The Planck constant (denoted h), also called Planck's constant, is a physical constant A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement used to describe the sizes of quanta In physics, a quantum is the minimum unit of any physical entity involved in an interaction. An example of an entity that is quantized is the energy transfer of elementary particles of matter (called fermions) and of photons and other bosons. The word comes from the Latin "quantus," for "how much." Behind this, one finds the of action in quantum mechanics Quantum mechanics , also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic scales. In advanced topics of QM, some of these, specifically as angular momentum In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system. The angular momentum L of a particle with respect to some point of origin is in the atomic structure. It is named after Max Planck Max Planck was a German physicist. He is considered to be the founder of the quantum theory, and thus one of the most important physicists of the twentieth century. Planck was awarded the Nobel Prize in Physics in 1918, one of the founders of quantum theory. It is an important proportionality constant when describing atoms The atom is a basic unit of matter that consists of a dense, central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons . The electrons of an atom are bound to the nucleus by the electromagnetic force. Likewise, a group of atoms can remain and the photons of electromagnetic radiation Electromagnetic radiation is a phenomenon that takes the form of self-propagating waves in a vacuum or in matter. It comprises electric and magnetic field components, which oscillate in phase perpendicular to each other and perpendicular to the direction of energy propagation. Electromagnetic radiation is classified into several types according to and in determining several physical constants. The Planck constant has dimensions In mathematics and science, dimensional analysis is a tool to understand the properties of physical quantities independent of the units used to measure them. Every physical quantity is some combination of mass, length, time, electric charge, and temperature, . For example, speed, which may be measured in meters per second (m/s), miles per hour (mi/ of energy In physics, energy is a quantity that is often understood as the ability to perform work. This quantity can be assigned to any particle, object, or system of objects as a consequence of its physical state multiplied by time Time has been defined as the continuum in which events occur in succession from the past to the present and on to the future. Time has also been defined as a one-dimensional quantity used to sequence events, to quantify the durations of events and the intervals between them, and to quantify and measure the motions of objects and other changes, which are also the dimensions of action In physics, action is an attribute of the dynamics of a physical system. It is a functional which takes the trajectory of the system as its argument and returns a real number as the result. In SI units The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. It is the world's most widely used system of measurement, both in everyday commerce and in science, the Planck constant is expressed in joule seconds (J·s). The dimensions may also be written as momentum In classical mechanics, momentum is the product of the mass and velocity of an object (p = mv). In relativistic mechanics, this quantity is multiplied by the Lorentz factor. Momentum is sometimes referred to as linear momentum to distinguish it from the related subject of angular momentum. Linear momentum is a vector quantity, since it has a multiplied by distance Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance. A metric is a function that behaves according to a specific (N The newton is the SI derived unit of force, named after Isaac Newton in recognition of his work on classical mechanics·m The metre , symbol m, is the base unit of length in the International System of Units (SI). Originally intended to be one ten-millionth of the distance from the Earth's equator to the North Pole, its definition has been periodically refined to reflect growing knowledge of metrology. Since 1983, it is defined as the distance travelled by light in·s The second , sometimes abbreviated sec., is the name of a unit of time, and is the International System of Units (SI) base unit of time. It may be measured using a clock), which are also the dimensions of angular momentum.

The value of the Planck constant is:^[1]

The value of the reduced Planck constant is:

The two digits between the parentheses denote the standard uncertainty In metrology, measurement uncertainty is a non-negative parameter characterizing the dispersion of the values attributed to a measured quantity. The uncertainty has a probabilistic basis and reflects incomplete knowledge of the quantity. All measurements are subject to uncertainty and a measured value is only complete if it is accompanied by a in the last two digits of the value. The figures cited here are the 2006 CODATA The Committee on Data for Science and Technology was established in 1966 as an interdisciplinary committee of the International Council of Science (ICSU), formerly the International Council of Scientific Unions. It seeks to improve the compilation, critical evaluation, storage, and retrieval of data of importance to science and technology recommended values for the constants and their uncertainties. 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.

History

The Planck constant was first discovered as the proportionality constant In mathematics, two quantities are said to be proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio between the energy In physics, energy is a quantity that is often understood as the ability to perform work. This quantity can be assigned to any particle, object, or system of objects as a consequence of its physical state (E) of a photon In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force. The effects of this force are easily observable at both the microscopic and macroscopic level, because the photon and the frequency Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency. The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency. Loosely speaking, 1 year is the period of the Earth's orbit around the Sun, and the Earth's rotation on its axis has of its associated electromagnetic wave Electromagnetic radiation is a phenomenon that takes the form of self-propagating waves in a vacuum or in matter. It consists of electric and magnetic field components which oscillate in phase perpendicular to each other and perpendicular to the direction of energy propagation. Electromagnetic radiation is classified into several types according (ν). This relation between the energy and frequency is called the Planck relation or the Planck–Einstein equation:

Since the frequency Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency. The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency. Loosely speaking, 1 year is the period of the Earth's orbit around the Sun, and the Earth's rotation on its axis has ν, wavelength In physics, the wavelength of a sinusoidal wave is the spatial period of the wave – the distance over which the wave's shape repeats. It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a characteristic of both traveling waves and λ, and speed of light The speed of light, usually denoted by c, is a physical constant important in many areas of physics. Light and all other electromagnetic radiation always travel at this speed in empty space , regardless of the motion of the source or the inertial frame of the observer. Its value is exactly 299,792,458 metres per second (approximately 186,282 miles c are related by λν = c, the Planck relation can also be expressed as

Louis de Broglie Louis-Victor-Pierre-Raymond, 7th duc de Broglie, FRS (English pronunciation: /dəˈbrɔɪ/; French: [də bʁœj] ; 15 August 1892 – 19 March 1987) was a French physicist and a Nobel laureate. He was the sixteenth member elected to occupy seat 1 of the Académie française in 1944, and served as Perpetual Secretary of the Académie des sciences, extended the meaning of the Planck constant to be a constant of proportionality between the energy and the quantum wavelength of not just the photon, but any particle.

A closely related constant is the reduced Planck constant, sometimes called the Dirac constant. It is equal to the Planck constant divided by (or reduced by) 2π π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.141593 in the usual decimal notation. Many formulae from mathematics, science, and engineering involve π, which, and denoted ħ Ħ is a letter of the Latin alphabet, derived from H with the addition of a bar. It is used in Maltese for a voiceless pharyngeal fricative consonant (corresponding to the letter heth of Semitic abjads). Lowercase ħ is used in the International Phonetic Alphabet for the same sound ("h-bar"):

The reduced Planck constant is used when frequency is expressed in terms of radians The radian is the standard unit of angular measure, used in many areas of mathematics. It describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit. The SI per second ("angular frequency In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency (or angular speed) is the magnitude of the vector quantity angular velocity. The term angular frequency vector is sometimes used as a synonym for the vector quantity angular velocity") instead of cycles per second With the organisation of the International System of Units in 1960, the cycle per second was officially replaced by the hertz, or reciprocal second—i.e. the cycle in 'cycle per second' was dropped. Perhaps because of the convenient brevity it brings to both speech and writing, this particular mandate has been so widely adopted as to render the. The energy of a photon with angular frequency ω is given by

Planck conjectured (correctly, as it later turned out) that under certain conditions, energy could not take on any indiscriminate value: instead, the energy must be some multiple of a very small quantity (later to be named a "quantum In physics, a quantum is the minimum unit of any physical entity involved in an interaction. An example of an entity that is quantized is the energy transfer of elementary particles of matter (called fermions) and of photons and other bosons. The word comes from the Latin "quantus," for "how much." Behind this, one finds the"). This is counterintuitive in the everyday world, where it is possible to "make things a little bit hotter" or "move things a little bit faster", because the quanta of energy are very, very small in comparison to everyday human experience. Nevertheless, it is impossible, as Planck found out, to explain some phenomena without accepting that energy is quantized; that is, it exists only in integer The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as a subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, multiples of some base value.

Black-body radiation

In the last years of the nineteenth century, Planck was investigating the problem of black-body radiation In physics, a black body is an idealized object that absorbs all electromagnetic radiation falling on it. Blackbodies absorb and incandescently re-emit radiation in a characteristic, continuous spectrum. Because no light is reflected or transmitted, the object appears black when it is cold. However, a black body emits a temperature-dependent first posed by Kirchhoff Gustav Robert Kirchhoff was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects. He coined the term "black body" radiation in 1862, and two sets of independent concepts in both circuit theory and thermal emission are named & some forty years earlier. It is well known that hot objects glow, and that hotter objects glow brighter than cooler ones. The reason is that the electromagnetic field obeys laws of motion just like a mass on a spring, and can come to thermal equilibrium with hot atoms. When a hot object is in equilibrium with light, the amount of light it absorbs is equal to the amount of light it emits. If the object is black, meaning it absorbs all the light that hits it, then it emits the maximum amount of thermal light too.

The assumption that blackbody radiation is thermal leads to an accurate prediction: the total amount of emitted energy goes up with the temperature according to a definite rule, the Stefan–Boltzmann law (1879–84). But it was also known that the colour of the light given off by a hot object changes with the temperature, so that "white hot" is hotter than "red hot". Nevertheless, Wilhelm Wien Wilhelm Carl Werner Otto Fritz Franz Wien (13 January 1864 – 30 August 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to deduce Wien's displacement law, which calculates the emission of a blackbody at any temperature from the emission at any one reference temperature discovered the mathematical relationship between the peaks of the curves at different temperatures, by using the principle of adiabatic invariance. At each different temperature, the curve is moved over by Wien's displacement law Wien's displacement law states that the wavelength distribution of radiated heat energy from a black body at any temperature has essentially the same shape as the distribution at any other temperature, except that each wavelength is displaced, or moved over, on the graph. The average heat energy in each mode with frequency ν only depends on the (1893). Wien made a guess for the spectrum of the object, which was correct at low frequencies (large wavelength) but not at high frequencies (low wavelength). It still wasn't clear why the spectrum of a hot object had the form that it has (see diagram).

Planck hypothesized that the equations of motion for light are a set of harmonic oscillators If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency, one for each possible frequency. He examined how the entropy Entropy is a macroscopic property of a system that is a measure of the microscopic disorder within the system. It is an important part of the second law of thermodynamics. Thermodynamic systems are made up of microscopic objects, e.g. atoms or molecules, which "carry" energy. According to the second law of thermodynamics, the of the oscillators varied with the temperature of the body, trying to match Wien's law, and was able to derive an approximate mathematical function for black-body spectrum.^[2]

However, 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,^[2]. To save his theory, Planck had to resort to using the then controversial theory of statistical mechanics Statistical mechanics is the application of probability theory (which contains mathematical tools for dealing with large populations) to study the thermodynamic behavior of systems of a large number of particles. It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties,^[2] which he described as "an act of despair … I was ready to sacrifice any of my previous convictions about physics."^[3] One of his new boundary conditions was

to interpret U_N [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; —^[2]

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,^[4] but one which would revolutionize physics. Applying this new approach to Wien's displacement law showed that the "energy element" must be proportional to the frequency of the oscillator, the first version of what is now termed "Planck's relation":

Planck was able to calculate the value of h from experimental data on black-body radiation: his result, 6.55 × 10⁻³⁴ J·s, is within 1.2% of the currently accepted value.^[2] He was also able to make the first determination of the Boltzmann constant The Boltzmann constant is the physical constant relating energy at the particle level with temperature observed at the bulk level. It is the gas constant R divided by the Avogadro constant NA: k_B from the same data and theory.^[5]

Prior to Planck's work, it had been assumed that the energy of a body could take on any value whatsoever – that it was a continuous variable In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". An intuitive though imprecise idea of continuity is. This is equivalent to saying that the energy element ε (the difference between allowed values of the energy) is zero, and therefore that h is zero. This is the origin of the often-quoted summary that "the Planck constant is zero in classical physics" or that "classical physics is quantum mechanics at the limit that the Planck constant tends to zero". The Planck constant, of course, is never zero, but it is so small compared to most human experience that its existence had been ignored prior to Planck's work.

The black-body problem was revisited in 1905, when Rayleigh John William Strutt, 3rd Baron Rayleigh OM was an English physicist who, with William Ramsay, discovered the element argon, an achievement for which he earned the Nobel Prize for Physics in 1904. He also discovered the phenomenon now called Rayleigh scattering, explaining why the sky is blue, and predicted the existence of the surface waves now and Jeans (on the one hand) and Einstein (on the other hand) independently proved that classical electromagnetism could never account for the observed spectrum. These proofs are commonly known as the "ultraviolet catastrophe The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power", a name coined by Paul Ehrenfest in 1911. They contributed greatly (along with Einstein's work on the photoelectric effect) to convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The very first Solvay Conference in 1911 was devoted to "the theory of radiation and quanta".^[6] Max Planck received the 1918 Nobel Prize in Physics The Nobel Prize in Physics is awarded once a year by the Royal Swedish Academy of Sciences. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895 and awarded since 1901; the others are the Nobel Prize in chemistry, Nobel Prize in literature, Nobel Peace Prize, and Nobel Prize in physiology or medicine. The first Nobel "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

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