Planck–Einstein relation

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The Planck–Einstein relation[1][2] is also referred to as the Einstein relation,[1][3][4] Planck's energy–frequency relation,[5] the Planck relation,[6] and the Planck equation.[7] Also the eponym 'Planck formula'[8] belongs on this list, but also often refers instead to Planck's law[9][10] These various eponyms are far from standard; they are used only sporadically, neither regularly nor very widely. They refer to a formula integral to quantum mechanics, which states that the energy of a photon, E, is proportional to its frequency, ν:

E = h \nu

The constant of proportionality, h, is known as the Planck constant. Several equivalent forms of the relation exist.

The relation accounts for quantized nature of light, and plays a key role in understanding phenomena such as the photoelectric effect, and Planck's law of black body radiation. See also the Planck postulate.

Spectral forms[edit]

Light can be characterized using several spectral quantities, such as frequency ν, wavelength λ, wavenumber \scriptstyle\tilde{\nu}, and their angular equivalents (angular frequency ω, angular wavelength y, and angular wavenumber k). These quantities are related through

\nu = \frac{c}{\lambda} = c \tilde \nu = \frac{\omega}{2 \pi} = \frac{2 \pi c}{y} = \frac{ck}{2 \pi},

so the Planck relation can take the following 'standard' forms

E = h \nu = \frac{hc}{\lambda} = h c \tilde \nu,

as well as the following 'angular' forms,

E = \hbar \omega = \frac{\hbar c}{y} = \hbar c k.

The standard forms make use of the Planck constant h. The angular forms make use of the reduced Planck constant ħ = h/. Here c is the speed of light.

de Broglie relation[edit]

The de Broglie relation,[4][11][12] also known as the de Broglie's momentum–wavelength relation,[5] generalizes the Planck relation to matter waves. Louis de Broglie argued that if particles had a wave nature, the relation E = would also apply to them, and postulated that particles would have a wavelength equal to λ = h/p. Combining de Broglie's postulate with the Planck–Einstein relation leads to

p = h \tilde \nu or
p = \hbar k.

The de Broglie's relation is also often encountered in vector form

\mathbf{p} = \hbar \mathbf{k},

where p is the momentum vector, and k is the angular wave vector.

Bohr's frequency condition[edit]

Bohr's frequency condition states that the frequency of a photon absorbed or emitted during an electronic transition is related to the energy difference (ΔE) between the two energy levels involved in the transition:[13]

\Delta E = h \nu.

This is a direct consequence of the Planck–Einstein relation.

References[edit]

  1. ^ a b French & Taylor (1978), pp. 24, 55.
  2. ^ Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11.
  3. ^ Messiah (1958/1961), p. 72.
  4. ^ a b Weinberg (1995), p. 3.
  5. ^ a b Schwinger (2001), p. 203.
  6. ^ Landsberg (1978), p. 199.
  7. ^ Landé (1951), p. 12.
  8. ^ Griffiths, D.J. (1995), pp. 143, 216.
  9. ^ Griffiths, D.J. (1995), pp. 217, 312.
  10. ^ Weinberg (2013), pp. 24, 28, 31.
  11. ^ Messiah (1958/1961), p. 14.
  12. ^ Cohen-Tannoudji, Diu & Laloë (1973/1977), p. 27.
  13. ^ van der Waerden (1967), p. 5.

Cited bibliography[edit]