Planck–Einstein relation

The Planck–Einstein relation^[1]^[2]^[3] is also referred to as the Einstein relation,^[1]^[4]^[5] Planck's energy–frequency relation,^[6] the Planck relation,^[7] and the Planck equation.^[8] Also the eponym 'Planck formula'^[9] belongs on this list, but also often refers instead to Planck's law^[10]^[11] 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, $E$ , known as photon energy, 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

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 {c}{2\pi y}}={\frac {ck}{2\pi }},

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

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

as well as the following 'angular' forms,

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

The standard forms make use of the Planck constant $h$ . The angular forms make use of the reduced Planck constant $ħ = h / 2π$ . Here $c$ is the speed of light.

de Broglie relation

The de Broglie relation,^[5]^[12]^[13] also known as the de Broglie's momentum–wavelength relation,^[6] generalizes the Planck relation to matter waves. Louis de Broglie argued that if particles had a wave nature, the relation $E = hν$ 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 }

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

Bohr's frequency condition states that the frequency of a photon absorbed or emitted during an electronic transition is related to the energy difference ( $Δ E$ ) between the two energy levels involved in the transition:^[14]

\Delta E=h\nu .

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

References

1 2 French & Taylor (1978), pp. 24, 55.
↑ Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11.
↑ Kalckar 1985, p. 39.
↑ Messiah (1958/1961), p. 72.
1 2 Weinberg (1995), p. 3.
1 2 Schwinger (2001), p. 203.
↑ Landsberg (1978), p. 199.
↑ Landé (1951), p. 12.
↑ Griffiths, D.J. (1995), pp. 143, 216.
↑ Griffiths, D.J. (1995), pp. 217, 312.
↑ Weinberg (2013), pp. 24, 28, 31.
↑ Messiah (1958/1961), p. 14.
↑ Cohen-Tannoudji, Diu & Laloë (1973/1977), p. 27.
↑ van der Waerden (1967), p. 5.

Cited bibliography

Cohen-Tannoudji, C., Diu, B., Laloë, F. (1973/1977). Quantum Mechanics, translated from the French by S.R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, ISBN 0471164321.
French, A.P., Taylor, E.F. (1978). An Introduction to Quantum Physics, Van Nostrand Reinhold, London, ISBN 0-442-30770-5.
Griffiths, D.J. (1995). Introduction to Quantum Mechanics, Prentice Hall, Upper Saddle River NJ, ISBN 0-13-124405-1.
Landé, A. (1951). Quantum Mechanics, Sir Isaac Pitman & Sons, London.
Landsberg, P.T. (1978). Thermodynamics and Statistical Mechanics, Oxford University Press, Oxford UK, ISBN 0-19-851142-6.
Messiah, A. (1958/1961). Quantum Mechanics, volume 1, translated from the French by G.M. Temmer, North-Holland, Amsterdam.
Schwinger, J. (2001). Quantum Mechanics: Symbolism of Atomic Measurements, edited by B.-G. Englert, Springer, Berlin, ISBN 3-540-41408-8.
van der Waerden, B.L. (1967). Sources of Quantum Mechanics, edited with a historical introduction by B.L. van der Waerden, North-Holland Publishing, Amsterdam.
Weinberg, S. (1995). The Quantum Theory of Fields, volume 1, Foundations, Cambridge University Press, Cambridge UK, ISBN 978-0-521-55001-7.
Weinberg, S. (2013). Lectures on Quantum Mechanics, Cambridge University Press, Cambridge UK, ISBN 978-1-107-02872-2.

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