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Display information for equation id:math.2547.67 on revision:2547

* Page found: Das ideale Fermigas (eq math.2547.67)

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Hash: f04c224d86bfa7cb0a468c9a907be058

TeX (original user input): 

\begin{align}

& U=\frac{2}{5}\left( \frac{V}{{{h}^{3}}} \right)4\pi \frac{\left( 2s+1 \right)}{2}{{\left( 2m \right)}^{\frac{3}{2}}}{{\left( \mu  \right)}^{\frac{5}{2}}}\left[ 1+\frac{5}{2}\frac{{{\pi }^{2}}}{4}{{\left( \frac{kT}{\mu } \right)}^{2}} \right] \\

& \approx \frac{2}{5}\left( \frac{V}{{{h}^{3}}} \right)4\pi \frac{\left( 2s+1 \right)}{2}{{\left( 2m \right)}^{\frac{3}{2}}}{{\left( {{E}_{F}} \right)}^{\frac{5}{2}}}{{\left[ 1-\frac{{{\pi }^{2}}}{12}{{\left( \frac{kT}{{{E}_{F}}} \right)}^{2}} \right]}^{\frac{5}{2}}}\left[ 1+\frac{5}{2}\frac{{{\pi }^{2}}}{4}{{\left( \frac{kT}{{{E}_{F}}} \right)}^{2}} \right] \\

\end{align}

TeX (checked): 

{\begin{aligned}&U={\frac {2}{5}}\left({\frac {V}{{h}^{3}}}\right)4\pi {\frac {\left(2s+1\right)}{2}}{{\left(2m\right)}^{\frac {3}{2}}}{{\left(\mu \right)}^{\frac {5}{2}}}\left[1+{\frac {5}{2}}{\frac {{\pi }^{2}}{4}}{{\left({\frac {kT}{\mu }}\right)}^{2}}\right]\\&\approx {\frac {2}{5}}\left({\frac {V}{{h}^{3}}}\right)4\pi {\frac {\left(2s+1\right)}{2}}{{\left(2m\right)}^{\frac {3}{2}}}{{\left({{E}_{F}}\right)}^{\frac {5}{2}}}{{\left[1-{\frac {{\pi }^{2}}{12}}{{\left({\frac {kT}{{E}_{F}}}\right)}^{2}}\right]}^{\frac {5}{2}}}\left[1+{\frac {5}{2}}{\frac {{\pi }^{2}}{4}}{{\left({\frac {kT}{{E}_{F}}}\right)}^{2}}\right]\\\end{aligned}}

LaTeXML (experimentell; verwendet MathML) rendering

MathML (44.764 KB / 4.673 KB) :

U = 2 5 ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ ( 2 ⁢ s + 1 ) 2 ⁢ ( 2 ⁢ m ) 3 2 ⁢ ( μ ) 5 2 ⁢ [ 1 + 5 2 ⁢ π 2 4 ⁢ ( k ⁢ T μ ) 2 ] ≈ 2 5 ⁢ ( V h 3 ) ⁢ 4 ⁢ π ⁢ ( 2 ⁢ s + 1 ) 2 ⁢ ( 2 ⁢ m ) 3 2 ⁢ ( E F ) 5 2 ⁢ [ 1 - π 2 12 ⁢ ( k ⁢ T E F ) 2 ] 5 2 ⁢ [ 1 + 5 2 ⁢ π 2 4 ⁢ ( k ⁢ T E F ) 2 ] absent 𝑈 2 5 𝑉 superscript ℎ 3 4 𝜋 2 𝑠 1 2 superscript 2 𝑚 3 2 superscript 𝜇 5 2 delimited-[] 1 5 2 superscript 𝜋 2 4 superscript 𝑘 𝑇 𝜇 2 absent absent 2 5 𝑉 superscript ℎ 3 4 𝜋 2 𝑠 1 2 superscript 2 𝑚 3 2 superscript subscript 𝐸 𝐹 5 2 superscript delimited-[] 1 superscript 𝜋 2 12 superscript 𝑘 𝑇 subscript 𝐸 𝐹 2 5 2 delimited-[] 1 5 2 superscript 𝜋 2 4 superscript 𝑘 𝑇 subscript 𝐸 𝐹 2 {\displaystyle{\displaystyle\begin{aligned} \par&\displaystyle U=\frac{2}{5}% \left(\frac{V}{{{h}^{3}}}\right)4\pi\frac{\left(2s+1\right)}{2}{{\left(2m% \right)}^{\frac{3}{2}}}{{\left(\mu\right)}^{\frac{5}{2}}}\left[1+\frac{5}{2}% \frac{{{\pi}^{2}}}{4}{{\left(\frac{kT}{\mu}\right)}^{2}}\right]\\ \par&\displaystyle\approx\frac{2}{5}\left(\frac{V}{{{h}^{3}}}\right)4\pi\frac{% \left(2s+1\right)}{2}{{\left(2m\right)}^{\frac{3}{2}}}{{\left({{E}_{F}}\right)% }^{\frac{5}{2}}}{{\left[1-\frac{{{\pi}^{2}}}{12}{{\left(\frac{kT}{{{E}_{F}}}% \right)}^{2}}\right]}^{\frac{5}{2}}}\left[1+\frac{5}{2}\frac{{{\pi}^{2}}}{4}{{% \left(\frac{kT}{{{E}_{F}}}\right)}^{2}}\right]\\ \par\end{aligned}}}

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MathML (6.069 KB / 558 B) :

U=25(Vh3)4π(2s+1)2(2m)32(μ)52[1+52π24(kTμ)2]≈25(Vh3)4π(2s+1)2(2m)32(EF)52[1−π212(kTEF)2]52[1+52π24(kTEF)2]

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

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  • μ
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