Planck's constant

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Planck's constant (denoted h) is a physical constant that is used to describe the sizes of quanta. It plays a central role in the theory of quantum mechanics, and is named after Max Planck, one of the founders of quantum theory. A closely-related quantity is the reduced Planck constant (denoted <math>\hbar</math>, pronounced as "h-bar"), which is sometimes called Dirac's constant, after Paul Dirac.

Planck's constant and the reduced Planck's constant are used to describe quantization, a phenomenon occurring in subatomic particles such as electrons and photons in which certain physical properties occur in fixed amounts rather than assuming a continuous range of possible values.

1 Units, value and symbols
2 Origins of Planck's constant and Dirac's constant
3 Usage
4 See also
5 Reference
6 External links

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Units, value and symbols

Planck's constant has units of energy multiplied by time, which are the units of action (J·s). These units may also be written as momentum times distance (N·m·s), which are also the units of angular momentum. However, often the unit of choice is eV·s, because of the small energies that are affected by the uncertainty principle.

The value of Planck's constant is:

<math>h =\,\,\, 6.626\ 0693(11) \times10^{-34}\ \mbox{J}\cdot\mbox{s} \,\,\, = \,\,\, 4.135\ 667\ 43(35) \times10^{-15}\ \mbox{eV}\cdot\mbox{s}</math>

(The digits in brackets signify the uncertainty (standard deviation) in the last digits of the value).

The value of the reduced Planck's constant is:

<math>\hbar\equiv\frac{h}{2\pi} = \,\,\, 1.054\ 571\ 68(18)\times10^{-34}\ \mbox{J}\cdot\mbox{s} \,\,\, = \,\,\, 6.582\ 119\ 15(56) \times10^{-16}\ \mbox{eV}\cdot\mbox{s}</math>

where π is the constant pi (3.141…).

The figures cited here are the 2002 CODATA-recommended values for the constants and their uncertainties. The 2002 CODATA results were made available in December 2003 and represent the best-known, internationally-accepted values for these constants, based on all data available as of 31 December 2002. New CODATA figures are scheduled to be published approximately every four years.

On some browsers, the Unicode symbol (ℎ) is rendered as Planck's constant, and the symbol (ℏ) is rendered as Dirac's constant.

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Origins of Planck's constant and Dirac's constant

Planck's constant, <math> h \ </math>, 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:

<math> E = h f = \hbar \omega \ </math>

<math> E \ </math> is the quantized energy of the photons of radiation having frequency (Hz) of <math> f \ </math> or angular frequency of <math> \omega \ </math> (radian/sec).

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.

Dirac's constant or the "reduced Planck's constant", <math> \hbar = \frac{h}{2 \pi} \ </math>, differs only from Planck's constant by a factor of <math>2 \pi \ </math>. The SI unit of measurement of Planck's constant is joule per hertz, or joule per (turn per second), while the unit of measurement of Dirac's constant is joule per (radian per second). The two constants are merely conversion factors between energy units and frequency units.

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 <math>\hat{x}</math> and the momentum operator <math>\hat{p}</math>:

<math>[\hat{p_i}, \hat{x_j}] = -i \hbar \delta_{ij}</math>

where <math>\delta_{ij}</math> is the Kronecker delta. For more information, see the mathematical formulation of quantum mechanics.

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Usage

Planck's constant is used to describe quantization. For instance, the energy (E)carried by a beam of light with constant frequency (ν) can only take on the values

<math>E = n h \nu \,,\quad n\in\mathbb{N}</math>

It is sometimes more convenient to use the angular frequency <math>\omega=2\pi\,\nu</math>, which gives

<math>E = n \hbar \omega \,,\quad n\in\mathbb{N}</math>

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

<math>\begin{matrix}

J^2 = j(j+1) \hbar^2, & j = 0, 1/2, 1, 3/2, \ldots \\ J_z = m \hbar, \qquad\quad & m = -j, -j+1, \ldots, j\end{matrix}</math>

Thus, <math>\hbar</math> may be said to be the "quantum of angular momentum".

Planck's constant also occurs in statements of Heisenberg's uncertainty principle. The uncertainty (more precisely: the standard deviation) in any position measurement, <math>\Delta x</math>, and the uncertainty in a momentum measurement along the same direction, <math>\Delta p</math>, obeys

<math> \Delta x \Delta p \ge \begin{matrix}\frac{1}{2}\end{matrix} \hbar</math>

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

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