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\begin{document}
\large
\begin{center}
	2015�N�x���v�͊wII �h��5 (5��21���o��A5��28����o)
	��
\end  {center}
\begin{flushright}
	�S�� �g�X ��
\end  {flushright}
 [���1.] \maru{1} 5/14�̏h��1��$\varepsilon _F$�ƃG�l���M�[�ƈ��͂����߂�	\\
\maru{2} ���Ƃŗ����������ϕ����v�Z����$E=3/2PV$�������B	

\bigskip

\noindent[��]\maru{1} �t�F���~�G�l���M�[$\varepsilon_F$�́A���̂悤�ɂ��ċ��߂�B
$N=\sum_l\braket{n_l}$�����Ԃ����ɋl�܂��Ă��鎞�A
\begin{align}
	N 	&= \int_{-\infty}^\infty D(\varepsilon)f(\varepsilon)d\varepsilon	\\
\intertext{$f(\varepsilon)$�̓t�F���~�̕��z�֐���\���B
 $D(\varepsilon)$�����R���q�Ɍ��炸��ʓI�ȏꍇ���l���āA�ϕ��̉�����$-\infty$�ɂ����B
 $T=0$�ł́A$f(\varepsilon)$�͊K�i�֐��ɂȂ�̂ŁA}
	N	&= \int_{-\infty}^{\varepsilon_F} D(\varepsilon)d\varepsilon	\\
\intertext{$D(\varepsilon)$��������ƁA}
	N	&= \int_0^{\varepsilon_F} \frac{V}
		{2\pi^2(c\hbar)^3}\varepsilon^2d\varepsilon	\\
		&= \frac{V}
		{2\pi^2(c\hbar)^3}\frac{\varepsilon_F^3}
				{3}	\label{NepsF}
\end{align}
 $\varepsilon_F$�ɂ��ĉ����΁A
\begin{equation}
	\varepsilon_F = \left\{6\pi^2(c\hbar)^3\frac{N}
					{V}\right\}^{1/3}
				\label{epsilonF}
\end{equation}

 �G�l���M�[�́A
\begin{align}
	E 	&= \int_{-\infty}^\infty \varepsilon D(\varepsilon)f(\varepsilon)d\varepsilon	\\
\intertext{�Ə����邩��A$f(\varepsilon)$�ɊK�i�֐��������āA}
	E	&= \int_{-\infty}^{\varepsilon_F} \varepsilon D(\varepsilon)d\varepsilon	\\
\intertext{$D(\varepsilon)$�������āA}
	E	&= \int_0^{\varepsilon_F}\varepsilon \frac{V}
		{2\pi^2(c\hbar)^3}\varepsilon^2d\varepsilon	\\
		&= \frac{V}
		{2\pi^2(c\hbar)^3}\frac{\varepsilon_F^4}
				{4}
\end{align}
����A\siki{NepsF}����
\begin{equation}
	E	= \frac{3}
			{4}N\varepsilon_F
\end{equation}

 ���͂́A�ȉ��̌���(�u��g��b�����w�V���[�Y 7 �u���v�͊w�v�A�����m�� ���A��g���X�vP54
    (2.75)���A��K�m�[�gP16(69)��\footnote{%
\www http://www.cmt.phys.kyushu-u.ac.jp/%
\texttt{\symbol{"7E}}A.Yoshimori/hoshu15.pdf
}
)
\begin{equation}
	P	= -\left(\frac{\pd E}
			{\pd V}\right)_{S,N} \label{UVbibun}
\end  {equation}
  ����v�Z�ł���B
 �������A$S$�̓G���g���s�[�����A���A$T=0$�Ȃ̂ŁA��3�@������$S=0$�ƂȂ�B
 ���������āA$T=0$�̎��������ʂ�$S$���Ƃ��������͖��������B
 �܂�A
\begin{equation}
	P	= -\left.\frac{\pd}
			{\pd V}\left(\frac{3}
			{4}N\varepsilon_F\right)\right|_{N���}
\end  {equation}
 \siki{epsilonF}����$\varepsilon_F$�́A$V^{-1/3}$�ɔ�Ⴗ��̂ŁA
\begin{equation}
	P	= \frac{1}
			{3}\left(\frac{3}
			{4}N\frac{\varepsilon_F}
				{V}\right)
		= \frac{1}
			{4}N\frac{\varepsilon_F}
				{V}
\end  {equation}

\bigskip

\noindent\maru{2} �v�����g�u���ƃm�[�g1�v(29)����(30)����
\begin{equation}
	\Xi = \prod_l\Xi_l
		= \prod_l(1+ze^{-\beta\varepsilon_l})
\end   {equation}
 ���v�͊wI�v�����g�u���v�A���T���u���̂܂Ƃ߁v(��K�m�[�g�ł�$J$)����
\begin{equation}
	\Omega =-\kB T\ln\Xi 
	= -\kB T\sum_l\ln(1+ze^{-\beta\varepsilon_l}) \\
\end   {equation}
 ���ʂ����ɋl�܂��Ă����
\begin{equation}
	\Omega	 =  -\kB T\int^\infty_{-\infty}D(\varepsilon)
			\ln(1+ze^{-\beta\varepsilon})d\varepsilon
\end   {equation}
 $\Omega=-PV$(��K�m�[�g(101)��)������A
\begin{equation}
	PV = \kB T\int^\infty_{-\infty}D(\varepsilon)
			\ln(1+ze^{-\beta\varepsilon})d\varepsilon
\end   {equation}
 ���̎���
\begin{equation}
	D(\varepsilon) = \begin{cases}
			\displaystyle\left(\frac{2m}
			{\hbar^2}\right)^{3/2}\frac{ V}
					{(2\pi)^2}g\varepsilon^{1/2}
		& \varepsilon\ge 0	\\
		0 	& \varepsilon<0
		\end{cases}		\label{jiyuuDeps}
\end{equation}
  ���������
\begin{equation}
	PV	= \kB T\int^\infty_0\left(\frac{2m}
			{\hbar^2}\right)^{3/2}\frac{ V}
					{(2\pi)^2}
			g\varepsilon^{1/2}
			\ln(1+ze^{-\beta\varepsilon})d\varepsilon
\end   {equation}

 ����𕔕��ϕ�����B
\begin{multline}
	PV = \kB T\left(\frac{2m}
			{\hbar^2}\right)^{3/2}\frac{ V}
					{(2\pi)^2}g
			\left\{\left[\frac{2}
							{3}\varepsilon^{3/2}\ln(1+ze^{-\beta\varepsilon})
		\right]^\infty_0\right.
				\\\left.
		-\int^\infty_0\frac{2}
							{3}\varepsilon^{3/2}
			\frac{z(-\beta)e^{-\beta\varepsilon}}
			{(1+ze^{-\beta\varepsilon})}
		d\varepsilon\right\}
\end   {multline}
 $[\dotsb]$�́A$\varepsilon=0$�ŁA$\varepsilon^{3/2}$�̂��߂�0�A
    $\varepsilon\to\infty$�́A$e^{-\beta\varepsilon}\to 0$��0�ɂȂ�B
\begin{align}
	PV &= \kB T\left(\frac{2m}
			{\hbar^2}\right)^{3/2}\frac{ V}
					{(2\pi)^2}g
		\beta\int^\infty_0\frac{2}
					{3}\varepsilon^{3/2}
			\frac{1}
			{z^{-1}e^{\beta\varepsilon}+1}
		d\varepsilon			\\
\intertext{$\beta=1/(\kB T)$������}
	 &= \left(\frac{2m}
			{\hbar^2}\right)^{3/2}\frac{ V}
					{(2\pi)^2}g
	\int^\infty_0\frac{2}
			{3}\varepsilon^{3/2}
			\frac{1}
			{z^{-1}e^{\beta\varepsilon}+1}
		d\varepsilon		\label{PVepsi}
\end   {align}

 ����A�G�l���M�[$E$�́A$f(\varepsilon_l)$���t�F���~���z�֐��Ƃ���ƁA
\begin{align}
	E &= \sum_l\varepsilon_lf(\varepsilon_l)	\\
	&= \int^\infty_{-\infty}D(\varepsilon)\frac{\varepsilon}
			{z^{-1}e^{\beta\varepsilon}+1}
		d\varepsilon \\
\intertext{\siki{jiyuuDeps}��������}
	&=\left(\frac{2m}
			{\hbar^2}\right)^{3/2}\frac{ V}
					{(2\pi)^2}g
		\int^\infty_0\frac{\varepsilon \varepsilon^{1/2}}
			{z^{-1}e^{\beta\varepsilon}+1}
		d\varepsilon \\
	 &= \left(\frac{2m}
			{\hbar^2}\right)^{3/2}\frac{ V}
					{(2\pi)^2}g
		\int^\infty_0\frac{\varepsilon^{3/2}}
			{z^{-1}e^{\beta\varepsilon}+1}
		d\varepsilon	\label{Eeq}
\end   {align}
 \siki{PVepsi}��\eqref{Eeq}�����ׂ�΁A$E=(3/2)PV$��������B




\bigskip

\noindent [���2.] $T$���܂܂Ȃ�$X(\varepsilon)$�ɂ�
��$\int_{0}^{\infty}X(\varepsilon)f(\varepsilon)d
\varepsilon$��$T^{2}$�܂Ńe�[���[�W�J���Ȃ����B$f(\varepsilon)$�̓t�F��
�~���z�֐��B


\bigskip

\noindent[��] �܂��A
\begin{equation}
	\largetilde{X}(\varepsilon) 	= \int_0^{\varepsilon}X(\varepsilon')d\varepsilon'
					\label{tildeteigi}
\end{equation}
  �Ƃ����ƁA
\begin{equation}
	X(\varepsilon) 	= \frac{d\largetilde{X}(\varepsilon)}
			{d\varepsilon}
\end{equation}
  �ł���A�^���𕔕��ϕ�����ƁA
\begin{align}
	\int_{0}^{\infty}X(\varepsilon)f(\varepsilon)d\varepsilon
	 &= \int_{0}^{\infty}\frac{d\largetilde{X}(\varepsilon)}
			{d\varepsilon}f(\varepsilon)d\varepsilon	\\
		&= \left[\largetilde{X}(\varepsilon)f(\varepsilon)
			\right]_0^\infty
			-\int_{0}^{\infty}\largetilde{X}(\varepsilon)
			\frac{df(\varepsilon)}
			{d\varepsilon}d\varepsilon		\\
\intertext{$\largetilde{X}(0)=0$�A$f(\infty)=0$������}
		&= -\int_{0}^{\infty}\largetilde{X}(\varepsilon)
			\frac{df(\varepsilon)}
			{d\varepsilon}d\varepsilon	\label{bubunn}
\end  {align}

 $-df(\varepsilon)/d\varepsilon$���v�Z����ƁA
\begin{equation}
	-\frac{df(\varepsilon)}
			{d\varepsilon}
	= \frac{1}
		{\kB T}\frac{1}
			{(\exp[(\varepsilon-\mu)/\kB T]+1)
			(\exp[-(\varepsilon-\mu)/\kB T]+1)}
						\label{fbibun}
\end   {equation}
 ���̊֐���$\varepsilon=\mu$�̂܂���$\kB T$���x�̕��̃s�[�N�����B
 ���������āA$\largetilde{X}(\varepsilon)$��$\varepsilon=\mu$�̂܂��œW�J����̂́A
  $T$�œW�J����̂Ɠ���������A2���܂œW�J����ƁA
\begin{equation}
	\largetilde{X}(\varepsilon)
	= \largetilde{X}(\mu)
	+(\varepsilon-\mu)\left.\frac{d\largetilde{X}(\varepsilon)}
			{d\varepsilon}\right|_{\varepsilon=\mu}
	+\frac{(\varepsilon-\mu)^2}
		{2}\left.\frac{d^2\largetilde{X}(\varepsilon)}
			{d\varepsilon^2}\right|_{\varepsilon=\mu}
\end   {equation}
 �����\siki{bubunn}�ɑ��
\begin{multline}
	-\int_{0}^{\infty}\largetilde{X}(\varepsilon)
			\frac{df(\varepsilon)}
			{d\varepsilon}d\varepsilon
		= -\int_{0}^{\infty}\left(\largetilde{X}(\mu)
	+(\varepsilon-\mu)\left.\frac{d\largetilde{X}(\varepsilon)}
			{d\varepsilon}\right|_{\varepsilon=\mu}\right. \\
	\left.+\frac{(\varepsilon-\mu)^2}
		{2}\left.\frac{d^2\largetilde{X}(\varepsilon)}
			{d\varepsilon^2}\right|_{\varepsilon=\mu}\right)
			\frac{df(\varepsilon)}
			{d\varepsilon}d\varepsilon
\end  {multline}
 1���ڂ́A
\begin{align}
	-\int_{0}^{\infty}\largetilde{X}(\mu)\frac{df(\varepsilon)}
			{d\varepsilon}d\varepsilon
	&= -\largetilde{X}(\mu)\int_{0}^{\infty}\frac{df(\varepsilon)}
			{d\varepsilon}d\varepsilon		\\
\intertext{$-df(\varepsilon)/d\varepsilon$�́A�s���s�[�N�̊֐�������A
  ������$-\infty$�ɂ��āA}
	&= -\largetilde{X}(\mu)\int_{-\infty}^{\infty}\frac{df(\varepsilon)}
			{d\varepsilon}d\varepsilon		\\
	&= \largetilde{X}(\mu)
\end   {align}
 2���ڂ́A$-df(\varepsilon)/d\varepsilon$��$\varepsilon=\mu$�̎��ɑ΂���
   �Ώ̂Ȃ̂ŁA0�ɂȂ�B
 3���ڂ́A
\begin{align}
	&-\int_{0}^{\infty}\frac{(\varepsilon-\mu)^2}
		{2}\left.\frac{d^2\largetilde{X}(\varepsilon)}
			{d\varepsilon^2}\right|_{\varepsilon=\mu}\frac{df(\varepsilon)}
			{d\varepsilon}d\varepsilon		\\
	&= -\frac{1}
		{2}\left.\frac{d^2\largetilde{X}(\varepsilon)}
			{d\varepsilon^2}\right|_{\varepsilon=\mu}
	\int_{0}^{\infty}(\varepsilon-\mu)^2\frac{df(\varepsilon)}
			{d\varepsilon}d\varepsilon		\\
\intertext{$-df(\varepsilon)/d\varepsilon$��\siki{fbibun}�������A
  �ϕ��̉�����$-\infty$�ɂ��āA
  ����(�Ⴆ�΁A��g��b�����w�V���[�Y 7 �u���v�͊w�v�A�����m�� ���A��g���XP278(A.10))
  ���g���΁A}
	&= \frac{1}
		{2}\left.\frac{d^2\largetilde{X}(\varepsilon)}
			{d\varepsilon^2}\right|_{\varepsilon=\mu}
	\frac{\pi^2}
		{3}(\kB T)^2	\\
\intertext{$\largetilde{X}(\varepsilon)$�̒�`��^����\siki{tildeteigi}����A}
	&= \frac{1}
		{2}\left.\frac{dX(\varepsilon)}
			{d\varepsilon}\right|_{\varepsilon=\mu}
	\frac{\pi^2}
		{3}(\kB T)^2
\end   {align}

 ����
 \begin{equation}
	\int_{0}^{\infty}X(\varepsilon)f(\varepsilon)d\varepsilon
		= \largetilde{X}(\mu)
	+\frac{\pi^2}
		{6}(\kB T)^2\left.\frac{dX(\varepsilon)}
			{d\varepsilon}\right|_{\varepsilon=\mu}
\end  {equation}

\bigskip

\begin{center}
	2015�N�x���v�͊wII �h��4 (5��14���o��A5��28�����ߐ؂艄��)
	��
\end  {center}

\noindent [���2.] (���ƂŐ�������)���R���q(�����I�A�������R�x$g$�A�񑊑Θ_)��
  $T=0$�̂Ƃ��Ⴂ���ʂ��珇��$N$�̗��q���߂čő�g��$k_F$�����߂�B
   $\varepsilon_F$���g���Ă͂����Ȃ��B

\bigskip


\noindent[��]	�܂��A���a$k_F$�̋����̏�Ԑ��𐔂���B
 ����͎��ƂŐ��������̂Ɠ����ŁA�������R�x���l����ƁA
\begin{align}
	\text{��Ԑ�} &= ���ak_F�̋��̓����ɂ���_�̐�\times g	\\
		&= g\frac{���̑̐�}
			{(�Ԋu)^3}		\\
		&= g\frac{\frac{4\pi}
				{3}k_F^3}
			{\left(\frac{2\pi}
			{L}\right)^3}		\\
\intertext{$L^3=V$������}
		&= g\frac{4\pi}
				{3}k_F^3
			\frac{V}
			{(2\pi)^3}
\end  {align}
 ���ꂪ$N$�ɓ������̂ŁA$k_F$�ɂ��ĉ�����
\begin{equation}
	k_F	= \left(6\pi^2\frac{N}
			{gV}\right)^{1/3}
\end   {equation}
 $k_F$��\textbf{�t�F���~�g��}�ƌĂ΂��B

\end  {document}