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Planck constant

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</style><div role="note" class="hatnote navigation-not-searchable">For the law governing black-body radiation, see <a href="Planck's_law" title="Planck's law">Planck's law</a>.</div>

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Planck constant</th></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label"><a href="SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="Joule" title="Joule">joule</a> per <a href="Hertz" title="Hertz">hertz</a> (J/Hz)</td></tr><tr><th scope="row" class="infobox-label">In <a href="SI_base_unit" title="SI base unit">SI base units</a></th><td class="infobox-data"><a href="Kilogram_(unit)" class="mw-redirect" title="Kilogram (unit)">kg</a>⋅<a href="Metre_(unit)" class="mw-redirect" title="Metre (unit)">m</a>2⋅<a href="Second" title="Second">s</a>−1</td></tr><tr><th scope="row" class="infobox-label"><a href="Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}</annotation>

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</math><img src="./0484fb7c4e4306a940c19093b216b3a88974f49f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.263ex; height:2.676ex;" alt="{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label"><a href="Quantity_value" class="mw-redirect" title="Quantity value">Value</a></th><td class="infobox-data">6.62607015×10−34 J⋅Hz−1 4.135667696...×10−15 eV⋅Hz−1</td></tr></tbody></table>

<table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Reduced Planck constant</th></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

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<annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation>

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</math><img src="./de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label"><a href="SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data">joule-second (J·s)</td></tr><tr><th scope="row" class="infobox-label">In <a href="SI_base_unit" title="SI base unit">SI base units</a></th><td class="infobox-data"><a href="Kilogram_(unit)" class="mw-redirect" title="Kilogram (unit)">kg</a>⋅<a href="Metre_(unit)" class="mw-redirect" title="Metre (unit)">m</a>2⋅<a href="Second" title="Second">s</a>−1</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from other quantities</div></th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">

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</style><div class="plainlist"><ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar ={\frac {h}{2\pi }}}">

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<annotation encoding="application/x-tex">{\displaystyle \hbar ={\frac {h}{2\pi }}}</annotation>

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</math><img src="./df05180652a87fe1af83af4bba3402117bd18466.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.736ex; height:5.343ex;" alt="{\displaystyle \hbar ={\frac {h}{2\pi }}}" loading="lazy"></li></ul></div></td></tr><tr><th scope="row" class="infobox-label"><a href="Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}</annotation>

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</math><img src="./0484fb7c4e4306a940c19093b216b3a88974f49f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.263ex; height:2.676ex;" alt="{\displaystyle {\mathsf {M}}{\mathsf {L}}^{2}{\mathsf {T}}^{-1}}" loading="lazy"></td></tr><tr><th scope="row" class="infobox-label"><a href="Quantity_value" class="mw-redirect" title="Quantity value">Value</a></th><td class="infobox-data">1.054571817...×10−34 J⋅s 6.582119569...×10−16 eV⋅s</td></tr></tbody></table>

The Planck constant, or Planck's constant, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">, is a fundamental <a href="Physical_constant" title="Physical constant">physical constant</a> of foundational importance in <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>: a <a href="Photon" title="Photon">photon</a>'s energy is equal to its <a href="Frequency" title="Frequency">frequency</a> multiplied by the Planck constant, and a particle's <a href="Momentum" title="Momentum">momentum</a> is equal to the <a href="Wavenumber" title="Wavenumber">wavenumber</a> of the associated <a href="Matter_wave" title="Matter wave">matter wave</a> (the reciprocal of its <a href="Wavelength" title="Wavelength">wavelength</a>) multiplied by the Planck constant.

The constant was postulated by <a href="Max_Planck" title="Max Planck">Max Planck</a> in 1900 as a <a href="Proportionality_constant" class="mw-redirect" title="Proportionality constant">proportionality constant</a> needed to explain experimental <a href="Black-body" class="mw-redirect" title="Black-body">black-body</a> radiation.<a href="#cite_note-Planck01-1">[1]</a> Planck later referred to the constant as the "quantum of <a href="Action_(physics)" title="Action (physics)">action</a>".<a href="#cite_note-2">[2]</a> In 1905, <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a> associated the "quantum" or minimal element of the energy to the electromagnetic wave itself. Max Planck received the 1918 <a href="Nobel_Prize_in_Physics" title="Nobel Prize in Physics">Nobel Prize in Physics</a> "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

In <a href="Metrology" title="Metrology">metrology</a>, the Planck constant is used, together with other constants, to define the <a href="Kilogram" title="Kilogram">kilogram</a>, the <a href="SI_unit" class="mw-redirect" title="SI unit">SI unit</a> of mass.<a href="#cite_note-3">[3]</a> The SI units are defined such that it has the exact value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> = 6.62607015×10−34 J⋅Hz−1‍<a href="#cite_note-physconst-h-4">[4]</a> when the Planck constant is expressed in SI units.

The closely related reduced Planck constant, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \hbar }">

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<annotation encoding="application/x-tex">{\textstyle \hbar }</annotation>

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</math><img src="./f9d68c123c2f0f7e66f5a7a890e93cca598247a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\textstyle \hbar }" loading="lazy"> (h-bar), equal to the Planck constant divided by <a href="Tau_(mathematics)" title="Tau (mathematics)">2π</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \hbar ={\frac {h}{2\pi }}}">

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<annotation encoding="application/x-tex">{\textstyle \hbar ={\frac {h}{2\pi }}}</annotation>

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</math><img src="./788c1a8a0024c8e4f4e46bb932e1dabb5c1ae78d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.005ex; height:3.676ex;" alt="{\textstyle \hbar ={\frac {h}{2\pi }}}" loading="lazy">, is commonly used in quantum physics equations. It relates the energy of a photon to its <a href="Angular_frequency" title="Angular frequency">angular frequency</a>, and the linear momentum of a particle to the <a href="Angular_wavenumber" class="mw-redirect" title="Angular wavenumber">angular wavenumber</a> of its associated matter wave. As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> has an exact defined value, the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \hbar }">

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<annotation encoding="application/x-tex">{\textstyle \hbar }</annotation>

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</math><img src="./f9d68c123c2f0f7e66f5a7a890e93cca598247a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\textstyle \hbar }" loading="lazy"> can be calculated to arbitrary precision: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

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<annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation>

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</math><img src="./de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"> = 1.054571817...×10−34 J⋅s.<a href="#cite_note-physconst-hbar-5">[5]</a> As a <a href="Proportionality_constant" class="mw-redirect" title="Proportionality constant">proportionality constant</a> in relationships involving angular quantities, the unit of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \hbar }">

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<annotation encoding="application/x-tex">{\textstyle \hbar }</annotation>

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</math><img src="./f9d68c123c2f0f7e66f5a7a890e93cca598247a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\textstyle \hbar }" loading="lazy"> may be given as J·s/rad, with the same numerical value, as the <a href="Radian" title="Radian">radian</a> is the natural <a href="Dimensionless_quantity" title="Dimensionless quantity">dimensionless</a> unit of <a href="Angle" title="Angle">angle</a>.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Origin_of_the_constant">Origin of the constant</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Planck's_law" title="Planck's law">Planck's law</a></div>

The Planck constant was formulated as part of Max Planck's successful effort to produce a mathematical expression that accurately predicted the observed spectral distribution of <a href="Black-body_radiation" title="Black-body radiation">black-body radiation</a>.<a href="#cite_note-Bitter-6">[6]</a> This 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 <a href="Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a> some 40 years earlier. Every <a href="Physical_body" class="mw-redirect" title="Physical body">physical body</a> spontaneously and continuously emits <a href="Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>. There was no expression or explanation for the overall shape of the observed emission spectrum. At the time, <a href="Wien_approximation" title="Wien approximation">Wien's law</a> fit the data for short wavelengths and high temperatures, but failed for long wavelengths.<a href="#cite_note-Bitter-6">[6]</a>: 141 Also around this time, but unknown to Planck, <a href="Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Lord Rayleigh</a> had derived theoretically a formula, now known as the <a href="Rayleigh%E2%80%93Jeans_law" title="Rayleigh–Jeans law">Rayleigh–Jeans law</a>, 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 <a href="Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillators</a>, one for each possible frequency. He examined how the <a href="Entropy" title="Entropy">entropy</a> 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,<a href="#cite_note-Planck01-1">[1]</a> 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, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">, which is thought to be for Hilfsgröße (auxiliary quantity),<a href="#cite_note-7">[7]</a> and subsequently became known as the Planck constant. The expression formulated by Planck showed that the spectral radiance per unit frequency of a body for <a href="Frequency" title="Frequency">frequency</a> ν at <a href="Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a> T is given by

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<annotation encoding="application/x-tex">{\displaystyle B_{\nu }(\nu ,T)d\nu ={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{k_{\mathrm {B} }T}}-1}}d\nu ,}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\text{B}}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\text{B}}}</annotation>

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</math><img src="./9582c7c795d2def2c061f0dfa3a6f0fb3dd2de44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\text{B}}}" loading="lazy"> is the <a href="Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> is the Planck constant, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is the <a href="Speed_of_light" title="Speed of light">speed of light</a> in the medium, whether material or vacuum.<a href="#cite_note-Planck_1914_6_168-8">[8]</a><a href="#cite_note-Chan8-9">[9]</a><a href="#cite_note-Rybicki_1979_22-10">[10]</a>

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.<a href="#cite_note-Planck01-1">[1]</a> To save his theory, Planck resorted to using the then-controversial theory of <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>,<a href="#cite_note-Planck01-1">[1]</a> which he described as "an act of desperation".<a href="#cite_note-Kragh-11">[11]</a> One of his new boundary conditions was

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</style><blockquote class="templatequote">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;</blockquote><div class="templatequotecite">— Planck, "On the Law of Distribution of Energy in the Normal Spectrum"<a href="#cite_note-Planck01-1">[1]</a></div>

With this new condition, Planck had imposed the quantization of the energy of the oscillators, in his own words, "a purely formal assumption ... actually I did not think much about it",<a href="#cite_note-12">[12]</a> 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 <a href="Planck%E2%80%93Einstein_relation" class="mw-redirect" title="Planck–Einstein relation">Planck–Einstein relation</a>:

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Planck was able to calculate the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> from experimental data on black-body radiation: his result, 6.55×10−34 J⋅s, is within 1.2% of the currently defined value.<a href="#cite_note-Planck01-1">[1]</a> He also made the first determination of the <a href="Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\text{B}}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{\text{B}}}</annotation>

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</math><img src="./9582c7c795d2def2c061f0dfa3a6f0fb3dd2de44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\text{B}}}" loading="lazy"> from the same data and theory.<a href="#cite_note-PlanckNobel-13">[13]</a>

<div class="mw-heading mw-heading3"><h3 id="Development_and_application">Development and application</h3></div>

The black-body problem was revisited in 1905, when <a href="John_William_Strutt%2C_3rd_Baron_Rayleigh" class="mw-redirect" title="John William Strutt, 3rd Baron Rayleigh">Lord Rayleigh</a> and <a href="James_Jeans" title="James Jeans">James Jeans</a> (together) and <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a> independently proved that classical electromagnetism could never account for the observed spectrum. These proofs are commonly known as the "<a href="Ultraviolet_catastrophe" title="Ultraviolet catastrophe">ultraviolet catastrophe</a>", a name coined by <a href="Paul_Ehrenfest" title="Paul Ehrenfest">Paul Ehrenfest</a> in 1911. They contributed greatly (along with Einstein's work on the <a href="Photoelectric_effect" title="Photoelectric effect">photoelectric effect</a>) in convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The first <a href="Solvay_Conference" title="Solvay Conference">Solvay Conference</a> in 1911 was devoted to "the theory of radiation and quanta".<a href="#cite_note-14">[14]</a>

<div class="mw-heading mw-heading4"><h4 id="Photoelectric_effect">Photoelectric effect</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Photoelectric_effect" title="Photoelectric effect">Photoelectric effect</a></div>

The photoelectric effect is the emission of electrons (called "photoelectrons") from a surface when light is shone on it. It was first observed by <a href="Alexandre_Edmond_Becquerel" class="mw-redirect" title="Alexandre Edmond Becquerel">Alexandre Edmond Becquerel</a> in 1839, although credit is usually reserved for <a href="Heinrich_Hertz" title="Heinrich Hertz">Heinrich Hertz</a>,<a href="#cite_note-Nobel21-15">[15]</a> who published the first thorough investigation in 1887. Another particularly thorough investigation was published by <a href="Philipp_Lenard" title="Philipp Lenard">Philipp Lenard</a> (Lénárd Fülöp) in 1902.<a href="#cite_note-Lenard-16">[16]</a> Einstein's 1905 paper<a href="#cite_note-17">[17]</a> discussing the effect in terms of light quanta would earn him the Nobel Prize in 1921,<a href="#cite_note-Nobel21-15">[15]</a> after his predictions had been confirmed by the experimental work of <a href="Robert_Andrews_Millikan" title="Robert Andrews Millikan">Robert Andrews Millikan</a>.<a href="#cite_note-Millikan-18">[18]</a> 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.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a>

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 <a href="Intensity_(physics)" title="Intensity (physics)">intensity</a>. The light from a theatre spotlight is more intense than the light from a domestic lightbulb; that is to say that the spotlight gives out more energy per unit time and per unit space (and hence consumes more electricity) than the ordinary bulb, even though the color of the light might be very similar. Other waves, such as sound or the waves crashing against a seafront, also have their intensity. However, the energy account of the photoelectric effect did not seem to agree with the wave description of light.

The "photoelectrons" emitted as a result of the photoelectric effect have a certain <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a>, which can be measured. This kinetic energy (for each photoelectron) is independent of the intensity of the light,<a href="#cite_note-Lenard-16">[16]</a> but depends linearly on the frequency;<a href="#cite_note-Millikan-18">[18]</a> and if the frequency is too low (corresponding to a photon energy that is less than the <a href="Work_function" title="Work function">work function</a> 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).<a href="#cite_note-21">[21]</a><a href="#cite_note-22">[22]</a> 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.<a href="#cite_note-Lenard-16">[16]</a>

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 <a href="Photon" title="Photon">photons</a>, was to be the same as Planck's "energy element", giving the modern version of the Planck–Einstein relation:

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<annotation encoding="application/x-tex">{\displaystyle E=hf.}</annotation>

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</math>

Einstein's postulate was later proven experimentally: the constant of proportionality between the frequency of incident light <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> and the kinetic energy of photoelectrons <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> was shown to be equal to the Planck constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">.<a href="#cite_note-Millikan-18">[18]</a>

<div class="mw-heading mw-heading4"><h4 id="Atomic_structure">Atomic structure</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Bohr_model" title="Bohr model">Bohr model</a></div>

In 1912 <a href="John_William_Nicholson" title="John William Nicholson">John William Nicholson</a> developed<a href="#cite_note-23">[23]</a> an atomic model and found the angular momentum of the electrons in the model were related by h/2π.<a href="#cite_note-HeilbronPath-24">[24]</a><a href="#cite_note-McCormmach-25">[25]</a>

Nicholson's nuclear quantum atomic model influenced the development of <a href="Niels_Bohr" title="Niels Bohr">Niels Bohr</a> 's atomic model<a href="#cite_note-26">[26]</a><a href="#cite_note-27">[27]</a><a href="#cite_note-McCormmach-25">[25]</a> and Bohr quoted him in his 1913 paper of the Bohr model of the atom.<a href="#cite_note-Bohr-28">[28]</a> Bohr's model went beyond Planck's abstract harmonic oscillator concept: an electron in a Bohr atom could only have certain defined energies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{n}}</annotation>

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</math><img src="./ad6b82f2a00af6c9efd4c16d4e99329605645c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.934ex; height:2.509ex;" alt="{\displaystyle E_{n}}" loading="lazy">, defined by

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

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</msub>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{n}=-{\frac {hcR_{\infty }}{n^{2}}},}</annotation>

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</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is the speed of light in vacuum, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{\infty }}</annotation>

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</math><img src="./639146705058e9d335aed47b55a9f802bc290b5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.639ex; height:2.509ex;" alt="{\displaystyle R_{\infty }}" loading="lazy"> is an experimentally determined constant (the <a href="Rydberg_constant" title="Rydberg constant">Rydberg constant</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \{1,2,3,...\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \{1,2,3,...\}}</annotation>

</semantics>

</math><img src="./a3e97f8c9bf0deb3e92c46a93ea68c9ea970330e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.251ex; height:2.843ex;" alt="{\displaystyle n\in \{1,2,3,...\}}" loading="lazy">. This approach also allowed Bohr to account for the <a href="Rydberg_formula" title="Rydberg formula">Rydberg formula</a>, an empirical description of the atomic spectrum of hydrogen, and to account for the value of the Rydberg constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\infty }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{\infty }}</annotation>

</semantics>

</math><img src="./639146705058e9d335aed47b55a9f802bc290b5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.639ex; height:2.509ex;" alt="{\displaystyle R_{\infty }}" loading="lazy"> in terms of other fundamental constants.

In discussing angular momentum of the electrons in his model Bohr introduced the quantity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {h}{2\pi }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {h}{2\pi }}}</annotation>

</semantics>

</math><img src="./d91adb268f891436e5499d126c8a68414506b99f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.6ex; height:3.676ex;" alt="{\textstyle {\frac {h}{2\pi }}}" loading="lazy">, now known as the <a href="#Reduced_Planck_constant">reduced Planck constant</a> as the quantum of <a href="Angular_momentum" title="Angular momentum">angular momentum</a>.<a href="#cite_note-Bohr-28">[28]</a>

<div class="mw-heading mw-heading4"><h4 id="Uncertainty_principle">Uncertainty principle</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Uncertainty_principle" title="Uncertainty principle">Uncertainty principle</a></div>

The Planck constant also occurs in statements of <a href="Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>'s uncertainty principle. Given numerous particles prepared in the same state, the <a href="Uncertainty" title="Uncertainty">uncertainty</a> in their position, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation>

</semantics>

</math><img src="./f3890eb866b6258d7a304fc34c70ee3fb3a81a70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}" loading="lazy">, and the uncertainty in their momentum, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta p_{x}}</annotation>

</semantics>

</math><img src="./a84011ce3f13232ac79d38f73b7b48c463144cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.278ex; height:2.509ex;" alt="{\displaystyle \Delta p_{x}}" loading="lazy">, obey

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta x\,\Delta p_{x}\geq {\frac {\hbar }{2}},}</annotation>

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</math>

where the uncertainty is given as the <a href="Standard_deviation" title="Standard deviation">standard deviation</a> of the measured value from its <a href="Expected_value" title="Expected value">expected value</a>. There are several other such pairs of physically measurable <a href="Conjugate_variable" class="mw-redirect" title="Conjugate variable">conjugate variables</a> 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 <a href="Commutator" title="Commutator">commutator</a> relationship between the <a href="Position_operator" title="Position operator">position operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {x}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {x}}}</annotation>

</semantics>

</math><img src="./18d95a7845e4e16ffb7e18ab37a208d0ab18e0e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\hat {x}}}" loading="lazy"> and the <a href="Momentum_operator" title="Momentum operator">momentum operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}}</annotation>

</semantics>

</math><img src="./8bd4c026f1b3413adc58b9b65e89e62bce92c85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}" loading="lazy">:

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [{\hat {p}}_{i},{\hat {x}}_{j}]=-i\hbar \delta _{ij},}</annotation>

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</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation>

</semantics>

</math><img src="./fa75d04c11480d976e1396951e02cbb3c4f71568.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}" loading="lazy"> is the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>.

<div class="mw-heading mw-heading4"><h4 id="Photon_energy">Photon energy</h4></div>

The <a href="Planck_relation" title="Planck relation">Planck relation</a> connects the particular <a href="Photon_energy" title="Photon energy">photon energy</a> E with its associated wave frequency f:

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<annotation encoding="application/x-tex">{\displaystyle E=hf.}</annotation>

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</math>

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

Since the frequency f, <a href="Wavelength" title="Wavelength">wavelength</a> λ, and <a href="Speed_of_light" title="Speed of light">speed of light</a> c are related by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\frac {c}{\lambda }}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f={\frac {c}{\lambda }}}</annotation>

</semantics>

</math><img src="./4dc0fb0b3be8b879ff12bb534c0b82ff36f7eb7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.568ex; height:4.843ex;" alt="{\displaystyle f={\frac {c}{\lambda }}}" loading="lazy">, the relation can also be expressed as

<mfrac>

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</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E={\frac {hc}{\lambda }}.}</annotation>

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</math>

<div class="mw-heading mw-heading4"><h4 id="de_Broglie_wavelength">de Broglie wavelength</h4></div>

In 1923, <a href="Louis_de_Broglie" title="Louis de Broglie">Louis de Broglie</a> 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 <a href="Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>. The <a href="De_Broglie_wavelength" class="mw-redirect" title="De Broglie wavelength">de Broglie wavelength</a> λ of the particle is given by

<mfrac>

</mfrac>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {h}{p}},}</annotation>

</semantics>

</math>

where p denotes the linear <a href="Momentum" title="Momentum">momentum</a> of a particle, such as a photon, or any other <a href="Elementary_particle" title="Elementary particle">elementary particle</a>.

The <a href="Photon_energy" title="Photon energy">energy of a photon</a> with angular frequency ω = 2πf is given by

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E=\hbar \omega ,}</annotation>

</semantics>

</math>

while its linear momentum relates to

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=\hbar k,}</annotation>

</semantics>

</math>

where k is an <a href="Wavenumber" title="Wavenumber">angular wavenumber</a>.

These two relations are the temporal and spatial parts of the special relativistic expression using <a href="Four-Vector" class="mw-redirect" title="Four-Vector">4-vectors</a>.

<msup>

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</msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

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<mover>

</mover>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle P^{\mu }=\left({\frac {E}{c}},{\vec {p}}\right)=\hbar K^{\mu }=\hbar \left({\frac {\omega }{c}},{\vec {k}}\right).}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Statistical_mechanics">Statistical mechanics</h4></div>

Classical <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a> requires the existence of h (but does not define its value).<a href="#cite_note-29">[29]</a> Eventually, following upon Planck's discovery, it was speculated that physical <a href="Action_(physics)" title="Action (physics)">action</a> could not take on an arbitrary value, but instead was restricted to integer multiples of a very small quantity, the "[elementary] <a href="Quantum" title="Quantum">quantum</a> of action", now called the Planck constant.<a href="#cite_note-30">[30]</a> This was a significant conceptual part of the so-called "<a href="Old_quantum_theory" title="Old quantum theory">old quantum theory</a>" developed by physicists including <a href="Niels_Bohr" title="Niels Bohr">Bohr</a>, <a href="Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld</a>, and <a href="Jun_Ishiwara" title="Jun Ishiwara">Ishiwara</a>, in which particle trajectories exist but are <a href="Hidden_variable_theory" class="mw-redirect" title="Hidden variable theory">hidden</a>, 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.<a href="#cite_note-31">[31]</a>: 373 Related to this is the concept of energy quantization which existed in old quantum theory and also exists in altered form in modern quantum physics. Classical physics cannot explain quantization of energy.

<div class="mw-heading mw-heading2"><h2 id="Dimension_and_value">Dimension and value</h2></div>

The Planck constant has the same <a href="Dimension_(physics)" class="mw-redirect" title="Dimension (physics)">dimensions</a> as <a href="Action_(physics)" title="Action (physics)">action</a> and as <a href="Angular_momentum" title="Angular momentum">angular momentum</a> (both with unit J·s = kg·m2·s−1). The Planck constant is fixed at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> = 6.62607015×10−34 J⋅Hz−1‍<a href="#cite_note-physconst-h-4">[4]</a> as part of the definition of the <a href="International_System_of_Units" title="International System of Units">SI</a> units.<a href="#cite_note-SIbrochure9th-32">[32]</a>

Alternatively, if the <a href="Radian" title="Radian">radian</a> were considered a <a href="Base_unit_of_measurement" title="Base unit of measurement">base unit</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> would have the dimension of action (unit J·s), while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation>

</semantics>

</math><img src="./de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"> would have the dimension of angular momentum (unit J·s·rad−1), instead.<a href="#cite_note-r763-33">[33]</a>

This value is used to define the SI unit of mass, the <a href="Kilogram" title="Kilogram">kilogram</a>: "the kilogram [...] is defined by taking the fixed numerical value of h to be 6.62607015×10−34 when expressed in the unit J⋅s, which is equal to kg⋅m2⋅s−1, where the <a href="Metre" title="Metre">metre</a> and the <a href="Second" title="Second">second</a> are defined in terms of <a href="Speed_of_light" title="Speed of light">speed of light</a> c and duration of <a href="Hyperfine_structure" title="Hyperfine structure">hyperfine transition</a> of the <a href="Ground_state" title="Ground state">ground state</a> of an unperturbed <a href="Caesium-133" class="mw-redirect" title="Caesium-133">caesium-133</a> atom ΔνCs."<a href="#cite_note-SIbrochure9th-32">[32]</a> Technologies of mass <a href="Metrology" title="Metrology">metrology</a> such as the <a href="Kibble_balance" title="Kibble balance">Kibble balance</a> measure the kilogram by fixing the Planck constant.

As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> has an exact defined value, the value of the reduced Planck constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation>

</semantics>

</math><img src="./de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"> can be calculated to arbitrary precision without any limiting uncertainty: <blockquote><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar ={\frac {h}{2\pi }}={\frac {6.626\,070\,15}{2\pi }}\times 10^{-34}\,\mathrm {J{\cdot }s} =1.054\,571\,817...\times 10^{-34}\,\mathrm {J{\cdot }s} }">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \hbar ={\frac {h}{2\pi }}={\frac {6.626\,070\,15}{2\pi }}\times 10^{-34}\,\mathrm {J{\cdot }s} =1.054\,571\,817...\times 10^{-34}\,\mathrm {J{\cdot }s} }</annotation>

</semantics>

</math><img src="./d59c27be238ce99df9de173cea811910add21809.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:64.568ex; height:5.343ex;" alt="{\displaystyle \hbar ={\frac {h}{2\pi }}={\frac {6.626\,070\,15}{2\pi }}\times 10^{-34}\,\mathrm {J{\cdot }s} =1.054\,571\,817...\times 10^{-34}\,\mathrm {J{\cdot }s} }" loading="lazy"></blockquote>As a <a href="Proportionality_constant" class="mw-redirect" title="Proportionality constant">proportionality constant</a> in relationships involving angular quantities, the unit of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \hbar }">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle \hbar }</annotation>

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</math><img src="./f9d68c123c2f0f7e66f5a7a890e93cca598247a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\textstyle \hbar }" loading="lazy"> may be given as J·s/rad, with the same numerical value, as the radian is the natural <a href="Dimensionless_quantity" title="Dimensionless quantity">dimensionless</a> unit of <a href="Angle" title="Angle">angle</a>. This is analogous to the use of <a href="Hertz" title="Hertz">hertz</a> (Hz) for ordinary <a href="Frequency" title="Frequency">frequency</a> and <a href="Radian_per_second" title="Radian per second">radians per second</a> (rad/s) for <a href="Angular_frequency" title="Angular frequency">angular frequency</a>, both dimensionally equal to s−1.

<div class="mw-heading mw-heading3"><h3 id="Significance_of_the_value">Significance of the value</h3></div>

The Planck constant is one of the smallest constants used in physics. This reflects the fact that on a <a href="Human_scale" title="Human scale">scale adapted to humans</a>, where energies are typical of the order of kilojoules and times are typical of the order of seconds or minutes, the Planck constant is very small. When the <a href="Action_(physics)" title="Action (physics)">product of energy and time</a> for a physical event approaches the Planck constant, <a href="Quantum_effects" class="mw-redirect" title="Quantum effects">quantum effects</a> dominate.<a href="#cite_note-FeynmanII-34">[34]</a>

Equivalently, the order of the Planck constant reflects the fact that everyday objects and systems are made of a large number of <a href="Microscopic_scale" title="Microscopic scale">microscopic particles</a>. For example, in <a href="Green" title="Green">green</a> light (with a <a href="Wavelength" title="Wavelength">wavelength</a> of 555 <a href="Nanometre" title="Nanometre">nanometres</a> or a <a href="Frequency" title="Frequency">frequency</a> of 540 THz) each <a href="Photon" title="Photon">photon</a> has an <a href="Energy" title="Energy">energy</a> E = hf = 3.58×10−19 J. This is a very small amount of energy in terms of everyday experience, but everyday experience is not concerned with individual photons any more than with individual <a href="Atom" title="Atom">atoms</a> or <a href="Molecule" title="Molecule">molecules</a>. An amount of <a href="Light" title="Light">light</a> more typical in everyday experience (though much larger than the smallest amount perceivable by the <a href="Human_eye" title="Human eye">human eye</a>) is the energy of one <a href="Mole_(unit)" title="Mole (unit)">mole</a> of photons, which can be computed by multiplying the photon energy by the <a href="Avogadro_constant" title="Avogadro constant">Avogadro number</a>, 6.02214076×1023,<a href="#cite_note-physconst-NA-35">[35]</a> with the result of 216 kJ, about equal to the <a href="Food_energy" title="Food energy">food energy</a> in a small fresh <a href="Apple" title="Apple">apple</a>.<a href="#cite_note-36">[36]</a>

<div class="mw-heading mw-heading2"><h2 id="Reduced_Planck_constant">Reduced Planck constant</h2></div>

Many equations in quantum physics are customarily written using the reduced Planck constant,<a href="#cite_note-37">[37]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BmhQxWwk0K4C&dq=%22The%20more%20commonly%20used%20constant%2C%20however%2C%20is%22&pg=PA105">104</a> also known as the Dirac constant, equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {h}{2\pi }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

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<annotation encoding="application/x-tex">{\textstyle {\frac {h}{2\pi }}}</annotation>

</semantics>

</math><img src="./d91adb268f891436e5499d126c8a68414506b99f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.6ex; height:3.676ex;" alt="{\textstyle {\frac {h}{2\pi }}}" loading="lazy"> and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \hbar }">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle \hbar }</annotation>

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</math><img src="./f9d68c123c2f0f7e66f5a7a890e93cca598247a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\textstyle \hbar }" loading="lazy"> (pronounced h-bar<a href="#cite_note-Chabay_and_Sherwood-38">[38]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zMWHEAAAQBAJ&dq=%22pronounced+h-bar%22&pg=PA336">336</a> ).

<div class="mw-heading mw-heading3"><h3 id="History_2">History</h3></div>

The combination <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {h}{2\pi }}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\textstyle {\frac {h}{2\pi }}}</annotation>

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</math><img src="./d91adb268f891436e5499d126c8a68414506b99f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.6ex; height:3.676ex;" alt="{\textstyle {\frac {h}{2\pi }}}" loading="lazy"> appeared in <a href="Niels_Bohr" title="Niels Bohr">Niels Bohr</a>'s 1913 paper,<a href="#cite_note-bohr1-39">[39]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=g1k3AQAAMAAJ&q=%22If+we+therefore+assume+that+the+orbit+of+the+electron+in%22">15</a> where it was denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M_{0}}">

<msub>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\textstyle M_{0}}</annotation>

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</math><img src="./feb4ebe89e459609fa9e97bf72ee561acb3f7836.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\textstyle M_{0}}" loading="lazy">.<a href="#cite_note-McCormmach-25">[25]</a>: 169 <a href="#cite_note-40">[a]</a> For the next 15 years, the combination continued to appear in the literature, but normally without a separate symbol.<a href="#cite_note-Mehra_and_Rechenberg_v1p1-41">[40]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MvxAAQAAIAAJ&q=%22electron+ring+in+protofluorine%22">180</a> <a href="#cite_note-53">[b]</a> Then, in 1926, in their seminal papers, <a href="Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger</a> and <a href="Paul_Dirac" title="Paul Dirac">Dirac</a> again introduced special symbols for it: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle K}</annotation>

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</math><img src="./985dcc2532a2d4d91b9a9610139216c63cf832d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\textstyle K}" loading="lazy"> in the case of Schrödinger,<a href="#cite_note-Schrodinger_1926_Erste-54">[52]</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle h}</annotation>

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</math><img src="./13fda070627ca694f85f588a432f8158cc4df1e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\textstyle h}" loading="lazy"> in the case of Dirac.<a href="#cite_note-55">[53]</a> Dirac continued to use <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle h}</annotation>

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</math><img src="./13fda070627ca694f85f588a432f8158cc4df1e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\textstyle h}" loading="lazy"> in this way until 1930,<a href="#cite_note-Mehra_and_Rechenberg_v6-56">[54]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=warvAAAAMAAJ&q=%22Until+1930,+Dirac+always+wrote%22">291</a> when he introduced the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \hbar }">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle \hbar }</annotation>

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</math><img src="./f9d68c123c2f0f7e66f5a7a890e93cca598247a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\textstyle \hbar }" loading="lazy"> in his book <a href="The_Principles_of_Quantum_Mechanics" title="The Principles of Quantum Mechanics">The Principles of Quantum Mechanics</a>.<a href="#cite_note-Mehra_and_Rechenberg_v6-56">[54]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=warvAAAAMAAJ&q=%22Until+1930,+Dirac+always+wrote%22">291</a> <a href="#cite_note-57">[55]</a>

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<ul><li><a href="Committee_on_Data_of_the_International_Science_Council" title="Committee on Data of the International Science Council">Committee on Data of the International Science Council</a></li>

<li><a href="International_System_of_Units" title="International System of Units">International System of Units</a></li>

<li><a href="Introduction_to_quantum_mechanics" title="Introduction to quantum mechanics">Introduction to quantum mechanics</a></li>

<li><a href="List_of_scientists_whose_names_are_used_in_physical_constants" title="List of scientists whose names are used in physical constants">List of scientists whose names are used in physical constants</a></li>

<li><a href="Planck_units" title="Planck units">Planck units</a></li>

<li><a href="Wave%E2%80%93particle_duality" title="Wave–particle duality">Wave–particle duality</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-40"><a href="#cite_ref-40">^</a> Bohr denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M}">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle M}</annotation>

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</math><img src="./913ace920108f7552777e36ac0b7ee3f5093a088.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\textstyle M}" loading="lazy"> the angular momentum of the electron around the nucleus, and wrote the quantization condition as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M=\tau M_{0}}">

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\textstyle M=\tau M_{0}}</annotation>

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</math><img src="./8f73e06ca931a8ca87a1600656b536fcee08bd8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.051ex; height:2.509ex;" alt="{\textstyle M=\tau M_{0}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \tau }">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle \tau }</annotation>

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</math><img src="./d590a3e8735feb2d65c6fa0c4bc71ff946cd8bec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\textstyle \tau }" loading="lazy"> is a positive integer (see <a href="Bohr_model" title="Bohr model">Bohr model</a>).

</li>

<li id="cite_note-53"><a href="#cite_ref-53">^</a> Here are some papers that are mentioned in<a href="#cite_note-Mehra_and_Rechenberg_v1p1-41">[40]</a> and in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {h}{2\pi }}}">

<mfrac>

<mrow>

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</mfrac>

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<annotation encoding="application/x-tex">{\textstyle {\frac {h}{2\pi }}}</annotation>

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</math><img src="./d91adb268f891436e5499d126c8a68414506b99f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.6ex; height:3.676ex;" alt="{\textstyle {\frac {h}{2\pi }}}" loading="lazy"> appeared without a separate symbol:<a href="#cite_note-42">[41]</a>: 428 <a href="#cite_note-43">[42]</a>: <a rel="nofollow" class="external text" href="https://www.biodiversitylibrary.org/item/93032#page/599/mode/1up">549</a> <a href="#cite_note-44">[43]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=B6lJAAAAYAAJ&dq=%22we+must+demand+that+p+can+take+no+other+values+than%22&pg=PA508">508</a> <a href="#cite_note-45">[44]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=2ZPqCAAAQBAJ&dq=%22festgelegt+,+w%C3%A4hrend+das+Valenzelektron+mit%22&pg=PA102">230</a> <a href="#cite_note-46">[45]</a>: 458 <a href="#cite_note-47">[46]</a><a href="#cite_note-48">[47]</a>: <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KF2oBgAAQBAJ&dq=%22gegeben+ist,+hat+das+ganze+Atom+den+Drehimpuls%22&pg=PA137">276</a> <a href="#cite_note-49">[48]</a><a href="#cite_note-50">[49]</a><a href="#cite_note-51">[50]</a>.<a href="#cite_note-52">[51]</a>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

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<li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20111006162543/http://theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf">"On the Law of Distribution of Energy in the Normal Spectrum"</a> (PDF). Archived from <a rel="nofollow" class="external text" href="http://theochem.kuchem.kyoto-u.ac.jp/Ando/planck1901.pdf">the original</a> (PDF) on 6 October 2011. Retrieved 13 October 2011.</cite></li></ul>

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<li id="cite_note-Nobel21-15">^ <a href="#cite_ref-Nobel21_15-0">a</a> <a href="#cite_ref-Nobel21_15-1">b</a> See, e.g., <cite id="CITEREFArrhenius1922" class="citation web cs1"><a href="Svante_Arrhenius" title="Svante Arrhenius">Arrhenius, Svante</a> (10 December 1922). <a rel="nofollow" class="external text" href="https://nobelprize.org/nobel_prizes/physics/laureates/1921/press.html">"Presentation speech of the 1921 Nobel Prize for Physics"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110904232203/http://www.nobelprize.org/nobel_prizes/physics/laureates/1921/press.html">Archived</a> from the original on 4 September 2011. Retrieved 13 December 2008.</cite>

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<li id="cite_note-49"><a href="#cite_ref-49">^</a> <cite id="CITEREFKramersPauli1923" class="citation journal cs1"><a href="Hans_Kramers" title="Hans Kramers">Kramers, H. A.</a>; <a href="Wolfgang_Pauli" title="Wolfgang Pauli">Pauli, W.</a> (December 1923). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=9PpMAQAAIAAJ&dq=%22negativen+Halogenionen+ein+resultierendes%22&pg=RA1-PA351">"Zur Theorie der Bandenspektren"</a>. Zeitschrift für Physik. 13 (1): 351–367. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1923ZPhy...13..351K">1923ZPhy...13..351K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01328226">10.1007/BF01328226</a>.</cite>

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<div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3></div>

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<ul><li><cite id="CITEREFBarrow2002" class="citation cs2"><a href="John_D._Barrow" title="John D. Barrow">Barrow, John D.</a> (2002), <a rel="nofollow" class="external text" href="https://archive.org/details/constantsofnatur0000barr">The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe</a>, Pantheon Books, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-375-42221-8</bdi></cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=yhgb23tAFFs&list=PL-vj-3_a7wTDeKEupZSX7Tw42yReNgJLl">"The role of the Planck constant in physics" – presentation at 26th CGPM meeting at Versailles, France, November 2018 when voting took place.</a></li>

<li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=0nV9f5uqul0&t=84s">"The Planck constant and its units" – presentation at the 35th Symposium on Chemical Physics at the University of Waterloo, Waterloo, Ontario, Canada, 3 November 2019.</a></li></ul>

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