Constanta Planck

Constanta Planck (notată cu h) este o constantă fizică folosită pentru a descrie dimensiunile cuantelor. Joacă un rol central în teoria mecanicii cuantice, şi îşi trage numele de la Max Planck, unul din fondatorii teoriei cuantice. O cantitate strâns legată de aceasta este constanta Planck redusă (cunoscută şi sub numele de Constanta lui Dirac şi notată cu ħ, pronunţat "h-bar"). Constanta Planck se foloseşte şi la măsurarea energiei emise de fotoni, ca în ecuaţia E=h $\nu$ , unde E este energia, h este constanta lui Planck, iar $\nu$ este frecvenţa.

Constanta Planck şi constanta Planck redusă sunt utilizate pentru a descrie cuantificarea, un fenomen ce are loc la nivelul particulelor subatomice, cum ar fi electronul şi fotonul unde anumite proprietăţi fizice apar în cantităţi fixate, şi nu pot lua valori dintr-un spectru continuu.

Unităţi, valoare şi simboluri

Constanta Planck are dimensiune de energie înmulţită cu timp, care sunt şi dimensiunile acţiunii. În Sistemul Internaţional de Unităţi, constanta Planck este exprimată în joule-secunde. Dimensiunea constantei poate fi scrisă impuls ori distanţă (N·m·s), care sunt dimensiunile impulsului unghiular. Adesea, unitatea aleasă este eV·s, datorită energiilor mici adesea întâlnite în fizica cuantică.

Valoarea constantei Planck constant este:

h=\,\,\,6.626\ 068\ 96(33)\times 10^{-34}\ {\mbox{J}}\cdot {\mbox{s}}\,\,\,=\,\,\,4.135\ 667\ 33(10)\times 10^{-15}\ {\mbox{eV}}\cdot {\mbox{s}}

Cele două cifre dintre paranteze reprezintă incertitudinea standard a ultimelor cifre ale valorii.

Valoarea constantei Dirac este:

\hbar \ \equiv {\frac {h}{2\pi }}=\,\,\,1.054\ 571\ 628(53)\times 10^{-34}\ {\mbox{J}}\cdot {\mbox{s}}\,\,\,=\,\,\,6.582\ 118\ 99(16)\times 10^{-16}\ {\mbox{eV}}\cdot {\mbox{s}}

Cifrele citate aici sunt valorile recomandate de CODATA în 2006 pentru constante şi incertitudinile lor. Rezultatele CODATA din 2006 au fost publicate în Martie 2007 şi reprezintă cele mai bine cunoscute, internaţional acceptate valori pentru aceste constante, bazate pe toate datele disponibile la data de 31 Decembrie 2006. Noi cifre CODATA sunt publicate la intervale de patru ani.

Unicode rezervă poziţiile U+210E (ℎ) pentru constanta Planck, şi U+210F (ℏ) pentru constanta Dirac.

Originile constantei Planck

The Planck constant, $h\$ , was proposed in reference to the problem of black-body radiation. The underlying assumption to Planck's law of black body radiation was that the electromagnetic radiation emitted by a black body could be modeled as a set of harmonic oscillators with quantized energy of the form:

E=h\nu =h\omega /(2\pi )=\hbar \omega \

$E\$ is the quantized energy of the photons of radiation having frequency (Hz) of $\nu \$ (nu) or angular frequency (rad/s) of $\omega \$ (omega).

This model proved extremely accurate, but it provided an intellectual stumbling block for theoreticians who did not understand where the quantization of energy arose — Planck himself only considered it "a purely formal assumption". This line of questioning helped lead to the formation of quantum mechanics.

In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental corner-stones to the entire theory lies in the commutator relationship between the position operator ${\hat {x}}$ and the momentum operator ${\hat {p}}$ :

[{\hat {p_{i}}},{\hat {x_{j}}}]=-i\hbar \delta _{ij}

where $\delta _{ij}$ is the Kronecker delta. For more information, see the mathematical formulation of quantum mechanics.

Folosire

Constanta lui Planck este folosită pentru a descrie cuantificarea. For instance, the energy (E) carried by a beam of light with constant frequency (ν) can only take on the values

E=nh\nu \,,\quad n\in \mathbb {N} .

It is sometimes more convenient to use the angular frequency $\omega =2\pi \,\nu$ , which gives

E=n\hbar \omega \,,\quad n\in \mathbb {N} .

Many such "quantization conditions" exist. A particularly interesting condition governs the quantization of angular momentum. Let J be the total angular momentum of a system with rotational invariance, and J_z the angular momentum measured along any given direction. These quantities can only take on the values

{\begin{aligned}J^{2}=j(j+1)\hbar ^{2},\quad &j=0,1/2,1,3/2,\ldots ,\\J_{z}=m\hbar ,\qquad \quad \quad &m=-j,-j+1,\ldots ,j.\end{aligned}}

Thus, $\hbar$ may be said to be the "quantum of angular momentum".

The Planck constant also occurs in statements of Heisenberg's uncertainty principle. Given a large number of particles prepared in the same state, the uncertainty in their position, $\Delta x$ , and the uncertainty in their momentum (in the same direction), $\Delta p$ , obey

\Delta x\,\Delta p\geq {\begin{matrix}{\frac {1}{2}}\end{matrix}}\hbar

where the uncertainty is given as the standard deviation of the measured value from its expected value.

There are a number of other such pairs of physically measurable values which obey a similar rule.

Constanta Dirac

The Dirac constant or the "reduced Planck constant", $\hbar ={\frac {h}{2\pi }}\$ , differs only from the Planck constant by a factor of $2\pi$ . The Planck constant is stated in SI units of measurement, joules per hertz, or joules per (cycle per second), while the Dirac constant is the same value stated in joules per (radian per second). Both constants are conversion factors between energy units and frequency units.

In essence, the Dirac constant is a conversion factor between phase (in radians) and action (in joule-seconds) as seen in the Schrödinger equation. All other uses of Planck's constant and Dirac's constant follow from that.

Semnificaţia dimensiunii constantei Planck

Expressed in the SI units of J·s, the Planck constant is one of the smallest constants used in physics. The significance of this is that it reflects the extremely small scales at which quantum mechanical effects are observed, and hence why we are not familiar with quantum physics in our everyday lives in the way that we are with classical physics. Indeed, classical physics can essentially be defined as the limit of quantum mechanics as the Planck constant tends to zero. However, in the natural units describing physics at the atomic scale, the Planck constant is taken as 1, reflecting the fact that physics at the atomic scale is dominated by quantum effects.

Bibliografie

NIST link to CODATA value
Barrow, John D. (2002). The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe (în English). Pantheon Books. ISBN 0-375-42221-8.

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