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The Planck constant (denoted , also called Planck's constant) is a physical constant that is the quantum of action, central in quantum mechanics.
First recognized in 1900 by Max Planck, it was originally the proportionality constant between the minimal increment of energy, , of a hypothetical electrically charged oscillator in a cavity that contained black body radiation, and the frequency, , of its associated electromagnetic wave. In 1905, the value , the minimal energy increment of a hypothetical oscillator, was theoretically associated by Einstein with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle, as opposed to an electromagnetic wave. It was eventually called the photon.
The Planck–Einstein relation connects the particulate photon energy with its associated wave frequency :
E = hf
This energy is extremely small in terms of ordinarily perceived everyday objects.
Since the frequency , wavelength , and speed of light are related by $f= \frac{c}{\lambda}$, the relation can also be expressed as
$$E = \frac{hc}{\lambda} .$$
This leads to another relationship involving the Planck constant. With denoting the linear momentum of a particle (not only a photon, but other fine particles as well), the de Broglie wavelength of the particle is given by
$$\lambda = \frac{h}{p} .$$
In applications where it is natural to use the angular frequency (i.e. where the frequency is expressed in terms of radians per second instead of cycles per second or hertz) it is often useful to absorb a factor of into the Planck constant. The resulting constant is called the reduced Planck constant or Dirac constant. It is equal to the Planck constant divided by , and is denoted (pronounced "h-bar"):
$$\hbar = \frac{h}{2 \pi} .$$
The energy of a photon with angular frequency , where , is given by
E = ℏω,
while its linear momentum relates to
p = ℏk,
where k is a wavenumber. In 1923, Louis de Broglie generalized the Planck–Einstein relation by postulating that the Planck constant represents the proportionality between the momentum and the quantum wavelength of not just the photon, but the quantum wavelength of any particle. This was confirmed by experiments soon afterwards. This holds throughout quantum theory, including electrodynamics.
These two relations are the temporal and spatial component parts of the special relativistic expression using 4-Vectors.
$$P^\mu = \left(\frac{E}{c}, \vec{p}\right) = \hbar K^\mu = \hbar\left(\frac{\omega}{c}, \vec{k}\right)$$
Classical statistical mechanics requires the existence of (but does not define its value). Eventually, following upon Planck's discovery, it was recognized that physical action cannot take on an arbitrary value. Instead, it must be some multiple of a very small quantity, the " quantum of action", now called the Planck constant. Classical physics cannot explain this fact. In many cases, such as for monochromatic light or for atoms, this quantum of action also implies that only certain energy levels are allowed, and values in between are forbidden.