Planck constant - Citizendia

A commemoration plaque for Max Planck on his discovery of Planck's constant, in front of Humboldt University, Berlin. For other universities in Berlin see List of Universities in Berlin. Berlin is the capital city and one of sixteen states of Germany. English translation: "Max Planck, discoverer of the elementary quantum of action h, taught in this building from 1889 to 1928. "

The Planck constant (denoted $h$ ) is a physical constant that is used to describe the sizes of quanta. A physical Constant is a Physical quantity that is generally believed to be both universal in nature and constant in time It plays a central part in the theory of quantum mechanics, and is named after Max Planck, one of the founders of quantum theory. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Planck's constant divided by $2π$ is called the reduced Planck constant, also known as Dirac's constant and denoted $\scriptstyle \hbar$ , pronounced "h-bar".

The Planck constant is the conversion factor for the energy of a photon from its frequency, $E = h f$ , where E is energy, h is Planck's constant, and f is frequency. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena The reduced Planck constant is used in the same formula when the frequency is in radians instead of cycles, $\scriptstyle E=\hbar\omega$ , where the frequency in radians per second is given by $ω = 2π f$ . The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57

The Planck constant and the reduced Planck constant are used to describe quantization, a phenomenon occurring in subatomic particles such as electrons and photons in which certain physical properties occur in fixed amounts rather than assuming a continuous range of possible values. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena

1 Significance of the size of Planck's constant
2 Units, value and symbols
- 2.1 Recent values published after CODATA 2006
3 Origins of Planck's constant
4 Usage
5 Dirac constant
6 See also
7 References
8 External links

Significance of the size of Planck's constant

Expressed in the SI units of joule seconds (J·s), the Planck constant is one of the smallest constants used in physics. The joule-second is a unit equal to a Joule multiplied by a Second, used to measure action or Angular momentum. The significance of this is that it reflects the extremely small scales at which quantum mechanical effects are observed, and hence why we are not familiar with quantum physics in our everyday lives in the way that we are with classical physics. Indeed, classical physics can essentially be defined as the limit of quantum mechanics as the Planck constant tends to zero.

In natural units, the Dirac constant is taken as 1 (i. In Physics, natural units are Physical units of Measurement defined in terms of universal Physical constants, such that some chosen physical e. , the Planck constant is 2·π), as is convenient for describing physics at the atomic scale dominated by quantum effects.

Units, value and symbols

The Planck constant has dimensions of energy multiplied by time, which are also the dimensions of action. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation In SI units, the Planck constant is expressed in joule seconds (J·s). The joule-second is a unit equal to a Joule multiplied by a Second, used to measure action or Angular momentum. The dimensions may also be written as momentum times distance (N·m·s), which are also the dimensions of angular momentum. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Distance is a numerical description of how far apart objects are The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The value of the Planck constant is:

$h =\,\,\, 6.626\ 068\ 96(33) \times 10^{-34}\ \mbox{J}\cdot\mbox{s} \,\,\, = \,\,\, 4.135\ 667\ 33(10) \times10^{-15}\ \mbox{eV}\cdot\mbox{s}.$ ^[1]^[2]

The two digits between the parentheses denote the standard uncertainty in the last two digits of the value. Brackets are Punctuation marks used in pairs to set apart or interject text within other text

The value of the Dirac constant is:

$\hbar\ = \frac{h}{2\pi} = \,\,\, 1.054\ 571\ 628(53)\times10^{-34}\ \mbox{J}\cdot\mbox{s} \,\,\, = \,\,\, 6.582\ 118\ 99(16) \times10^{-16}\ \mbox{eV}\cdot\mbox{s}$

The figures cited here are the 2006 CODATA-recommended values for the constants and their uncertainties. CODATA ( Committee on Data for Science and Technology) was established in 1966 as an interdisciplinary committee of the International Council of Science (ICSU formerly The 2006 CODATA results were made available in March 2007 and represent the best-known, internationally-accepted values for these constants, based on all data available as of 31 December 2006. New CODATA figures are scheduled to be published approximately every four years.

Unicode reserves codepoints U+210E (ℎ) for the Planck constant, and U+210F (ℏ) for the Dirac constant. In Computing, Unicode is an Industry standard allowing Computers to consistently represent and manipulate text expressed in most of the world's

Recent values published after CODATA 2006

In October 2005, the National Physical Laboratory (NPL) reported initial measurements of the Planck constant using a newly improved watt balance. The watt balance is an experimental electromechanical apparatus that may one day provide a definition of the Kilogram based on Electronics. They report a value of:

$h =\,\,\, 6.626\ 070\ 95(44) \times 10^{-34}\ \mbox{J}\cdot\mbox{s} \,\,\,$

which is significantly different (statistically) from the 2006 CODATA value above (Robinson and Kibble 2007). The NPL value was published after 2006 CODATA, and along with other future measurements will be taken into account in the next CODATA published value.

Origins of Planck's constant

The Planck constant, $h \$ , was proposed in reference to the problem of black-body radiation. In Physics, a black body is an object that absorbs all light that falls on it Radiation, as in Physics, is Energy in the form of waves or moving Subatomic particles emitted by an atom or other body as it changes from a higher energy The underlying assumption to Planck's law of black body radiation was that the electromagnetic radiation emitted by a black body could be modeled as a set of harmonic oscillators with quantized energy of the form:

$E = h \nu = h \omega /(2 \pi) = \hbar \omega \$

$E \$ is the quantized energy of the photons of radiation having frequency (Hz) of $\nu \$ (nu) or angular frequency (rad/s) of $\omega \$ (omega). For a general introduction see Black body. In Physics, Planck's law describes the spectral radiance of Electromagnetic radiation This article is about the harmonic oscillator in classical mechanics In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena Frequency is a measure of the number of occurrences of a repeating event per unit Time. The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second. Nu (uppercase Ν, lowercase ν; Νι Ni is the 13th letter of the Greek alphabet. Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 OMEGA is the premier Counter-terrorism unit of Latvia. Founded in 1992 OMEGA cooperates with many other counter-terrorism units over the world

This model proved extremely accurate, but it provided an intellectual stumbling block for theoreticians who did not understand where the quantization of energy arose — Planck himself only considered it "a purely formal assumption". This line of questioning helped lead to the formation of quantum mechanics.

In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental corner-stones to the entire theory lies in the commutator relationship between the position operator $\hat{x}$ and the momentum operator $\hat{p}$ :

$[\hat{p}_i, \hat{x}_j] = -i \hbar \delta_{ij}$

where $\delta_{ij}\,$ is the Kronecker delta. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two For more information, see the mathematical formulation of quantum mechanics. The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics.

Usage

The Planck constant is used to describe quantization. For instance, the energy (E) carried by a beam of light with constant frequency ( $\nu\,$ ) can only take on the values

$E = n h \nu \,,\quad n\in\mathbb{N}.$

It is sometimes more convenient to use the angular frequency $\omega=2\pi\,\nu$ , which gives

$E = n \hbar \omega \,,\quad n\in\mathbb{N}.$

Many such "quantization conditions" exist. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 Frequency is a measure of the number of occurrences of a repeating event per unit Time. Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency A particularly interesting condition governs the quantization of angular momentum. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Let J be 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

$\begin{align} J^2 = j(j+1) \hbar^2,\quad & j = 0, 1/2, 1, 3/2, \ldots, \\ J_z = m \hbar, \qquad\quad\quad & m = -j, -j+1, \ldots, j. \end{align}$

Thus, $\hbar$ may be said to be the "quantum of angular momentum".

The Planck constant also occurs in statements of Heisenberg's uncertainty principle. Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain Given a large number of particles prepared in the same state, the uncertainty in their position, $Δ x$ , and the uncertainty in their momentum (in the same direction), $Δ p$ , obey

$\Delta x\, \Delta p \ge \begin{matrix}\frac{1}{2}\end{matrix} \hbar$

where the uncertainty is given as the standard deviation of the measured value from its expected value. Uncertainty is a term used in subtly different ways in a number of fields including Philosophy, Statistics, Economics, Finance, Insurance In Probability and Statistics, the standard deviation is a measure of the dispersion of a collection of values

There are a number of other such pairs of physically measurable values which obey a similar rule.

Dirac constant

The Dirac constant or the "reduced Planck constant", $\hbar = \frac{h}{2 \pi} \$ , differs only from the Planck constant by a factor of $2π$ . The Planck constant is stated in SI units of measurement, joules per hertz, or joules per (cycle per second), while the Dirac constant is the same value stated in joules per (radian per second). The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second. The cycle per second was a once-common unit of Frequency. With the organisation of the International System of Units (abbreviated SI from the French in The radian per second (symbol rad·s&minus1 or rad/s) is the SI unit of Angular velocity.

In essence, the Dirac constant is a conversion factor between phase (in radians) and action (in joule-seconds) as seen in the Schrödinger equation. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system The Planck constant is similarly a conversion factor between phase (in cycles) and action. All other uses of Planck's constant and Dirac's constant follow from that relationship.

References

^ Planck constant. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. In Physics, natural units are Physical units of Measurement defined in terms of universal Physical constants, such that some chosen physical Planck units are Units of measurement named after the German physicist Max Planck, who first proposed them in 1899 The quantum Hall effect (or integer quantum Hall effect) is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and 2006 CODATA recommended values. CODATA ( Committee on Data for Science and Technology) was established in 1966 as an interdisciplinary committee of the International Council of Science (ICSU formerly NIST. Retrieved on 2007-08-08. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1220 - Sweden is defeated by Estonian tribes in the Battle of Lihula.
^ Planck constant in eV s. 2006 CODATA recommended values. CODATA ( Committee on Data for Science and Technology) was established in 1966 as an interdisciplinary committee of the International Council of Science (ICSU formerly NIST. Retrieved on 2007-08-08. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1220 - Sweden is defeated by Estonian tribes in the Battle of Lihula.

Barrow, John D. (2002). John David Barrow FRS (born November 29, 1952, London) is an English cosmologist, theoretical physicist, and The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe (in English). Pantheon Books. ISBN 0-375-42221-8.
Robinson, I. A. ; B. P. Kibble (October 2007). "An initial measurement of Planck’s constant using the NPL Mark II watt balance". Metrologia 44: 427–440. doi:10.1088/0026-1394/44/6/001. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.

External links

Planck's original 1901 paper

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