niels Henrik David Bohr (; 7 October 1885 – 18 November 1962) was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. B ...
,
Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
, and Ishiwara, in which particle trajectories exist but are hidden, but quantum laws constrain them based on their action. This view has been replaced by fully modern quantum theory, in which definite trajectories of motion do not even exist; rather, the particle is represented by a wavefunction spread out in space and in time. 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. 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.

# Reduced Planck constant

Implicit in the dimensions of the Planck constant is the fact that the SI unit of frequency, the
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
, represents one complete cycle, 360 degrees or radians, per second. In applications where it is natural to use the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
(i.e. where the frequency is expressed in terms of
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s per second instead of cycles per second or
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
) it is often useful to absorb a factor of into the Planck constant. The resulting constant is called the reduced Planck constant or Dirac constant. It is equal to the Planck constant divided by , and is denoted by $\hbar$ (pronounced "h-bar"): :$\hbar = \frac .$

# Value

The Planck constant has dimensions of
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. In SI units, the Planck constant is expressed with the unit
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
per
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
(J⋅Hz) or joule-second (J⋅s). :$h = \mathrm$ :$\hbar= = 1.054\ 571\ 817...\times 10^\ \text\text = 6.582\ 119\ 569...\times 10^\ \text\text.$ The above values have been adopted as fixed in the 2019 redefinition of the SI base units.

## Understanding the 'fixing' of the value of ''h''

Since 2019, the numerical value of the Planck constant has been fixed, with a finite decimal representation. Under the present definition of the kilogram, which states that "The kilogram ..is defined by taking the fixed numerical value of to be when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the
metre The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pre ...
and the second are defined in terms of
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
and duration of
hyperfine transition In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the ...
of the ground state of an unperturbed caesium-133 atom ." This implies that mass metrology aims to find the value of one kilogram, and the kilogram is ''compensating''. Every experiment aiming to measure the kilogram (such as the
Kibble balance A Kibble balance is an electromechanical measuring instrument that measures the weight of a test object very precisely by the electric current and voltage needed to produce a compensating force. It is a metrological instrument that can real ...
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 to be exact was taken in 2010, when its measured value was , thus the present definition of kilogram was also enforced. In the future, the value of one kilogram must be refined to times the mass of the International Prototype of the Kilogram (IPK).

## 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 In physics, an electronvolt (symbol eV, also written electron-volt and electron volt) is the measure of an amount of kinetic energy gained by a single electron accelerating from rest through an electric potential difference of one volt in vacu ...
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 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 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 In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
of 555  nanometres (a wavelength that can be perceived by the human eye to be
green Green is the color between cyan and yellow on the visible spectrum. It is evoked by light which has a dominant wavelength of roughly 495570 nm. In subtractive color systems, used in painting and color printing, it is created by a combin ...
) has a frequency of (). Each
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
has an energy . 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 The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining ...
, , with the result of , about the food energy in three apples.

# Determination

In principle, the Planck constant can 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. 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. All of the methods given below ''except'' the X-ray crystal density method rely on the theoretical basis of the Josephson effect and the quantum Hall effect.

## Josephson constant

The Josephson constant ''K''J 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 . :$K_ = \frac = \frac.$ 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, ''K''J is related to the Planck constant by :$h = \frac.$

## Kibble balance

A Kibble balance (formerly known as a watt balance) is an instrument for comparing two powers, one of which is measured in SI
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James ...
s and the other of which is measured in conventional electrical units. From the definition of the ''conventional'' watt ''W''90, this gives a measure of the product ''K''J2''R''K in SI units, where ''R''K 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 , the measurement of ''K''J2''R''K is a direct determination of the Planck constant. :$h = \frac .$

## Magnetic resonance

The gyromagnetic ratio ''γ'' of an object is the ratio of its magnetic moment to its angular momentum, which is directly related to the constant of proportionality between the frequency ''ν'' of nuclear magnetic resonance (or
electron paramagnetic resonance Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spin ...
for electrons) and the applied magnetic field ''B'': . It is difficult to measure gyromagnetic ratios precisely because of the difficulties in precisely measuring ''B'', but the value for protons in
water Water (chemical formula ) is an inorganic, transparent, tasteless, odorless, and nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known living organisms (in which it acts as ...
at is known to an uncertainty of better than 10−6. 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, ''γ''′p. The gyromagnetic ratio is related to the shielded proton magnetic moment ''μ''′p, the
spin number In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
''I'' ( for protons) and the reduced Planck constant. :$\gamma^_\text = \frac = \frac.$ The ratio of the shielded proton magnetic moment ''μ''′p to the electron magnetic moment ''μ''e 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 ''μ''e in
Bohr magneton In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum. The Bohr magneton, in SI units is defined as \mu_\m ...
s is also known: it is half the electron ''g''-factor ''g''e. Hence :$\mu^_\text = \frac \frac$ :$\gamma^_\text = \frac \frac.$ A further complication is that the measurement of ''γ''′p 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 Γ′p-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 Γ′p-90(hi) is of interest in determining the Planck constant. :$\gamma^_\text = \frac \Gamma^_\text\left(\text\right) = \frac \Gamma^_\text\left(\text\right).$ Substitution gives the expression for the Planck constant in terms of Γ′p-90(hi): :$h = \frac \frac .$

## Faraday constant

The Faraday constant ''F'' is the charge of one mole of electrons, equal to the Avogadro constant ''N''A multiplied by the elementary charge ''e''. It can be determined by careful electrolysis experiments, measuring the amount of
silver Silver is a chemical element with the symbol Ag (from the Latin ', derived from the Proto-Indo-European ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, white, lustrous transition metal, it exhibits the highest electrical ...
dissolved from an electrode in a given time and for a given electric current. Substituting the definitions of ''N''A and ''e'' gives the relation to the Planck constant. :$h = \frac \frac.$

## X-ray crystal density

The X-ray crystal density method is primarily a method for determining the Avogadro constant ''N''A 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 ''N''A as the ratio between the volume of the unit cell of a
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
, measured by X-ray crystallography, and the molar volume of the substance. Crystals of
silicon Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic ...
are used, as they are available in high quality and purity by the technology developed for the
semiconductor A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way ...
industry. The unit cell volume is calculated from the spacing between two crystal planes referred to as ''d''220. The molar volume ''V''m(Si) requires a knowledge of the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of the crystal and the atomic weight of the silicon used. The Planck constant is given by :$h = \frac \frac .$

## Particle accelerator

The experimental measurement of the Planck constant in the Large Hadron Collider laboratory was carried out in 2011.

# See also

*
CODATA 2018 The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the International ...
* International System of Units * Introduction to quantum mechanics * Planck units *
Wave–particle duality Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical physics, classical concepts "particle" or "wave" to fu ...

*

# External links

"The role of the Planck constant in physics" – presentation at 26th CGPM meeting at Versailles, France, November 2018 when voting took place.
{{Authority control Fundamental constants 1900 in science Max Planck