$$ \begin{aligned} & f_{5 / 2}(z)=\frac{8}{15 \sqrt{\pi}}(\ln z)^{5 / 2}\left(1+\frac{5}{8} \pi^{2}(\ln z)^{-2}+\ldots .\right) \\ & f_{3 / 2}(z)=\frac{4}{3 \sqrt{\pi}}(\ln z)^{3 / 2}\left(1+\frac{1}{8} \pi^{2}(\ln z)^{-2}+\ldots .\right) \\ & f_{1 / 2}(z)=\frac{2}{\sqrt{\pi}}(\ln z)^{5 / 2}\left(1-\frac{1}{24} \pi^{2}(\ln z)^{-2}+\ldots .\right) \end{aligned} $$

$$ \begin{aligned} & f_{r}(z)=\frac{(\ln z)^{r}}{\Gamma(r+1)}\left(1+r(r-1) \frac{\pi^{2}}{6} \frac{1}{(\ln z)^{2}}+\ldots\right) \\ & f_{5 / 2}(z)=\frac{(\ln z)^{5 / 2}}{\Gamma(7 / 2)}\left(1+\frac{5}{2} \frac{3}{2} \frac{\pi^{2}}{6} \frac{1}{(\ln z)^{2}}+\ldots .\right) \\ & \Rightarrow f_{5 / 2}(z)=\frac{8}{15 \sqrt{\pi}}(\ln z)^{5 / 2}\left(1+\frac{15 \pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots .\right) \\ & f_{3 / 2}(z)=\frac{(\ln z)^{3 / 2}}{\Gamma(5 / 2)}\left(1+\frac{3}{2} \frac{1}{2} \frac{\pi^{2}}{6} \frac{1}{(\ln z)^{2}}+\ldots .\right) \\ & \Rightarrow \quad f_{3 / 2}(z)=\frac{4}{3 \sqrt{\pi}}(\ln z)^{3 / 2}\left(1+\frac{\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots .\right) \\ & f_{1 / 2}(z)=\frac{(\ln z)^{1 / 2}}{\Gamma(3 / 2)}\left(1+\frac{1}{2}\left(-\frac{1}{2}\right) \frac{\pi^{2}}{6} \frac{1}{(\ln z)^{2}}+\ldots .\right)\\ \Rightarrow f_{1 / 2}(z)&=\frac{2(\ln z)^{1 / 2}}{\sqrt{\pi}}\left(1-\frac{\pi^{2}}{24} \frac{1}{(\ln z)^{2}}+\ldots.\right) \end{aligned} $$

$$E_{F}=\left(\frac{3 N}{4 \pi V}\right)^{2 / 3}\left(\frac{h^{2}}{2 m}\right)$$

$$\mu=E_{F}\left(1-\frac{\pi^{2}}{12} \frac{1}{(\ln z)^{2}}+\ldots.\right)$$\\

$$ \begin{aligned} n&=\frac{N}{V}\\ &=\frac{f_{3 / 2}(z)}{\lambda^{3}}\\ & =\frac{4}{\lambda^{3} 3 \sqrt{\pi}}(\ln z)^{3 / 2}\left(1+\frac{\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots\right) \\ & =\frac{(2 \pi m k T)^{3 / 2} 4}{h^{3} 3 \sqrt{\pi}}(\ln z)^{3 / 2}\left(1+\frac{\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots .\right) \\ & =\left(\frac{2 m}{h^{2}}\right)^{3 / 2}\left(\frac{4 \pi}{3}\right)(k T \ln z)^{3 / 2}\left(1+\frac{\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots .\right) \\ & =\left(\frac{2 m}{h^{2}}\right)^{3 / 2}\left(\frac{4 \pi}{3}\right)(\mu)^{3 / 2}\left(1+\frac{\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots .\right) \quad \because \ln z=\ln e^{\mu / k T} \\ & =\left(\frac{2 m}{h^{2}}\right)^{3 / 2}\left(\frac{4 \pi}{3}\right)(\mu)^{3 / 2}\left(1+\frac{\pi^{2}}{8} \frac{k^{2} T^{2}}{\mu^{2}}+\ldots .\right)\\ &\operatorname{At} \mathbf{T}=\mathbf{0 K}, \mu=E_{F}\\ &\Rightarrow \frac{N}{V}=\left(\frac{2 m}{h^{2}}\right)^{3 / 2}\left(\frac{4 \pi}{3}\right) E_{F}^{3 / 2}(1+0+\ldots)\\ & \Rightarrow E_{F}=\left(\frac{3 N}{4 \pi V}\right)^{2 / 3}\left(\frac{h^{2}}{2 m}\right) \end{aligned} $$

$$ \begin{aligned} & \frac{N}{V}=\left(\frac{2 m}{h^{2}}\right)^{3 / 2}\left(\frac{4 \pi}{3}\right)(\mu)^{3 / 2}\left(1+\frac{\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots .\right) \\ & \left(\frac{3 N}{4 \pi V}\right)^{2 / 3}\left(\frac{h^{2}}{2 m}\right)=(\mu)^{3 / 2}\left(1+\frac{\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots .\right) \\ & \Rightarrow E_{F}^{3 / 2}=\mu^{3 / 2}\left(1+\frac{\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots .\right) \\ & \Rightarrow \mu=E_{F}\left(1+\frac{\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots .\right)^{-2 / 3} \end{aligned} $$

$$ \begin{aligned} & \Rightarrow \mu=E_{F}\left(1-\frac{2}{3} \frac{\pi^{2}}{8} \frac{1}{(\ln z)^{2}}+\ldots .\right)^{-2 / 3} \\ & \mu=E_{F}\left(1-\frac{\pi^{2}}{12} \frac{1}{(\ln z)^{2}}+\ldots .\right) \end{aligned} $$

$$\frac{U}{N}=\frac{3}{5} E_{F}\left(1+\frac{5 \pi^{2}}{12}\left(\frac{k T}{E_{F}}\right)^{2}+\ldots\right)$$

$$ \begin{aligned} \frac{U}{N}&=\frac{3}{2} k T \frac{f_{5 / 2}(z)}{f_{3 / 2}(z)}\\ & =\frac{3}{2} k T \frac{8 / 15 \sqrt{\pi}}{4 / 3 \sqrt{\pi}} \frac{(\ln z)^{5 / 2}}{(\ln z)^{3 / 2}} \frac{\left(1+\frac{5}{8} \pi^{2}(\ln z)^{-2}\right)}{\left(1+\frac{1}{8} \pi^{2}(\ln z)^{-2}\right)} \\ & =\frac{3}{2} k T \frac{2}{5} \ln z\left(1+\frac{5}{8} \pi^{2}(\ln z)^{-2}\right)\left(1+\frac{1}{8} \pi^{2}(\ln z)^{-2}\right)^{-1} \\ & =\frac{3}{5} k T \ln z\left(1+\frac{5}{8} \pi^{2}(\ln z)^{-2}\right)\left(1-\frac{1}{8} \pi^{2}(\ln z)^{-2}\right) \\ & =\frac{3}{5} k T \ln z\left(1-\frac{1}{8} \pi^{2}(\ln z)^{-2}+\frac{5}{8} \pi^{2}(\ln z)^{-2}-\frac{5}{64} \pi^{4}(\ln z)^{-4}\right) \\ & =\frac{3}{5} k T \ln z\left(1+\frac{\pi^{2}}{2}(\ln z)^{-2}+\ldots\right) \\ & =\frac{3}{5} \mu\left(1+\frac{\pi^{2}}{2}(\ln z)^{-2}+\ldots\right)=\frac{3}{5} E_{F}\left(1+\frac{5 \pi^{2}}{12}\left(\frac{k T}{E_{F}}\right)^{2}+\ldots\right)\left(1+\frac{\pi^{2}}{2}(\ln z)^{-2}+\ldots\right) \\ & =\frac{3}{5} E_{F}\left(1+\frac{5 \pi^{2}}{12}\left(\frac{k T}{E_{F}}\right)^{2}\right) \end{aligned} $$

\( P=\frac{2}{5} n E_{F}\left(1+\frac{5 \pi^{2}}{12}\left(\frac{k T}{E_{F}}\right)^{2}\right) \)

$$ \begin{aligned} P&=\frac{2}{3} \frac{U}{V}\\ &=\frac{2}{3} \frac{U / N}{V / N} \\ & =\frac{2}{3} \frac{\frac{3}{5} E_{F}\left(1+\frac{5 \pi^{2}}{12}\left(\frac{k T}{E_{F}}\right)^{2}\right)}{V / N}\\ &=\frac{2}{5} n E_{F}\left(1+\frac{5 \pi^{2}}{12}\left(\frac{k T}{E_{F}}\right)^{2}\right) \end{aligned} $$

. $$\dfrac{C_{V}}{N k}=\dfrac{\pi^{2} k T}{2 E_{F}}$$

$$ \begin{aligned} \frac{C_{V}}{N k}&=\frac{\frac{d}{d T} U}{N k}\\ & =\frac{1}{k} \frac{d}{d T} \frac{U}{N}=\frac{1}{k} \frac{d}{d T}\left(\frac{3}{5} E_{F}\left(1+\frac{5 \pi^{2}}{12}\left(\frac{k T}{E_{F}}\right)^{2}\right)\right) \\ & =\frac{3}{5 k} E_{F}\left(0+\frac{5 \pi^{2}}{6} \frac{k^{2} T}{E_{F}^{2}}\right) \\ & =\frac{\pi^{2} k T}{2 E_{F}} \\ \Rightarrow C_{V}&=0 \quad \text { Whe } n T \rightarrow 0 \end{aligned} $$

$$ \frac{S}{k}=\sum g_{i}\left(\frac{\beta E_{i}-\ln z}{z^{-1} e^{\beta E_{i}}+1}\right)+\ln \left(1+z e^{-\beta E_{i}}\right) $$

$$ \begin{aligned} \frac{S}{k}&=\ln W_{\max }=\ln \left(\prod \frac{g_{i}!}{n_{i}!\left(g_{i}-n_{i}\right)!}\right) \\ & =\sum\left(g_{i} \ln g_{i}-g_{i}-n_{i} \ln n_{i}+n_{i}-\left(g_{i}-n_{i}\right) \ln \left(g_{i}-n_{i}\right)+\left(g_{i}-n_{i}\right)\right) \\ & =\sum\left(g_{i} \ln g_{i}-g_{i}-n_{i} \ln n_{i}+n_{i}-g_{i} \ln \left(g_{i}-n_{i}\right)+n_{i} \ln \left(g_{i}-n_{i}\right)+\left(g_{i}-n_{i}\right)\right) \\ & =\sum\left(g_{i} \ln g_{i}-n_{i} \ln n_{i}-g_{i} \ln \left(g_{i}-n_{i}\right)+n_{i} \ln \left(g_{i}-n_{i}\right)\right) \\ & =\sum\left(g_{i} \ln g_{i}-n_{i} \ln n_{i}-g_{i} \ln \left(g_{i}-n_{i}\right)+n_{i} \ln \left(g_{i}-n_{i}\right)\right) \\ & =\sum\left(n_{i} \ln \frac{\left(g_{i}-n_{i}\right)}{n_{i}}-g_{i} \ln \frac{\left(g_{i}-n_{i}\right)}{g_{i}}\right) \\ & =\sum\left(n_{i} \ln \left(\frac{g_{i}}{n_{i}}-1\right)-g_{i} \ln \left(1-\frac{n_{i}}{g_{i}}\right)\right) \\ & =\sum\left(n_{i} \ln \left(e^{\alpha+\beta E_{i}}+1-1\right)-g_{i} \ln \left(1-\frac{1}{e^{\alpha+\beta E_{i}}+1}\right)\right) \end{aligned} $$

$$ \begin{aligned} & =\sum\left(n_{i} \ln \left(e^{\alpha+\beta E_{i}}\right)-g_{i} \ln \left(\frac{e^{\alpha+\beta E_{i}}}{e^{\alpha+\beta E_{i}}+1}\right)\right) \\ & =\sum\left(\frac{g_{i}}{e^{\alpha+\beta E_{i}}+1} \ln \left(e^{\alpha+\beta E_{i}}\right)-g_{i} \ln \left(\frac{1}{1+e^{-\alpha-\beta E_{i}}}\right)\right) \\ & =\sum\left(\frac{g_{i}\left(\alpha+\beta E_{i}\right)}{e^{\alpha+\beta E_{i}}+1}+g_{i} \ln \left(1+e^{-\alpha-\beta E_{i}}\right)\right) \\ & =\sum g_{i}\left(\frac{\left(\beta E_{i}-\ln z\right)}{z^{-1} e^{\beta E_{i}}+1}+\ln \left(1+z e^{-\beta E_{i}}\right)\right) \end{aligned} $$

$$ \frac{P V}{k T}=\log Z_{F D}=\sum \log \left(1+z e^{-\beta E_{i}}\right) $$

Proof:

$$ \begin{aligned} & \frac{S}{k}-\sum g_{i}\left(\frac{\left(\beta E_{i}-\ln z\right)}{z^{-1} e^{\beta E_{i}}+1}\right)=\sum g_{i} \ln \left(1+z e^{-\beta E_{i}}\right) \\ \Rightarrow & \frac{S}{k}-\sum n_{i}\left(\left(\beta E_{i}-\ln z\right)\right)=\sum g_{i} \ln \left(1+z e^{-\beta E_{i}}\right) \\ \Rightarrow & \frac{S}{k}-\sum\left(n_{i} E_{i} / k T-n_{i} \mu / k T\right)=\sum g_{i} \ln \left(1+z e^{-\beta E_{i}}\right) \\ \Rightarrow & \frac{S}{k}-\frac{\sum n_{i} E_{i}-\sum n_{i} \mu}{k T}=\sum g_{i} \ln \left(1+z e^{-\beta E_{i}}\right) \\ \Rightarrow & \frac{T S-E+G}{k T}=\sum g_{i} \ln \left(1+z e^{-\beta E_{i}}\right) \\ \Rightarrow & \frac{G-(E-T S)}{k T}=\sum g_{i} \ln \left(1+z e^{-\beta E_{i}}\right) \\ \Rightarrow & \frac{H-T S-U+T S}{k T}=\sum g_{i} \ln \left(1+z e^{-\beta E_{i}}\right) \\ \Rightarrow & \frac{P V}{k T}=\sum g_{i} \ln \left(1+z e^{-\beta E_{i}}\right)=\sum \log Z_{F e r m i} \end{aligned} $$

$$ \frac{P}{k T}=\frac{f_{5 / 2}(z)}{\lambda^{3}}$$

Where $$f_{r}(z)=z-\frac{z^{2}}{2^{n}}+\frac{z^{3}}{3^{n}}-\frac{z^{4}}{4^{n}}+\ldots .=\sum(-1)^{n+1} \frac{z^{n}}{n^{r}}$$ is Fermi Function\\

$$ \begin{aligned} \frac{P V}{k T}&=\sum g_{i} \ln \left(1+z e^{-\beta E_{i}}\right)\\ \Rightarrow \frac{P V}{k T}&=\int_{0}^{\infty} \ln \left(1+z e^{-\beta E_{i}}\right) \frac{V}{h^{3}} 4 \pi p^{2} dp\\ & =\frac{4 \pi V}{h^{3}} \int_{0}^{\infty} \ln \left(1+z e^{-\beta E_{i}}\right) p^{2} d p \\ & =\frac{4 \pi V}{h^{3}} \int_{0}^{\infty} \sum_{1}^{n}(-1)^{n+1} \frac{\left(z e^{-\beta E_{i}}\right)^{n}}{n} p^{2} d p \because \ln (1+y)=\sum_{n=1}^{\infty}(-1)^{n+1} \frac{y^{n}}{n} \\ & =\frac{4 \pi V}{h^{3}} \int_{0}^{\infty} \sum_{1}^{n}(-1)^{n+1} \frac{z^{n}}{n} e^{-n \beta E} p^{2} d p \\ & =\frac{4 \pi V}{h^{3}} \sum_{1}^{n}(-1)^{n+1} \frac{z^{n}}{n} \int_{0}^{\infty} e^{-n p^{2} / 2 m k T} p^{2} d p \\ & =\frac{4 \pi V}{h^{3}} \sum_{1}^{n}(-1)^{n+1} \frac{z^{n}}{n} \int_{0}^{\infty} e^{-x} p^{2} d p\\ &(Taking n p^{2} / 2 m k T=x \Rightarrow p^{2}=\dfrac{2 m k T x}{n}, d p\\ &=\dfrac{\sqrt{2 m k T} d x}{2 \sqrt{n} \sqrt{x}})\\ &=\frac{4 \pi V}{h^{3}} \sum_{1}^{n}(-1)^{n+1} \frac{z^{n}}{n} \int_{0}^{\infty} e^{-x}\left(\frac{2 m k T x}{n}\right)\left(\frac{\sqrt{2 m k T} d x}{2 \sqrt{n} \sqrt{x}}\right) \\ &=\frac{4 \pi V}{h^{3}} \sum_{1}^{n}(-1)^{n+1} \frac{z^{n}}{n} \int_{0}^{\infty} e^{-x} x^{1 / 2} d x\left(\frac{2 m k T}{n}\right)^{3 / 2} \frac{1}{2}\\ &=\frac{4 \pi V}{h^{3}} \sum_{1}^{n}(-1)^{2 n+1} \frac{z^{n}}{n} \Gamma(3 / 2)\left(\frac{2 m k T}{n}\right)^{3 / 2} \frac{1}{2} \\ &=\frac{4 \pi V}{h^{3}} \sum_{1}^{n}(-1)^{2 n+1} \frac{z^{n}}{n} \frac{\sqrt{\pi}}{2}\left(\frac{2 m k T}{n}\right)^{3 / 2} \frac{1}{2} \\ &=\frac{V}{h^{3}} \sum_{1}^{n}(-1)^{2 n+1} \frac{z^{n}}{n}\left(\frac{2 m \pi k T}{n}\right)^{3 / 2}\\ &=\frac{V}{h^{3}}(2 m \pi k T)^{3 / 2} \sum_{1}^{n}(-1)^{2 n+1} \frac{z^{n}}{n^{5 / 2}}\\ &=\frac{V}{h^{3}}(2 m \pi k T)^{3 / 2} f_{5 / 2}(z)\\ &=\frac{V}{\lambda^{3}} f_{5 / 2}(z)\\ \frac{P V}{k T}&=\frac{V}{\lambda^{3}} f_{5 / 2}(z) \\ \Rightarrow \frac{P}{k T}&=\frac{f_{5 / 2}(z)}{\lambda^{3}} \end{aligned} $$

$$ N=\frac{V f_{3 / 2}(z)}{\lambda^{3}} $$

$$ \begin{aligned} & N=\sum n_{i}=\sum \frac{g_{i}}{e^{\alpha+\beta E_{i}}+1}=\sum \frac{g_{i} e^{-\alpha-\beta E_{i}}}{1+e^{-\alpha-\beta E_{i}}} \\ & =\sum \frac{g_{i} z e^{-\beta E_{i}}}{1+z e^{-\beta E_{i}}}=z \sum \frac{\partial}{\partial z} g_{i} \ln \left(1+z e^{-\beta E_{i}}\right) \\ & =z \sum \frac{\partial}{\partial z} g_{i} \ln Z \\ & =z \sum \frac{\partial}{\partial z}\left(\frac{P V}{k T}\right) \\ & =z \frac{\partial}{\partial z}\left(\frac{V}{\lambda^{3}} f_{5 / 2}(z)\right) \\ & =\frac{V}{\lambda^{3}} f_{3 / 2}(z) \end{aligned} $$

$$ U=\frac{3}{2} N k T \frac{f_{5 / 2}(z)}{f_{3 / 2}(z)} $$

$$ \begin{aligned} & U=k T^{2} \frac{\partial}{\partial T} \sum g_{i} \ln Z \\ & =k T^{2} \frac{\partial}{\partial T}\left(\frac{P V}{k T}\right) \\ & =k T^{2} \frac{\partial}{\partial T}\left(\frac{V}{\lambda^{3}} f_{5 / 2}(z)\right)=k T^{2} V f_{5 / 2}(z) \frac{\partial}{\partial T}\left(\frac{1}{\lambda^{3}}\right) \\ & =k T^{2} V f_{5 / 2}(z)\left(-\frac{3}{\lambda^{4}} \frac{d \lambda}{d T}\right) \\ & =k T^{2} V f_{5 / 2}(z)\left(-\frac{3}{\lambda^{4}} \frac{d}{d T}\left(\frac{h}{\sqrt{2 \pi m k T}}\right)\right) \\ & =k T^{2} V f_{5 / 2}(z)\left(-\frac{3}{\lambda^{4}}\left(\frac{h}{\sqrt{2 \pi m k}}\left(\frac{-1}{2} T^{-3 / 2}\right)\right)\right) \\ & =\frac{3}{2} k T^{2} V f_{5 / 2}(z) \frac{1}{\lambda^{4}} \frac{h}{\sqrt{2 \pi m k T}} \frac{1}{T} \\ & =\frac{3}{2} k T V f_{5 / 2}(z) \frac{1}{\lambda^{3}}=\frac{3}{2} k T \frac{V}{\lambda^{3}} f_{5 / 2}(z)\\ & =\frac{3}{2} k T \frac{N}{f_{3 / 2}(z)} f_{5 / 2}(z) \\ & =\frac{3}{2} N k T \frac{f_{5 / 2}(z)}{f_{3 / 2}(z)} \end{aligned} $$

\begin{align} U &= -\frac{\partial}{\partial \beta} \ln Z \\ &= k T^2 \frac{\partial}{\partial T} \ln Z \\ &= k T^2 \frac{\partial \beta}{\partial T} \cdot \frac{\partial}{\partial \beta} \ln Z \\ &= k T^2 \frac{\partial}{\partial T} \left( \frac{1}{kT} \right) \cdot \frac{\partial}{\partial \beta} \ln Z \\ &= k T^2 \left( -\frac{1}{k T^2} \right) \cdot \frac{\partial}{\partial \beta} \ln Z \\ &= -\frac{\partial}{\partial \beta} \ln Z \end{align}

$$N=\frac{z V}{\lambda^{3}}$$

$$ \begin{aligned} & n_{i}=g_{i} e^{-\alpha} e^{-\beta E_{i}}=g_{i} z e^{-\beta E_{i}} \\ & \Rightarrow N=\sum n_{i}=z \sum g_{i} e^{-\beta E_{i}} \\ & =z \sum_{0}^{\infty} \frac{V}{h^{3}} e^{-\frac{p^{2}}{2 m k T}} 4 \pi p^{2} d p \\ & =\frac{4 \pi z V}{h^{3}} \int_{0}^{\infty} e^{-\left(\frac{p^{2}}{2 m k T}\right)} p^{2} d p \\ & =\frac{4 \pi z V}{h^{3}} \int_{0}^{\infty} e^{-x}(2 m k T) x\left(\sqrt{2 m k T} \frac{1}{2 \sqrt{x}} d x\right) \text { Let } \frac{p^{2}}{2 m k T}=x \Rightarrow d p=\sqrt{2 m k T} \frac{1}{2 \sqrt{x}} d x \\ & =\frac{2 \pi z V}{h^{3}}(2 m k T)^{3 / 2} \int_{0}^{\infty} e^{-x} x^{-1 / 2} d x \\ & =\frac{2 \pi z V}{h^{3}}(2 m k T)^{3 / 2} \sqrt{\pi}=\frac{z V}{h^{3}}(2 \pi m k T)^{3 / 2} \\ & =\frac{z V}{\lambda^{3}} \end{aligned} $$

\begin{align*} n(p)\, dp &= \frac{g(p)\, dp}{e^{\dfrac{\alpha + E}{kT}} + 1} \\ &= \frac{g(p)\, dp}{e^{\dfrac{\alpha + E}{0}} + 1} \\ &= \frac{g(p)\, dp}{e^{\infty} + 1} \\ &= \frac{g(p)\, dp}{\infty + 1} \\ &= g(p)\, dp \\ \Rightarrow \quad n(p)\, dp &= 2\left(\frac{V \cdot 4\pi p^{2} dp}{h^{3}}\right) \\ &= \frac{8\pi V p^{2} dp}{h^{3}} \end{align*}

\begin{align*} n(E)\, dE &= \frac{8 \pi V (2mE) \cdot \frac{\sqrt{2m}}{2\sqrt{E}}\, dE}{h^{3}} \\ &= \frac{8 \sqrt{2} \pi V m^{3/2} E^{1/2}\, dE}{h^{3}} \\ \end{align*}

\begin{align*} n(v)\, dv &= \frac{8 \sqrt{2} \pi V m^{3/2} \left(\frac{1}{2}mv^{2}\right)^{1/2} \cdot \frac{1}{2}m \cdot 2v\, dv}{h^{3}} \\ \Rightarrow \quad n(v)\, dv &= \frac{8 \pi V m^{3} v^{2}\, dv}{h^{3}} \end{align*}

$$ \begin{aligned} N&=\int_{0}^{E_{F}} n(E) d E \\ & =\int_{0}^{E_{F}} \frac{8 \sqrt{2} \pi V m^{3 / 2} E^{1 / 2} d E}{h^{3}} \\ & =\frac{8 \sqrt{2} \pi V m^{3 / 2}}{h^{3}} \int_{0}^{E_{F}} E^{1 / 2} d E \\ & =\frac{8 \sqrt{2} \pi V m^{3 / 2}}{h^{3}} \times \frac{2}{3} E_{F}^{3 / 2} \\ & =\left(\frac{16 \sqrt{2} \pi V m^{3 / 2}}{3 h^{3}}\right) E_{F}^{3 / 2} \\ \Rightarrow E_{F}^{3}&=\frac{9 N^{2} h^{6}}{256 \times 2 \pi^{2} V^{2} m^{3}} \\ \Rightarrow E_{F} & =\left(\frac{9 N^{2} h^{6}}{256 \times 2 \pi^{2} V^{2} m^{3}}\right)^{1 / 3}\\ &=\frac{h^{2}}{2 m}\left(\frac{3 N}{8 \pi V}\right)^{1 / 3} \\ \Rightarrow E_{F} & =\frac{h^{2}}{2 m}\left(\frac{3 \rho}{8 \pi}\right)^{1 / 3} \end{aligned} $$

$$ \begin{aligned} & d x d p_{x} d y d p_{y} d z d p_{z}=h^{3} \\ \Rightarrow & d x d y d z\left(d p_{x} d p_{y} d p_{z}\right)=h^{3} \\ \Rightarrow & \left(d p_{x} d p_{y} d p_{z}\right)=\frac{h^{3}}{d x d y d z}=\frac{h^{3}}{V}\\ \Rightarrow \dfrac{4}{3} \pi p_{F}^{3}&=\dfrac{h^{3}}{V} \frac{N}{2} \end{aligned} $$

$$ \begin{aligned} \Rightarrow p_{F} &=\left(\frac{3 h^{3} N}{8 \pi V}\right)^{1 / 3} \\ \Rightarrow\left(2 m E_{F}\right)^{1 / 2} &=\left(\frac{3 h^{3} N}{8 \pi V}\right)^{1 / 3} \\ \Rightarrow E_{F} & =\frac{1}{2 m}\left(\frac{3 h^{3} N}{8 \pi V}\right)^{2 / 3} \\ \Rightarrow E_{F} &=\frac{h^{2}}{2 m}\left(\frac{3 N}{8 \pi V}\right)^{2 / 3} \\ \Rightarrow E_{F}&=\frac{h^{2}}{2 m}\left(\frac{3 \rho}{8 \pi}\right)^{2 / 3} \end{aligned} $$

\begin{align*} \Rightarrow \quad V_{F} &= \sqrt{\frac{2 E_{F}}{m}} \\ &= \sqrt{\frac{2}{m} \cdot \frac{h^{2}}{2m} \left( \frac{3\rho}{8\pi} \right)^{2/3}} \\ \Rightarrow \quad V_{F} &= \frac{h}{m} \left( \frac{3\rho}{8\pi} \right)^{1/3} \end{align*}

\begin{align*} p_{F} &= m V_{F} \\ & = m \cdot \frac{h}{m} \left( \frac{3\rho}{8\pi} \right)^{1/3} \\ &= h \left( \frac{3\rho}{8\pi} \right)^{1/3} \end{align*}

\begin{align*} T_{F} &= \frac{E_{F}}{k} \\ &= \frac{1}{k} \cdot \frac{h^{2}}{2m} \left( \frac{3\rho}{8\pi} \right)^{2/3} \\ &= \frac{h^{2}}{2mk} \left( \frac{3\rho}{8\pi} \right)^{2/3} \end{align*}

$$ \langle E\rangle=\frac{\int_{0}^{E_{F}} E n(E) d E}{\int_{0}^{E_{F}} n(E) d E}=\frac{\int_{0}^{E_{F}} E n(E) d E}{\int_{0}^{E_{F}} n(E) d E} $$

$$ \begin{aligned} & \int_{0}^{E_{F}} E \frac{8 \sqrt{2} \pi V m^{3 / 2}}{h^{3}} E^{1 / 2} d E \\ & \int_{0}^{E_{F}} \frac{8 \sqrt{2} \pi V m^{3 / 2}}{h^{3}} E^{1 / 2} d E \\ &= \int_{0}^{E_{F}} E^{3 / 2} d E \\ &= \frac{(2 / 5) E_{F}^{5 / 2}}{(2 / 3) E_{F}^{3 / 2}} \\ &= \frac{3}{5} E_{F} \end{aligned} $$

$$ \begin{aligned} \langle E\rangle & =\int_{0}^{E_{F}} E n(E) d E\\ &=\int_{0}^{E_{F}} E \frac{8 \sqrt{2} \pi V m^{3 / 2}}{h^{3}} E^{1 / 2} d E \\ & =\frac{8 \sqrt{2} \pi V m^{3 / 2}}{h^{3}} \int_{0}^{E_{F}} E^{3 / 2} d E\\ &=\frac{8 \sqrt{2} \pi V m^{3 / 2}}{h^{3}} \frac{2}{5} E_{F}^{5 / 2} \\ & =\frac{16 \sqrt{2} \pi V m^{3 / 2}}{5 h^{3}} E_{F}^{5 / 2} \\ & =\frac{16 \sqrt{2} \pi V m^{3 / 2}}{5 h^{3}} E_{F}^{3 / 2} E_{F}\\ &=\frac{16 \sqrt{2} \pi V m^{3 / 2}}{5 h^{3}}\left(\frac{3 N h^{3}}{16 \sqrt{2} \pi V m^{3 / 2}}\right) E_{F} \\ & =\frac{3}{5} N E_{F} \end{aligned} $$

\begin{align*} \langle v \rangle &= \frac{\int_{0}^{v_{F}} v\, n(v)\, dE}{\int_{0}^{v_{F}} n(v)\, dE} \\ &= \frac{\int_{0}^{v_{F}} v \cdot \frac{8\pi V m^{3}}{h^{3}} v^{2}\, dv}{\int_{0}^{v_{F}} \frac{8\pi V m^{3}}{h^{3}} v^{2}\, dv} \\ &= \frac{\frac{8\pi V m^{3}}{h^{3}} \int_{0}^{v_{F}} v^{3}\, dv}{\frac{8\pi V m^{3}}{h^{3}} \int_{0}^{v_{F}} v^{2}\, dv} \\ &= \frac{\int_{0}^{v_{F}} v^{3}\, dv}{\int_{0}^{v_{F}} v^{2}\, dv} \\ &= \frac{\frac{1}{4} v_{F}^{4}}{\frac{1}{3} v_{F}^{3}} \\ &= \frac{3}{4} v_{F} \end{align*}

\begin{align*} f(E) &= \frac{1}{e^{-\infty} + 1} = \frac{1}{\frac{1}{e^{\infty}} + 1} = \frac{1}{0 + 1} = \frac{1}{1} = 1 \end{align*}

\begin{align*} f(E) &= \frac{1}{e^{\infty} + 1} = \frac{1}{\infty + 1} = \frac{1}{\infty} = 0 \end{align*}

\begin{align*} f(E) &= \frac{1}{e^{0} + 1} = \frac{1}{1 + 1} = \frac{1}{2} = 0.5 \end{align*}