Planck relation

<h2 id="spectral-forms">Spectral forms</h2>
Light can be characterized using several <a href="/facts/Spectrum/wPUjkpqI">spectral</a> quantities, such as <a href="/facts/Frequency/QUSbPaW8">frequency</a> ν, <a href="/facts/Wavelength/kcJU0ymz">wavelength</a> λ, <a href="/facts/Wavenumber/cLaLAeW5">wavenumber</a> 
 
 
 
 
 
 
 ν
 ~
 
 
 
 
 
 {\displaystyle {\tilde {\nu }}}
 
, and their angular equivalents (<a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> ω, <a href="/facts/Angular_wavelength/kcJU0ymz">angular wavelength</a> y, and <a href="/facts/Angular_wavenumber/cLaLAeW5">angular wavenumber</a> k). These quantities are related through

ν
        =
        
          
            c
            λ
          
        
        =
        c
        
          
            
              ν
              ~
            
          
        
        =
        
          
            ω
            
              2
              π
            
          
        
        =
        
          
            c
            
              2
              π
              y
            
          
        
        =
        
          
            
              c
              k
            
            
              2
              π
            
          
        
        ,
      
    
    {\displaystyle \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
        ν
        =
        
          
            
              h
              c
            
            λ
          
        
        =
        h
        c
        
          
            
              ν
              ~
            
          
        
        ,
      
    
    {\displaystyle E=h\nu ={\frac {hc}{\lambda }}=hc{\tilde {\nu }},}

as well as the following "angular" forms:

E
        =
        ℏ
        ω
        =
        
          
            
              ℏ
              c
            
            y
          
        
        =
        ℏ
        c
        k
        .
      
    
    {\displaystyle E=\hbar \omega ={\frac {\hbar c}{y}}=\hbar ck.}

The standard forms make use of the <a href="/facts/Planck_constant/qGHAe3qq">Planck constant</a> h. The angular forms make use of the <a href="/facts/Reduced_Planck_constant/qGHAe3qq">reduced Planck constant</a> ħ = ⁠h/2π⁠. Here c is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a>.

<h2 id="de-broglie-relation">de Broglie relation</h2>
See also: <a href="/facts/Matter_wave/Rpszb8f1">Matter wave § de Broglie relations</a>
The de Broglie relation,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> also known as de Broglie's momentum–wavelength relation,<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> generalizes the Planck relation to <a href="/facts/Matter_wave/Rpszb8f1">matter waves</a>. <a href="/facts/Louis_de_Broglie/v696wFwH">Louis de Broglie</a> argued that if <a href="/facts/Wave%25E2%2580%2593particle_duality/Au3UlcwQ">particles had a wave nature</a>, 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
        
          
            
              ν
              ~
            
          
        
      
    
    {\displaystyle p=h{\tilde {\nu }}}
  
 or 
  
    
      
        p
        =
        ℏ
        k
        .
      
    
    {\displaystyle p=\hbar k.}

The de Broglie relation is also often encountered in <a href="/facts/Vector_(physics)/U8mM7A5F">vector</a> form

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

where p is the momentum vector, and k is the <a href="/facts/Angular_wave_vector/Jk8wUEft">angular wave vector</a>.

<h2 id="bohrs-frequency-condition">Bohr's frequency condition</h2>
Bohr's frequency condition<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> states that the frequency of a photon absorbed or emitted during an <a href="/facts/Electronic_transition/zboUt1QR">electronic transition</a> is related to the energy difference (ΔE) between the two <a href="/facts/Energy_level/66x51Khc">energy levels</a> involved in the transition:<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

Δ
        E
        =
        h
        ν
        .
      
    
    {\displaystyle \Delta E=h\nu .}

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

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Compton_wavelength/og2h1J2q">Compton wavelength</a></li></ul>

<h2 id="cited-bibliography">Cited bibliography</h2>
<ul><li><a href="/facts/Claude_Cohen-Tannoudji/fjs27Pvx">Cohen-Tannoudji, C.</a>, 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, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0471164321.</li>
<li><a href="/facts/Anthony_French/Lr9JYIWb">French, A.P.</a>, <a href="/facts/Edwin_F._Taylor/WRCBubeB">Taylor, E.F.</a> (1978). An Introduction to Quantum Physics, Van Nostrand Reinhold, London, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-442-30770-5.</li>
<li>Griffiths, D.J. (1995). Introduction to Quantum Mechanics, Prentice Hall, Upper Saddle River NJ, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-13-124405-1.</li>
<li><a href="/facts/Alfred_Land%25C3%25A9/axkIBIx8">Landé, A.</a> (1951). Quantum Mechanics, Sir Isaac Pitman & Sons, London.</li>
<li>Landsberg, P.T. (1978). Thermodynamics and Statistical Mechanics, Oxford University Press, Oxford UK, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-851142-6.</li>
<li><a href="/facts/Albert_Messiah/XxCR7zw4">Messiah, A.</a> (1958/1961). <a href="https://archive.org/details/QuantumMechanicsVolumeI">Quantum Mechanics</a>, volume 1, translated from the French by G.M. Temmer, North-Holland, Amsterdam.</li>
<li><a href="/facts/Julian_Schwinger/BdepoI1r">Schwinger, J.</a> (2001). Quantum Mechanics: Symbolism of Atomic Measurements, edited by <a href="/facts/Berthold-Georg_Englert/txpMTr9w">B.-G. Englert</a>, Springer, Berlin, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-41408-8.</li>
<li><a href="/facts/Bartel_Leendert_van_der_Waerden/rITfcujK">van der Waerden, B.L.</a> (1967). Sources of Quantum Mechanics, edited with a historical introduction by B.L. van der Waerden, North-Holland Publishing, Amsterdam.</li>
<li><a href="/facts/Steven_Weinberg/G61D6kJw">Weinberg, S.</a> (1995). The Quantum Theory of Fields, volume 1, Foundations, Cambridge University Press, Cambridge UK, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-55001-7.</li>
<li><a href="/facts/Steven_Weinberg/G61D6kJw">Weinberg, S.</a> (2013). Lectures on Quantum Mechanics, Cambridge University Press, Cambridge UK, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-107-02872-2.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">French & Taylor (1978), pp. 24, 55. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Cohen-Tannoudji, Diu & Laloë (1973/1977), pp. 10–11. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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
 <a href="0 444 86712 0" target="_blank">0 444 86712 0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Schwinger (2001), p. 203. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Landsberg (1978), p. 199. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Landé (1951), p. 12. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Griffiths, D. J. (1995), pp. 143, 216. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Griffiths, D. J. (1995), pp. 217, 312. <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Weinberg (2013), pp. 24, 28, 31. <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Weinberg (1995), p. 3. <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Messiah (1958/1961), p. 14. <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Cohen-Tannoudji, Diu & Laloë (1973/1977), p. 27. <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Schwinger (2001), p. 203. <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Flowers et al. (n.d), 6.2 The Bohr Model <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">van der Waerden (1967), p. 5. <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
</ol>

Planck relation open-in-new

Planck relation