Planck quantum of action
the Planck quantum of action h is a fundamental natural constant of the physics, which is used for the description of the values of quantized sizes. It is of fundamental importance in quantum physics. The value of the Planck quantum of action amounts to about
- <math>h = 6 {,} 62607 \ 10^ cdot {- 34} \, \ rm {J \, s} = 4 {,} 13567 \ 10^ cdot {- 15} \ rm {eVs},< /math>
and therefore the dimension of energy has times time, thus an effect.
Becomes frequent instead of <math> h </math> also the size <math> \ hbar< /math> (speak „h-crosswise”) uses with:
- <math> \ hbar = \ frac {h} {2 \ pi} = 1 {,} 054572\ 10^ cdot {- 34} \, \ rm {J \, s},< /math>
whereby <math> \ pi< /math> the circle number (pi) is. <math> \ hbar< /math> sometimes also after Paul Dirac Dirac constant one calls.
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meaning the Planck quantum of action during the quantized description of phenomena. In particular micro-cosmic objects such as elementary particles (e.g. Electrons, photons and also different) have physical characteristics, those recognizably not anycontinuous value, but only certain discrete values to assume know.
porportionality factor between photon energy and frequency
the energy E of electromagnetic radiation of a given frequency <math> \ nu< /math> (gr. Letter “ny “) can be absorbed and emitted only in certain portions. The energy of a field can change thus only around the following amount:
- <math> \ Delta E = h \, \ nu </math>
Max Planck led the constant of h (of auxiliary variable) in the year 1900 first only as aid to the solution of the problem of the description of the radiation behavior of black bodies (also callsBlack body radiation or black body radiation). After the classical derivative (→ Rayleigh Jeans law) the intensity with rising frequency would have had to continue to increase (which contradicts the reality and ultraviolet disaster one calls).
Planck considered a then unknown auxiliary term by views to the entropy possible and receiveda radiation formula, which those already admitted of radiation values described correctly. It assumed therefore the fact that the found formula is correct and looked for an explanation. A certain similarity of the formula with the formula of the distribution of velocity in the statistic gas theory was noticeable to it. Therefore it tookon that radiation of the frequency <math> \ nu< /math> only in energy packages of the size <math> E = h \ nu< /math> to be emitted and absorbed can. The quantum of action is here a proportionality constant, whose size results from the adjustment at experimentally determined values the black body radiation.
Planck held the not-continuous characterthe energy first for a consequence of the characteristic of the radiation source. Only Albert Einstein postulated 1905 the light quantum hypothesis, which means that quantization is independent of the radiation source a characteristic of the radiation field. Cause for it were the experimental results to the photoelectric effect.
Frequently one replacesthe frequency <math> \ nu< /math> by the rotative frequency <math> \ omega=2 \ pi \ nu< /math>. Then math <E=h> becomes \ nu< /math> too
- <math> E= \ hbar \ omega< /math>
porportionality factor between angular momentum quantum number and angular momentum
of Planck motivation for the designation “quantum of action” was first alone the dimension of the constants. Only in that for 1913 of Niels bore set up Atom model of the hydrogen atom went the action integral of an electron circling around the atomic nucleus over a closed circulation as quantized size into action. From this quantization condition it follows that the angular momentum< math> \ vec L< /math> the electron only integral multiple of <math> \ hbar< /math> to assume can. (Beside the product of an energy alsoa time difference has also the product of an impulse with a distance the dimension of an effect, and concomitantly the angular momentum.)
more exact views of the amount of the angular momentum <math> \ vec {L}< /math> each system in any inertial system it resulted in later that these against the outdated Bohr atom model not asintegral multiple of <math> \ hbar< /math> arises. The relation reads rather:
- <math> |\ vec {L}| = \ {l (l+1) sqrt} \ hbar< /math>
<math> \ hbar< /math> appears thus further as proportionality constant.
The angular momentum quantum number <math> l< /math> knows integral values from 0 to <math> n-1< /math> assume, whereby <math> n< /math> the principal quantum number is. For the component of the angular momentum along anyAxle applies however that their amount an integral multiple of <math> \ hbar< /math> is. If the spin comes into the play, the quantum numbers can take also half-integral values.
porportionality factor between impulse and wave number vector
in the year 1924 wrote Louis de Broglie to also massive particles such as electrons wave characteristicstoo. It linked the impulse <math> \ vec p< /math> with the wavelength <math> \ lambda< /math>: <math> p = h \ lambda< /math>, and/or vectorially <math> \ vec p = \ hbar \, \ vec k< /math>, with the wave number vector <math> \ vec k< /math> of the amount <math>|\ vec k|=2 \ pi \ lambda< /math>. The direction of <math> \ vec k< /math> that of the particle corresponds, to its subject wave <math> \ vec k< /math> describes.
general meaning in quantum mechanics
in quantum mechanics developed into the 1920er years comes - originally to the solution of a thermodynamic problem imported - the quantum of action a general meaning. It steps z. B. in the impulse operator and energy operator in the Schroedinger equation, the fundamentalEquation of this theory, up. The Planck quantum of action <math> \ hbar< /math> the universal conversion factor is in quantum physics between energies and (circle) frequencies, not only for photons, as well as between Wellenzahlen and impulses. It is a meaningful aspect for quantum physics, energies with (circle) frequencies and impulses with Wellenzahlen too identify, by one <math> \ hbar< /math> with 1 identifies. That happens for example in atomic units or in Planck units, in particular in high-energy physics. Abstract-mathematically said: The energy impulse vector space is identified with the dual area of the Minkowski area time.
Heisenberg uncertainty relation and Heisenberg commutation relation
In the year 1927 the Planck' quantum of action arose also in the Heisenberg uncertainty relation :
- <math> \ delta x \ cdot \ delta p \ ge \ frac {h} {4 \ pi} = \ frac {\ hbar} {2}< /math>
Sometimes math <\> hbar /math< becomes> therefore as the more fundamental constant outstandingly.
In the Heisenberg commutation relation the Planck quantum of action places the constant in the commutator between impulse- and local operator :
- <math> \ left [\ {P has}, \ has {Q} \ right] = \ frac {\ hbar} {i} \ has {\ mathbf {1}}< /math>
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