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Display information for equation id:math.2547.63 on revision:2547

* Page found: Das ideale Fermigas (eq math.2547.63)

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Hash: bd5031afe7d4e054ed98964fd00a4169

TeX (original user input): 

\begin{align}

& \bar{N}=\frac{2}{3}\frac{\left( 2s+1 \right)}{2}\left( \frac{V}{{{h}^{3}}} \right)4\pi {{\left( 2m\mu  \right)}^{\frac{3}{2}}}\left[ 1+\frac{{{\pi }^{2}}}{8}{{\left( \frac{kT}{\mu } \right)}^{2}} \right]=\frac{2}{3}\frac{\left( 2s+1 \right)}{2}\left( \frac{V}{{{h}^{3}}} \right)4\pi {{\left( 2m{{E}_{F}} \right)}^{\frac{3}{2}}} \\

& \Rightarrow {{\left( \mu  \right)}^{\frac{3}{2}}}\left[ 1+\frac{{{\pi }^{2}}}{8}{{\left( \frac{kT}{\mu } \right)}^{2}} \right]\approx {{\left( {{E}_{F}} \right)}^{\frac{3}{2}}} \\

& \Rightarrow \mu \approx {{E}_{F}}{{\left[ 1+\frac{{{\pi }^{2}}}{8}{{\left( \frac{kT}{\mu } \right)}^{2}} \right]}^{-\frac{2}{3}}} \\

\end{align}

TeX (checked): 

{\begin{aligned}&{\bar {N}}={\frac {2}{3}}{\frac {\left(2s+1\right)}{2}}\left({\frac {V}{{h}^{3}}}\right)4\pi {{\left(2m\mu \right)}^{\frac {3}{2}}}\left[1+{\frac {{\pi }^{2}}{8}}{{\left({\frac {kT}{\mu }}\right)}^{2}}\right]={\frac {2}{3}}{\frac {\left(2s+1\right)}{2}}\left({\frac {V}{{h}^{3}}}\right)4\pi {{\left(2m{{E}_{F}}\right)}^{\frac {3}{2}}}\\&\Rightarrow {{\left(\mu \right)}^{\frac {3}{2}}}\left[1+{\frac {{\pi }^{2}}{8}}{{\left({\frac {kT}{\mu }}\right)}^{2}}\right]\approx {{\left({{E}_{F}}\right)}^{\frac {3}{2}}}\\&\Rightarrow \mu \approx {{E}_{F}}{{\left[1+{\frac {{\pi }^{2}}{8}}{{\left({\frac {kT}{\mu }}\right)}^{2}}\right]}^{-{\frac {2}{3}}}}\\\end{aligned}}

LaTeXML (experimentell; verwendet MathML) rendering

MathML (47.832 KB / 5.048 KB) :

N ¯ = 2 3 ⁢ ( 2 ⁢ s + 1 ) 2 ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ ( 2 ⁢ m ⁢ μ ) 3 2 ⁢ [ 1 + π 2 8 ⁢ ( k ⁢ T μ ) 2 ] = 2 3 ⁢ ( 2 ⁢ s + 1 ) 2 ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ ( 2 ⁢ m ⁢ E F ) 3 2 ⇒ ( μ ) 3 2 ⁢ [ 1 + π 2 8 ⁢ ( k ⁢ T μ ) 2 ] ≈ ( E F ) 3 2 ⇒ μ ≈ E F ⁢ [ 1 + π 2 8 ⁢ ( k ⁢ T μ ) 2 ] - 2 3 absent ¯ 𝑁 2 3 2 𝑠 1 2 𝑉 superscript ℎ 3 4 𝜋 superscript 2 𝑚 𝜇 3 2 delimited-[] 1 superscript 𝜋 2 8 superscript 𝑘 𝑇 𝜇 2 2 3 2 𝑠 1 2 𝑉 superscript ℎ 3 4 𝜋 superscript 2 𝑚 subscript 𝐸 𝐹 3 2 absent ⇒ absent superscript 𝜇 3 2 delimited-[] 1 superscript 𝜋 2 8 superscript 𝑘 𝑇 𝜇 2 superscript subscript 𝐸 𝐹 3 2 absent ⇒ absent 𝜇 subscript 𝐸 𝐹 superscript delimited-[] 1 superscript 𝜋 2 8 superscript 𝑘 𝑇 𝜇 2 2 3 {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle\bar{N}=\frac{2}% {3}\frac{\left(2s+1\right)}{2}\left(\frac{V}{{{h}^{3}}}\right)4\pi{{\left(2m% \mu\right)}^{\frac{3}{2}}}\left[1+\frac{{{\pi}^{2}}}{8}{{\left(\frac{kT}{\mu}% \right)}^{2}}\right]=\frac{2}{3}\frac{\left(2s+1\right)}{2}\left(\frac{V}{{{h}% ^{3}}}\right)4\pi{{\left(2m{{E}_{F}}\right)}^{\frac{3}{2}}}\\ \par&\displaystyle\Rightarrow{{\left(\mu\right)}^{\frac{3}{2}}}\left[1+\frac{{% {\pi}^{2}}}{8}{{\left(\frac{kT}{\mu}\right)}^{2}}\right]\approx{{\left({{E}_{F% }}\right)}^{\frac{3}{2}}}\\ \par&\displaystyle\Rightarrow\mu\approx{{E}_{F}}{{\left[1+\frac{{{\pi}^{2}}}{8% }{{\left(\frac{kT}{\mu}\right)}^{2}}\right]}^{-\frac{2}{3}}}\\ \par\end{aligned}}}

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MathML (6.04 KB / 580 B) :

N¯=23(2s+1)2(Vh3)4π(2mμ)32[1+π28(kTμ)2]=23(2s+1)2(Vh3)4π(2mEF)32⇒(μ)32[1+π28(kTμ)2]≈(EF)32⇒μ≈EF[1+π28(kTμ)2]−23

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Calculated based on the variables occurring on the entire Das ideale Fermigas page

Identifiers

  • N¯
  • s
  • V
  • h
  • π
  • m
  • μ
  • π
  • k
  • T
  • μ
  • s
  • V
  • h
  • π
  • m
  • EF
  • μ
  • π
  • k
  • T
  • μ
  • EF
  • μ
  • EF
  • π
  • k
  • T
  • μ

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