Planck relation

ThePlanck relation^[1]^[2]^[3](referred to asPlanck's energy–frequency relation,^[4]thePlanck–Einstein relation,^[5]Planck equation,^[6]andPlanck formula,^[7]though the latter might also refer toPlanck's law^[8]^[9]) is a fundamental equation inquantum mechanicswhich states that theenergy $E$ of aphoton,known asphoton energy,is proportional to itsfrequency $ν$ : $E=h\nu.$ Theconstant of proportionality, $h$ ,is known as thePlanck constant.Several equivalent forms of the relation exist, including in terms ofangular frequency $ω$ : $E=\hbar \omega,$ where $\hbar =h/2\pi$ .Written using the symbol $f$ for frequency, the relation is $E=hf.$

The relation accounts for thequantized nature of lightand plays a key role in understanding phenomena such as thephotoelectric effectandblack-body radiation(where the relatedPlanck postulatecan be used to derivePlanck's law).

Spectral forms

Light can be characterized using severalspectralquantities, such asfrequency $ν$ ,wavelength $λ$ ,wavenumber ${\tilde {\nu }}$ ,and their angular equivalents (angular frequency $ω$ ,angular wavelength $y$ ,andangular 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 thePlanck constant $h$ .The angular forms make use of thereduced Planck constant $ħ=.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠h/2π⁠$ .Here $c$ is thespeed of light.

de Broglie relation

The de Broglie relation,^[10]^[11]^[12]also known as de Broglie's momentum–wavelength relation,^[4]generalizes the Planck relation tomatter waves.Louis de Broglieargued that ifparticles 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 $p=h{\tilde {\nu }}$ or $p=\hbar k.$

The de Broglie relation is also often encountered invectorform $\mathbf {p} =\hbar \mathbf {k},$ where $p$ is the momentum vector, and $k$ is theangular wave vector.

Bohr's frequency condition

Bohr's frequency condition^[13]states that the frequency of a photon absorbed or emitted during anelectronic transitionis related to the energy difference ( $Δ E$ ) between the twoenergy levelsinvolved in the transition:^[14] $\Delta E=h\nu.$

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

References

^French & Taylor (1978), pp. 24, 55.
^Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11.
^Kalckar, J., ed. (1985), "Introduction",N. Bohr:Collected Works. Volume 6: Foundations of Quantum Physics $I$ ,(1926–1932),vol. 6, Amsterdam: North-Holland Publ., pp. 7–51,ISBN 0 444 86712 0^: 39
^^a ^bSchwinger (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.
^Weinberg (1995), p. 3.
^Messiah (1958/1961), p. 14.
^Cohen-Tannoudji, Diu & Laloë (1973/1977), p. 27.
^Flowers et al. (n.d), 6.2 The Bohr Model
^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 byB.-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.

[FT_24-1] French & Taylor (1978), pp. 24, 55.

[CT_10-2] Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11.

[3] Kalckar, J., ed. (1985), "Introduction",N. Bohr:Collected Works. Volume 6: Foundations of Quantum Physics $I$ ,(1926–1932),vol. 6, Amsterdam: North-Holland Publ., pp. 7–51,ISBN 0 444 86712 0^: 39

[Schwinger_203-4] Schwinger (2001), p. 203.

[5] Landsberg (1978), p. 199.

[6] Landé (1951), p. 12.

[7] Griffiths, D. J. (1995), pp. 143, 216.

[8] Griffiths, D. J. (1995), pp. 217, 312.

[9] Weinberg (2013), pp. 24, 28, 31.

[Weinberg_3-10] Weinberg (1995), p. 3.

[11] Messiah (1958/1961), p. 14.

[12] Cohen-Tannoudji, Diu & Laloë (1973/1977), p. 27.

[13] Flowers et al. (n.d), 6.2 The Bohr Model

[14] van der Waerden (1967), p. 5.

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