What are the uses of Plancks constant

Planck's quantum of action


The Planck's quantum of actionH is a fundamental natural constant in quantum physics. It occurs when describing quantum phenomena, in which physical properties cannot assume any arbitrary continuous value, but only certain discrete values.

Planck’s quantum of action links particle and wave properties, it is the ratio of energy and frequency of a light quantum and the ratio between mass, speed and wavelength of any particle, which is substantially below the speed of light.

The value of Planck's quantum of action is

, [1]

and therefore has the dimension of energy times time, i.e. an effect or an angular momentum.

Often takes place H also the so-called "reduced Planck quantum of action" (pronounced "h-quer") used with:

, [1]

where π is the circle number (pi). In the past it was sometimes referred to as "Dirac's constant" after Paul Dirac[2].

The Unicode symbols are U + 210E (ℎ) for Planck's quantum of action and U + 210F (ℏ) for Planck's reduced quantum of action.

Light quanta

The energy E.electromagnetic radiation of a given frequency ν (Greek letter “ny”) can only be absorbed and emitted in certain portions. The energy of a field can therefore only change by the following amount:

Max Planck ran the constant H (of Auxiliary variable) in 1900 initially only as an aid to solving the problem of describing the radiation behavior of black bodies (also referred to as black body radiation or cavity radiation)[3]. According to the classical derivation (→ Rayleigh-Jeans law), the intensity should have increased with increasing frequency (which contradicts reality and is referred to as an ultraviolet catastrophe).

By considering entropy, Planck considered an additional term unknown at the time to be possible and obtained a radiation formula that correctly described the radiation values ​​that were already known. He therefore assumed that the formula he had found was correct and looked for an explanation. He noticed a certain similarity between the formula and the formula for velocity distribution in statistical gas theory. Therefore he assumed that radiation of frequency ν only occurs in energy packets of size E. = Hν can be emitted and absorbed. The quantum of action here is a constant of proportionality, the size of which results from the adaptation to the experimentally determined values ​​of the cavity radiation.

Planck initially considered the discontinuous character of the energy to be a consequence of the properties of the radiation source. Albert Einstein was the first to postulate the light quantum hypothesis in 1905, which states that quantization is a property of the radiation field regardless of the radiation source. The reason for this was the experimental results on the photoelectric effect.

Often the frequency ν is replaced by the angular frequency ω = 2πν. Then it will be E. = Hν to

Matter waves

In 1923, Louis de Broglie also ascribed wave properties to massive particles such as electrons. He linked the impulse with the wavelength λ: p = H / λ, or vectorial , with the wavenumber vector of the amount . The direction of corresponds to that of the particle, its matter wave describes.

Quantization of the angular momentum

Planck's motivation for the designation “quantum of effect” was initially solely the dimension of the constants. Only in the atomic model of the hydrogen atom established by Niels Bohr in 1913 did the action integral of an electron circling around the atomic nucleus appear as a quantized quantity over a closed circuit. From this quantization condition it follows that the angular momentum of the electron are only integer multiples of can accept. (In addition to the product of an energy with a time difference, the product of an impulse with a distance also has the dimension of an effect, and thus also the angular momentum.)

More detailed considerations of the amount of the orbital angular momentum every system in any inertial system later showed that, contrary to the outdated Bohr model of the atom, this is not an integral multiple of occurs. Rather, the relation is:

thus continues to appear as a constant of proportionality.

The orbital angular momentum quantum number l can be integer values ​​from 0 to n - Assume 1, where n is the principal quantum number. For the component of the angular momentum along any axis, it is true that its amount is an integral multiple of is. When the spin comes into play, the quantum numbers can also assume half-integer values.

Quantum mechanics

In the quantum mechanics developed in the 1920s, the quantum of action - originally introduced to solve a thermodynamic problem - has a general meaning. It occurs z. B. in the momentum operator and energy operator in the Schrödinger equation, the fundamental equation of this theory. Planck's quantum of action is the universal conversion factor in quantum physics between energies and (circular) frequencies, not only for photons, but also between wave numbers and pulses. It is a sensible view of quantum physics, energies with (circular) frequencies and pulses with wave numbers too identifyby identified with 1. This happens, for example, in atomic units or in Planck units, especially in high-energy physics. In abstract-mathematical terms: The energy-momentum-vector space is identified with the dual space of the Minkowski space-time.

Planck's quantum of action also appears in Heisenberg's uncertainty relation:

Sometimes will therefore viewed as the more fundamental constant.

In Heisenberg's commutation relation, Planck's quantum of action represents the constant in the commutator between the momentum and position operator:


  1. ab P. J. Mohr and B. N. Taylor: CODATA recommended values ​​of the fundamental physical constants: 2002. In: Rev. Mod. Phys. Vol. 77, No. 1, 2005, pp. 1-107.
  2. The one-character notation for the reduced Planck quantum of action was introduced in 1926 by P. A. M. Dirac. A short section on history can be found e.g. in M. Jammer, "The Conceptual Development of Quantum Mechanics", McGraw-Hill, New York (1966), 294. Dirac's original work: P. A. M. Dirac, "Quantum mechanics and a preliminary investigation of the hydrogen atom", Proc. Roy. Soc. A, 110: 561-579 (1926).
  3. M. Planck: "On the theory of the law of energy distribution in the normal spectrum", Negotiations of the German Physical Society 2 (1900) No. 17, pp. 237 - 245, Berlin (presented on December 14, 1900)


  • Domenico Giulini, “Long live the audacity!” - Albert Einstein and the foundation of quantum theory, online, in: Herbert Hunziker, The young Einstein and Aarau, Birkhäuser 2005, ISBN 3764374446

Web links, sources

  • CODATA Internationally recommended values ​​of the Fundamental Physical Constants at NIST
  • How Max Planck came up with the idea of ​​quantum theory - from Armin Hermann
  • What is the quantum of action? - by Dr. Jörg Resag

Category: quantum physics