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The Planck constant (denoted Template:Math, also called Planck's constant) is equal to 6,626,070,040.

First recognized in 1900 by Max Planck, it was conceived as the proportionality constant between the minimal increment of energy, Template:Math, of a hypothetical electrically charged oscillator in a cavity that contained black body radiation, and the frequency, Template:Math, of its associated electromagnetic wave. In 1905, the value Template:Math, the minimal energy increment of a hypothetical oscillator, was theoretically associated by Albert 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 photon.

The Planck–Einstein relation connects the particular photon energy Template:Math with its associated wave frequency Template:Math:

$E = hf$

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

Since the frequency Template:Math, wavelength Template:Math, and speed of light Template:Math are related by $f= \frac{c}{\lambda}$, the relation can also be expressed as

$E = \frac{hc}{\lambda} .$

The de Broglie wavelength Template:Math of the particle is given by

$\lambda = \frac{h}{p} .$

Where Template:Math denotes the linear momentum of a particle, such as a photon, or any other elementary particle.

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 Template:Math into the Planck constant. The resulting constant is called the reduced Planck constant. It is equal to the Planck constant divided by Template:Math, and is denoted Template:Math (pronounced "h-bar"):

$\hbar = \frac{h}{2 \pi} .$

The energy of a photon with angular frequency Template:Math is given by

$E = \hbar \omega ,$

while its linear momentum relates to

$p = \hbar k ,$

where k is an angular 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 Template:Math (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. This is the so-called "old quantum theory" developed by Bohr and Sommerfeld, in which particle trajectories exist but are hidden, but quantum laws constrain them based on their action. This view has been largely replaced by fully modern quantum theory, in which definite trajectories of motion do not even exist, rather, the particle is represented by a wavefunction spread out in space and in time. Thus there is no value of the action as classically defined. Related to this is the concept of energy quantization which existed in old quantum theory and also exists in altered form in modern quantum physics. Classical physics cannot explain either quantization of energy or the lack of a classical particle motion.

In many cases, such as for monochromatic light or for atoms, quantization of energy also implies that only certain energy levels are allowed, and values in between are forbidden.

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