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blackbody_radiation [2021/01/29 07:24] – created adminblackbody_radiation [2021/01/29 23:06] (current) – [1.ii.4 The Quantum Postulate] admin
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 For those who do not like interactive things, here is a graph of the blackbody spectrum for various temperatures: For those who do not like interactive things, here is a graph of the blackbody spectrum for various temperatures:
  
 +{{ :blackbodyspectrum.jpg?direct&600 |The blackbody spectrum for various temperatures}}
 +**Image Credit**: [[https://openstax.org/books/university-physics-volume-3/pages/6-1-blackbody-radiation|Samuel J. Ling, Jeff Sanny, William Moebs, University Physics Volume 3 (OpenStax, 2016)]]
 +
 +Access for free at [[https://openstax.org/books/university-physics-volume-3/pages/1-introduction|https://openstax.org/books/university-physics-volume-3/pages/1-introduction]]
  
 The blackbody spectrum has the following features: The blackbody spectrum has the following features:
  
-There is a well-defined, continuous energy distribution that only depends on the temperature of the cavity walls. +  * There is a well-defined, continuous energy distribution that only depends on the temperature of the cavity walls. 
-Wien’s displacement law: There is a maximum at a particular wavelength:+  * **Wien’s displacement law**: There is a maximum at a particular wavelength: 
 +  \[\boxed{\lambda_{\max} \propto \frac{1}{T}.}\] 
 +  * **Stefan-Boltzmann Law**: In 1879, Stefan found experimentally that the total intensity (power per unit surface area obeys) 
 +  \[\boxed{I = a\sigma T^4,}\] 
 +  where 
 +    * $a$ is a constant that depends on the material 
 +    * $a=1$ for a blackbody 
 +    * $a<1$ for a non blackbody 
 +    * $\sigma = 5.67 \times 10^{-8}\,\text{Wm}^{-2}\text{K}^{-1}$ 
 +    * Boltzmann derived this theoretically in 1884.
  
- +====== 1.ii.3 Attempts to derive the blackbody spectrum ======
  
-Stefan-Boltzmann Law: In 1879, Stefan found experimentally that the total intensity (power per unit surface area obeys)+===== Wien's Law =====
  
- is a constant that depends on the material +  * Wien extended the Stefan-Boltzmann law to obtain 
- for a black-body +  \[\boxed{u(\nu,T) = A\nu^3 e^{(-\beta\nu/T)}.}\] 
- for a non black-body +  * This only works for large frequencies $\nu$. 
-Boltzmann derived this theoretically in 1884. +  * $A$ and $\beta$ and  are free parameters that have to be fixed by experiment.
-1.ii.3 Attempts to derive the blackbody spectrum +
-Wien's Law +
-Wien extended the Stefan-Boltzmann law to obtain+
  
-This only works for large frequencies . +===== Rayleigh-Jeans Law =====
- and  are free parameters that have to be fixed by experiment. +
-Rayleigh-Jeans Law +
-Rayleigh considered the statistical mechanics of radiation and derived+
  
-This only works for small . +  * Rayleigh considered the statistical mechanics of radiation and derived 
-Ultraviolet catastrophe: Total intensity  diverges for high frequency.  Implies that the cavity contains infinite energy +  \[\boxed{u(\nu,T) = \frac{8\pi \nu^2}{c^3} k T.}\] 
-Planck Distribution +  * This only works for small $\nu$. 
-Planck obtained+  * Ultraviolet catastrophe: Total intensity $\int_{-\infty}^{\infty} u(\nu,T)\,\mathrm{d}\nu$ diverges for high frequency.  Implies that the cavity contains infinite energy
  
-which agrees with experiment for+===== Planck Distribution =====
  
-. +  * Planck obtained 
- +  \[\boxed{u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1},}\] 
- is called Planck’s constant: A new fundamental constant of nature introduced by quantum theory.+  which agrees with experiment for 
 +  \[\boxed{h=6.626\times 10^{-34}\,\mathrm{m}^2\mathrm{kg}\mathrm{s}^{-1}.}\] 
 +  * $h$ is called //**Planck’s constant**//: A new fundamental constant of nature introduced by quantum theory.
  
 The following graph compares the three distributions. The following graph compares the three distributions.
  
-1.ii.4 The Quantum Postulate +[[https://commons.wikimedia.org/wiki/File:Mplwp_blackbody_nu_planck-wien-rj_5800K.svg|{{ https://upload.wikimedia.org/wikipedia/commons/thumb/1/18/Mplwp_blackbody_nu_planck-wien-rj_5800K.svg/512px-Mplwp_blackbody_nu_planck-wien-rj_5800K.svg.png |Geek3, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0 , via Wikimedia Commons}}]]  
-To derive his distribution, Planck assumed that the energy of the radiation emitted at frequency  by the oscillating electrons in the walls of the cavity could only come in integer multiples of .+**Image Credit**: [[https://commons.wikimedia.org/wiki/File:Mplwp_blackbody_nu_planck-wien-rj_5800K.svg|Wikimedia Commons]] 
 + 
 +====== 1.ii.4 The Quantum Postulate ====== 
 + 
 +To derive his distribution, Planck assumed that the energy of the radiation emitted at frequency $\nu$ by the oscillating electrons in the walls of the cavity could only come in integer multiples of $h\nu$.
  
-This is the quantum postulate. +\[\boxed{E(\nu)=nh\nu\qquad\qquad\mbox{for}\qquad\qquad n=0,1,2,3,\cdots.}\]
-For a derivation of the Planck distribution from this posulate, see http://disciplinas.stoa.usp.br/pluginfile.php/48089/course/section/16461/qsp_chapter10-plank.pdf +
-1.ii.5 Wien's Displacement Law +
-Wien’s displacement law says that the maximum energy density of the blackbody spectrum is at  for some constant . +
-We can derive this by finding the maximum of the Planck distribution expressed as a function of wavelength  that we found in the in-class activity. +
-Setting  yields a transcendental equation (i.e. an equation that has no closed-form solution so we have to solve it numerically).  Solving this numerically gives+
  
-with+This is the //**quantum postulate**//.
  
-.+Since not everyone taking this class has studied statistical mechanics yet, the derivation of the Planck distribution from the quantum postulate is beyond the scope of this course, but see [[http://disciplinas.stoa.usp.br/pluginfile.php/48089/course/section/16461/qsp_chapter10-plank.pdf|this link]] for a derivation.
  
-In Class Activities+====== 1.ii.5 Wien's Displacement Law ======
  
-Derive the Rayleigh-Jeans law  from the Planck formula  for small .+  * Wien’s displacement law says that the maximum energy density of the blackbody spectrum is at $\lambda_{\max} = b/T$ for some constant $b$. 
 +  * We can derive this by finding the maximum of the Planck distribution expressed as a function of wavelength $\tilde{u}(\lambda, T)$.  Expressing the Planck distribution in terms of wavelength is one of the in-class activities. 
 +  * Setting $\frac{\partial \tilde{u}}{\partial \lambda}=0$ yields a transcendental equation (i.e. an equation that has no closed-form solution so we have to solve it numerically).  Solving this numerically gives 
 +  \[\boxed{\lambda_{\max} = \frac{b}{T},}\] 
 +  with 
 +  \[\boxed{b = \frac{hc}{4.965 k}.}\]
  
-Derive Wien’s law  from the Planck formula  for large .  What are   and  in terms of , , and ?+{{:question-mark.png?nolink&50 |}} 
 +====== In Class Activities ======
  
-The energy density for frequencies between  and  is , where .   Derive the energy density  as a function of wavelength using .+  - Derive the Rayleigh-Jeans law $u(\nu,T) = \frac{8\pi \nu^2}{c^3} k T$ from the Planck formula $u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1}$ for small $\nu$. 
 +  - Derive Wien’s law $u(\nu,T) = A\nu^3 e^{(-\beta\nu/T)}$ from the Planck formula $u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1}$ for large $\nu$.  What are  $A$ and $\beta$ in terms of $h$, $c$, and $k$? 
 +  - The energy density for frequencies between $\nu$ and $\nu + \mathrm{d}\nu$ is $u(\nu, T)\mathrm{d}\nu$, where $u(\nu,T)=\frac{8\pi}{c^3} \frac{h \nu^3}{e^{h\nu/kT}-1}$.   Derive the energy density $\tilde{u}(\lambda,T)$ as a function of wavelength using $\lambda = \frac{c}{\nu}$.