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 microscopic particles such as electrons and photons in which certain physical properties occur in fixed amounts rather than assuming a continuous range of possible values.

Contents 1 Units, value and symbols 2 Usage 3 See also 4 Reference

Units, value and symbols

Planck's constant has units of energy (joules or J) multiplied by time (seconds or s), which are the units of action (joule seconds or J·s). These units may also be written as momentum times distance (newton·metre·second or N·m·s), which are the units of angular momentum.

The value of Planck's constant, in joule seconds, is:

<math>h=6.626\ 0693(11) \times10^{-34}\ \mbox{J}\cdot\mbox{s}</math>

or, with electronvolts (eV) as the unit of energy:

<math>h=4.135\ 667\ 43(35) \times10^{-15}\ \mbox{eV}\cdot\mbox{s}.</math>

The value of the reduced Planck's constant, in joule seconds, is: Paul Dirac):

<math>\hbar\equiv\frac{h}{2\pi}=1.054\ 571\ 68(18)\times10^{-34}\ \mbox{J}\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 at 31 December 2002. New CODATA figures are scheduled to be published approximately every four years.

On some browsers, the Unicode symbol &#8462; (ℎ) is rendered as Planck's constant, and the symbol &#8463; (ℏ) is rendered as Dirac's constant.

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.

See also

• Electromagnetic radiation
• Natural units
• Schrödinger equation
• Wave-particle duality
• Quantum Hall effect

Reference

• NIST link to CODATA value