Virial theorem about Hydrogen atom

\documentclass{jsarticle}
\parindent=0pt
\pagestyle{empty}
\begin{document}
\section{Gamma equation}
\begin{equation}
\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt
\end{equation}
\begin{equation}
\int_0^\infty t^{z-1}e^{-\alpha t}\,dt=\frac{\Gamma(z)}{\alpha^z}
\end{equation}

\section{spherical harmonics}

\begin{eqnarray}
Y_{0,0}&=&\sqrt{\frac1{4\pi}}\\
Y_{1,0}&=&\sqrt{\frac3{4\pi}}\cos \theta\\
Y_{1,\pm1}&=&\mp\sqrt{\frac3{8\pi}}\sin \theta e^{\pm i\varphi}\\
Y_{2,0}&=&\sqrt{\frac5{16\pi}}(3\cos^2 \theta -1)\\
Y_{2,\pm1}&=&\mp\sqrt{\frac{15}{8\pi}}\sin \theta\cos \theta e^{\pm i\varphi}\\
Y_{2,\pm2}&=&\mp\sqrt{\frac{15}{32\pi}}\sin^2 \theta e^{\pm i2\varphi}
\end{eqnarray}

\section{H\_atom}
\subsection{1s}
\begin{equation}
\psi_\mathrm{1s}=\frac1{\sqrt{\pi}}e^{-r}
\end{equation}
\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{1s}|\psi_\mathrm{1s}\right>&=&\frac1{\pi}\int e^{-2r}\,dV\\
&=&4\int r^2e^{-2r}\,dr\\
&=&4\frac{\Gamma(3)}{2^3}\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{1s}\left|-\frac12\nabla^2\right|\psi_\mathrm{1s}\right>\\
&=&-\frac1{2\pi}\int e^{-r}\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)e^{-r}\,dV\\
&=&2\int e^{-r}\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}\right)e^{-r}r^2\,dr\\
&=&-2\int e^{-r}\left(1-\frac{2}{r}\right)e^{-r}r^2\,dr\\
&=&-2\int (r^2-2r)e^{-2r}\,dr\\
&=&-2\left[\frac{\Gamma(3)}{2^3}-2\frac{\Gamma(2)}{2^2}\right]\\
&=&-2\left(\frac14-\frac12\right)\\
&=&\frac12
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{1s}\left|-\frac1{r}\right|\psi_\mathrm{1s}\right>\\
&=&-\frac1{\pi}\int e^{-r}\frac1{r}e^{-r}\,dV\\
&=&-4\int \frac1{r}e^{-2r}r^2\,dr\\
&=&-4\int r e^{-2r}\,dr\\
&=&-4\frac{\Gamma(2)}{2^2}\\
&=&-1
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac12-1\\
&=&-\frac12
\end{eqnarray*}

\newpage

\subsection{2s}
\begin{equation}
\psi_\mathrm{2s}=\frac1{\sqrt{32\pi}}(2-r)e^{-\frac{r}2}
\end{equation}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{2s}|\psi_\mathrm{2s}\right>&=&\frac1{32\pi}\int (2-r)^2e^{-r}\,dV\\
&=&\frac18\int (r^4-4r^3+4r^2)e^{-r}\,dr\\
&=&\frac18\left[\Gamma(5)-4\Gamma(4)+4\Gamma(3)\right]\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{2s}\left|-\frac12\nabla^2\right|\psi_\mathrm{2s}\right>\\
&=&-\frac1{64\pi}\int (2-r)e^{-\frac{r}2}\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)(2-r)e^{-\frac{r}2}\,dV\\
&=&-\frac1{16}\int r^2(2-r)e^{-\frac{r}2}\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}\right)(2-r)e^{-\frac{r}2}\,dr\\
&=&-\frac1{16}\int r^2(2-r)e^{-\frac{r}2}\left(1+\frac{2-r}4-\frac2{r}-\frac{2-r}{r}\right)e^{-\frac{r}2}\,dr\\
&=&-\frac1{16}\int \left(\frac14 r^4-3 r^3+9r^2-8r\right)e^{-r}\,dr\\
&=&-\frac1{16}\left[\frac14\Gamma(5)-3\Gamma(4)+9\Gamma(3)-8\Gamma(2)\right]\\
&=&-\frac{6-18+18-8}{16}\\
&=&\frac18
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{2s}\left|-\frac1{r}\right|\psi_\mathrm{2s}\right>\\
&=&-\frac1{32\pi}\int (2-r)e^{-\frac{r}2}\frac1{r}(2-r)e^{-\frac{r}2}\,dV\\
&=&-\frac18\int \frac{(2-r)^2}{r}e^{-r}r^2\,dr\\
&=&-\frac18\int (r^3-4r^2+4r)e^{-r}\,dr\\
&=&-\frac18\left[\Gamma(4)-4\Gamma(3)+4\Gamma(2)\right]\\
&=&-\frac{6-8+4}{8}\\
&=&-\frac14
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac18-\frac14\\
&=&-\frac18
\end{eqnarray*}

\newpage

\subsection{2px}
\begin{eqnarray}
\psi_\mathrm{2p_x}&=&\frac1{\sqrt{32\pi}}re^{-\frac{r}2}\sin \theta\cos \varphi\\
&=&\frac1{\sqrt{24}}re^{-\frac{r}2}\frac1{\sqrt 2}\left(-Y_{1,+1}+Y_{1,-1}\right)
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{2p_x}|\psi_\mathrm{2p_x}\right>&=&\frac1{32\pi}\int r^2e^{-r}\sin^2 \theta \cos^2 \varphi\,dV\\
&=&\frac1{32\pi}\int r^4e^{-r}\,dr\int_0^{\pi}\sin^3 \theta\,d\theta\int_0^{2\pi}\cos^2\varphi\,d\varphi\\
&=&\frac1{32\pi}\Gamma(5)\left[-\cos \theta+\frac13\cos^3 \theta\right]_0^\pi\left[\frac{1+\cos{2\varphi}}2\right]_0^{2\pi}\\
&=&\frac1{32\pi}\times 24\times \frac43 \times \pi\\
&=&1
\end{eqnarray*}


\subsubsection{$\left$}
\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{2p_x}\left|-\frac12\nabla^2\right|\psi_\mathrm{2p_x}\right>\\
&=&-\frac1{96}\int re^{-\frac{r}2}(-Y_{1,+1}+Y_{1,-1})^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)re^{-\frac{r}2}(-Y_{1,+1}+Y_{1,-1})\,dV\\
&=&-\frac1{96}\int (\frac14r^4-2r^3)e^{-r}\,dr \int\!\!\!\int|-Y_{1,+1}+Y_{1,-1}|^2\sin \theta\,d\theta\,d\varphi\\
&=&-\frac1{48}\left[\frac14\Gamma(5)-2\Gamma(4)\right]\\
&=&-\frac{6-12}{48}\\
&=&\frac1{8}
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{2p_x}\left|-\frac1{r}\right|\psi_\mathrm{2p_x}\right>\\
&=&-\frac1{48}\int re^{-r}|-Y_{1,+1}+Y_{1,-1}|^2\,dV\\
&=&-\frac1{48}\int r^3e^{-r}\,dr\int\!\!\!\int |-Y_{1,+1}+Y_{1,-1}|^2\sin \theta\,d\theta\,d\varphi\\
&=&-\frac1{24}\Gamma(4)\\
&=&-\frac14
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac18-\frac14\\
&=&-\frac18
\end{eqnarray*}

\newpage

\subsection{2py}
\begin{eqnarray}
\psi_\mathrm{2p_y}&=&\frac1{\sqrt{32\pi}}re^{-\frac{r}2}\sin \theta\sin \varphi\\
&=&\frac1{\sqrt{24}}re^{-\frac{r}2}\frac{i}{\sqrt 2}\left(Y_{1,+1}+Y_{1,-1}\right)
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{2p_y}|\psi_\mathrm{2p_y}\right>&=&\frac1{32\pi}\int r^2e^{-r}\sin^2 \theta \sin^2 \varphi\,dV\\
&=&\frac1{32\pi}\int r^4e^{-r}\,dr\int_0^{\pi}\sin^3 \theta\,d\theta\int_0^{2\pi}\sin^2\varphi\,d\varphi\\
&=&\frac1{32\pi}\Gamma(5)\left[-\cos \theta+\frac13\cos^3 \theta\right]_0^\pi\left[\frac{1-\cos{2\varphi}}2\right]_0^{2\pi}\\
&=&\frac1{32\pi}\times 24\times \frac43 \times \pi\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{2p_y}\left|-\frac12\nabla^2\right|\psi_\mathrm{2p_y}\right>\\
&=&-\frac1{96}\int re^{-\frac{r}2}(Y_{1,+1}+Y_{1,-1})^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)re^{-\frac{r}2}(Y_{1,+1}+Y_{1,-1})\,dV\\
&=&-\frac1{96}\int (\frac14r^4-2r^3)e^{-r}\,dr \int\!\!\!\int|Y_{1,+1}+Y_{1,-1}|^2\sin \theta\,d\theta\,d\varphi\\
&=&-\frac1{48}\left[\frac14\Gamma(5)-2\Gamma(4)\right]\\
&=&-\frac{6-12}{48}\\
&=&\frac1{8}
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{2p_y}\left|-\frac1{r}\right|\psi_\mathrm{2p_y}\right>\\
&=&-\frac1{48}\int re^{-r}|Y_{1,+1}+Y_{1,-1}|^2\,dV\\
&=&-\frac1{24}\int r^3e^{-r}\,dr\int\!\!\!\int \frac{|Y_{1,+1}+Y_{1,-1}|^2}2\sin \theta\,d\theta\,d\varphi\\
&=&-\frac1{24}\Gamma(4)\\
&=&-\frac14
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac18-\frac14\\
&=&-\frac18
\end{eqnarray*}

\newpage

\subsection{2pz}
\begin{eqnarray}
\psi_\mathrm{2p_z}&=&\frac1{\sqrt{32\pi}}re^{-\frac{r}2}\cos \theta\\
&=&\frac1{\sqrt{24}}re^{-\frac{r}2}Y_{1,0}
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{2p_z}|\psi_\mathrm{2p_z}\right>&=&\frac1{32\pi}\int r^2e^{-r}\cos^2 \theta\,dV\\
&=&\frac1{32\pi}\int r^4e^{-r}\,dr\int_0^{\pi}\cos^2 \theta\sin \theta\,d\theta\int_0^{2\pi} \,d\varphi\\
&=&\frac1{32\pi}\Gamma(5)\left[-\frac13\cos^3 \theta\right]_0^\pi \times 2\pi\\
&=&\frac1{32\pi}\times 24\times \frac23 \times 2\pi\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{2p_z}\left|-\frac12\nabla^2\right|\psi_\mathrm{2p_z}\right>\\
&=&-\frac1{48}\int re^{-\frac{r}2}Y_{1,0}^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)re^{-\frac{r}2}Y_{1,0}\,dV\\
&=&-\frac1{48}\int (\frac14r^4-2r^3)e^{-r}\,dr \int\!\!\!\int|Y_{1,0}|^2\sin \theta\,d\theta\,d\varphi\\
&=&-\frac1{48}\left[\frac14\Gamma(5)-2\Gamma(4)\right]\\
&=&-\frac{6-12}{48}\\
&=&\frac1{8}
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{2p_z}\left|-\frac1{r}\right|\psi_\mathrm{2p_z}\right>\\
&=&-\frac1{24}\int re^{-r}|Y_{1,0}|^2\,dV\\
&=&-\frac1{24}\int r^3e^{-r}\,dr\int\!\!\!\int |Y_{1,0}|^2\sin \theta\,d\theta\,d\varphi\\
&=&-\frac1{24}\Gamma(4)\\
&=&-\frac14
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac18-\frac14\\
&=&-\frac18
\end{eqnarray*}

\newpage

\subsection{3s}
\begin{equation}
\psi_\mathrm{3s}=\frac1{\sqrt{6561\times 3\pi}}\left(27-18r+2r^2\right)e^{-\frac{r}3}
\end{equation}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{3s}|\psi_\mathrm{3s}\right>&=&\frac1{6561\times 3\pi}\int \left(27-18r+2r^2\right)^2e^{-\frac{2r}3}\,dV\\
&=&\frac4{6561\times 3}\int \left(729-972r+432r^2-72r^3+4r^4\right)r^2 e^{-\frac{2r}3}\,dr\\
&=&\frac4{6561\times 3}\left[729\Gamma(3)\left(\frac32\right)^3-972\Gamma(4)\left(\frac32\right)^4+432\Gamma(5)\left(\frac32\right)^5-72\Gamma(6)\left(\frac32\right)^6+4\Gamma(7)\left(\frac32\right)^7\right]\\
&=&\frac4{3^9}\left(\frac{3^9}4-\frac{3^{10}}2+3^9\times 4-3^9\times 5+\frac52\times 3^9\right)\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3s}\left|-\frac12\nabla^2\right|\psi_\mathrm{3s}\right>\\
&=&-\frac1{6561\times 6\pi}\int \left(27-18r+2r^2\right)e^{-\frac{r}3}\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)\left(27-18r+2r^2\right)e^{-\frac{r}3}\,dV\\
&=&-\frac2{6561\times 3}\int \left(27-18r+2r^2\right)\left(\frac29r^2-6r+39-\frac{54}{r}\right)e^{-\frac23r} r^2\,dr\\
&=&-\frac2{6561\times 3}\int \left(\frac49r^6-16r^5+192r^4-4\times3^5r^3+25\times 3^4r^2-2\times 3^6r\right)e^{-\frac23r}\,dr\\
&=&-\frac2{6561\times 3}\left[\frac49\Gamma(7)\left(\frac32\right)^7-16\Gamma(6)\left(\frac32\right)^6+192\Gamma(5)\left(\frac32\right)^5\right.\\
&&{}-\left.4\times3^5\Gamma(4)\left(\frac32\right)^4+25\times 3^4\Gamma(3)\left(\frac32\right)^3-2\times 3^6\Gamma(2)\left(\frac32\right)^2\right]\\
&=&-\frac2{6561\times 3}\left(\frac52\times 3^7-10\times 3^7+16\times 3^7-\frac12\times 3^{10}+\frac{25}4\times 3^7-\frac12\times 3^8\right)\\
&=&\frac2{6561\times 3}\times\frac14\times 3^7\\
&=&\frac1{18}
\end{eqnarray*}

\newpage
\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3s}\left|-\frac1{r}\right|\psi_\mathrm{3s}\right>\\
&=&-\frac1{6561\times 3\pi}\int \frac1{r}\left(27-18r+2r^2\right)^2e^{-\frac23r}\,dV\\
&=&-\frac4{6561\times 3}\int \left(729r-972r^2+432r^3-72r^4+4r^5\right)e^{-\frac{2r}3}\,dr\\
&=&\frac4{6561\times 3}\left[729\Gamma(2)\left(\frac32\right)^2-972\Gamma(3)\left(\frac32\right)^3+432\Gamma(4)\left(\frac32\right)^4-72\Gamma(5)\left(\frac32\right)^5+4\Gamma(6)\left(\frac32\right)^6\right]\\
&=&\frac4{3^9}\left(\frac{3^8}4-3^8+2\times 3^8-2\times 3^8+\frac52\times 3^7\right)\\
&=&-\frac19
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac1{18}-\frac19\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage

\subsection{3px}
\begin{eqnarray}
\psi_\mathrm{3p_x}&=&\frac4{\sqrt{6561\times 4\pi}}r\left(6-r\right)e^{-\frac{r}3}\sin \theta\cos \varphi\\
&=&\frac4{\sqrt{6561\times 6}}r\left(6-r\right)e^{-\frac{r}3}\frac1{\sqrt 2}(-Y_{1,+1}+Y_{1,-1})
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{3p_x}|\psi_\mathrm{3p_x}\right>&=&\frac{16}{6561\times 6}\int r^2\left(6-r\right)^2e^{-\frac23r}\frac{|-Y_{1,+1}+Y_{1,-1}|^2}2\,dV\\
&=&\frac{16}{6561\times 6}\int (r^6-12r^5+36r^4)e^{-\frac23r}\,dr\int\!\!\!\int\frac{|-Y_{1,+1}+Y_{1,-1}|^2}2\sin \theta \,d\theta\,d\varphi\\
&=&\frac{16}{6561\times 6}\left[\Gamma(7)\left(\frac32\right)^7-12\Gamma(6)\left(\frac32\right)^6+36\Gamma(5)\left(\frac32\right)^5\right]\\
&=&\frac{16}{6561\times 6}\left(\frac58\times 3^9-\frac52\times 3^8+3^8\right)\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3p_x}\left|-\frac12\nabla^2\right|\psi_\mathrm{3p_x}\right>\\
&=&-\frac4{6561\times 6}\int r\left(6-r\right)e^{-\frac{r}3}(-Y_{1,+1}+Y_{1,-1})^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)r\left(6-r\right)e^{-\frac{r}3}(-Y_{1,+1}+Y_{1,-1})\,dV\\
&=&-\frac8{6561\times 6}\int (6r-r^2)\left(-\frac19r^4+\frac83r^3-12r^2\right)e^{-\frac23r}\,dr\int\!\!\!\int\frac{|-Y_{1,+1}+Y_{1,-1}|^2}2\sin \theta \,d\theta\,d\varphi\\
&=&-\frac8{6561\times 6}\int\left(\frac19r^6-\frac{10}3r^5+28r^4-72r^3\right)e^{-\frac23r}\,dr\\
&=&-\frac8{6561\times 6}\left[\frac19\Gamma(7)\left(\frac32\right)^7-\frac{10}3\Gamma(6)\left(\frac32\right)^6+28\Gamma(5)\left(\frac32\right)^5-72\Gamma(4)\left(\frac32\right)^4\right]\\
&=&-\frac8{6561\times 6}\left(\frac58\times 3^7-\frac{25}4\times 3^6+7\times 3^6-3^7\right)\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3p_x}\left|-\frac1{r}\right|\psi_\mathrm{3p_x}\right>\\
&=&-\frac{16}{6561\times 6}\int r^3(6-r)^2e^{-\frac23r}\,dr\int\!\!\!\int \frac{|-Y_{1,+1}+Y_{1,-1}|^2}2\,d\theta\,d\varphi\\
&=&-\frac{16}{6561\times 6}\int (r^5-12r^4+36r^3)e^{-\frac23r}\,dr\\
&=&-\frac{16}{6561\times 6}\left[\Gamma(6)\left(\frac32\right)^6-12\Gamma(5)\left(\frac32\right)^5+36\Gamma(4)\left(\frac32\right)^4\right]\\
&=&-\frac{16}{6561\times 6}\left(\frac58\times 3^7-3^7+\frac12\times 3^7 \right)\\
&=&-\frac19
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac1{18}-\frac19\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage

\subsection{3py}
\begin{eqnarray}
\psi_\mathrm{3p_y}&=&\frac4{\sqrt{6561\times 4\pi}}r\left(6-r\right)e^{-\frac{r}3}\sin \theta\sin \varphi\\
&=&\frac4{\sqrt{6561\times 6}}r\left(6-r\right)e^{-\frac{r}3}\frac{i}{\sqrt 2}(Y_{1,+1}+Y_{1,-1})
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{3p_y}|\psi_\mathrm{3p_y}\right>&=&\frac{16}{6561\times 6}\int r^2\left(6-r\right)^2e^{-\frac23r}\frac{|Y_{1,+1}+Y_{1,-1}|^2}2\,dV\\
&=&\frac{16}{6561\times 6}\int (r^6-12r^5+36r^4)e^{-\frac23r}\,dr\int\!\!\!\int\frac{|Y_{1,+1}+Y_{1,-1}|^2}2\sin \theta \,d\theta\,d\varphi\\
&=&\frac{16}{6561\times 6}\left[\Gamma(7)\left(\frac32\right)^7-12\Gamma(6)\left(\frac32\right)^6+36\Gamma(5)\left(\frac32\right)^5\right]\\
&=&\frac{16}{6561\times 6}\left(\frac58\times 3^9-\frac52\times 3^8+3^8\right)\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3p_y}\left|-\frac12\nabla^2\right|\psi_\mathrm{3p_y}\right>\\
&=&-\frac4{6561\times 6}\int r\left(6-r\right)e^{-\frac{r}3}(Y_{1,+1}+Y_{1,-1})^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)r\left(6-r\right)e^{-\frac{r}3}(Y_{1,+1}+Y_{1,-1})\,dV\\
&=&-\frac8{6561\times 6}\int (6r-r^2)\left(-\frac19r^4+\frac83r^3-12r^2\right)e^{-\frac23r}\,dr\int\!\!\!\int\frac{|Y_{1,+1}+Y_{1,-1}|^2}2\sin \theta \,d\theta\,d\varphi\\
&=&-\frac8{6561\times 6}\int\left(\frac19r^6-\frac{10}3r^5+28r^4-72r^3\right)e^{-\frac23r}\,dr\\
&=&-\frac8{6561\times 6}\left[\frac19\Gamma(7)\left(\frac32\right)^7-\frac{10}3\Gamma(6)\left(\frac32\right)^6+28\Gamma(5)\left(\frac32\right)^5-72\Gamma(4)\left(\frac32\right)^4\right]\\
&=&-\frac8{6561\times 6}\left(\frac58\times 3^7-\frac{25}4\times 3^6+7\times 3^6-3^7\right)\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3p_y}\left|-\frac1{r}\right|\psi_\mathrm{3p_y}\right>\\
&=&-\frac{16}{6561\times 6}\int r^3(6-r)^2e^{-\frac23r}\,dr\int\!\!\!\int \frac{|Y_{1,+1}+Y_{1,-1}|^2}2\,d\theta\,d\varphi\\
&=&-\frac{16}{6561\times 6}\int (r^5-12r^4+36r^3)e^{-\frac23r}\,dr\\
&=&-\frac{16}{6561\times 6}\left[\Gamma(6)\left(\frac32\right)^6-12\Gamma(5)\left(\frac32\right)^5+36\Gamma(4)\left(\frac32\right)^4\right]\\
&=&-\frac{16}{6561\times 6}\left(\frac58\times 3^7-3^7+\frac12\times 3^7 \right)\\
&=&-\frac19
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac1{18}-\frac19\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage

\subsection{3pz}
\begin{eqnarray}
\psi_\mathrm{3p_z}&=&\frac4{\sqrt{6561\times 8\pi}}r\left(6-r\right)e^{-\frac{r}3}\cos \theta\\
&=&\frac4{\sqrt{6561\times 6}}r\left(6-r\right)e^{-\frac{r}3}Y_{1,0}
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{3p_z}|\psi_\mathrm{3p_z}\right>&=&\frac{16}{6561\times 6}\int r^2\left(6-r\right)^2e^{-\frac23r}|Y_{1,0}|^2\,dV\\
&=&\frac{16}{6561\times 6}\int (r^6-12r^5+36r^4)e^{-\frac23r}\,dr\int\!\!\!\int|Y_{1,0}|^2\sin \theta \,d\theta\,d\varphi\\
&=&\frac{16}{6561\times 6}\left[\Gamma(7)\left(\frac32\right)^7-12\Gamma(6)\left(\frac32\right)^6+36\Gamma(5)\left(\frac32\right)^5\right]\\
&=&\frac{16}{6561\times 6}\left(\frac58\times 3^9-\frac52\times 3^8+3^8\right)\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3p_z}\left|-\frac12\nabla^2\right|\psi_\mathrm{3p_z}\right>\\
&=&-\frac4{6561\times 6}\int r\left(6-r\right)e^{-\frac{r}3}Y_{1,0}^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)r\left(6-r\right)e^{-\frac{r}3}Y_{1,0}\,dV\\
&=&-\frac8{6561\times 6}\int (6r-r^2)\left(-\frac19r^4+\frac83r^3-12r^2\right)e^{-\frac23r}\,dr\int\!\!\!\int |Y_{1,0}|^2\sin \theta \,d\theta\,d\varphi\\
&=&-\frac8{6561\times 6}\int\left(\frac19r^6-\frac{10}3r^5+28r^4-72r^3\right)e^{-\frac23r}\,dr\\
&=&-\frac8{6561\times 6}\left[\frac19\Gamma(7)\left(\frac32\right)^7-\frac{10}3\Gamma(6)\left(\frac32\right)^6+28\Gamma(5)\left(\frac32\right)^5-72\Gamma(4)\left(\frac32\right)^4\right]\\
&=&-\frac8{6561\times 6}\left(\frac58\times 3^7-\frac{25}4\times 3^6+7\times 3^6-3^7\right)\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3p_z}\left|-\frac1{r}\right|\psi_\mathrm{3p_z}\right>\\
&=&-\frac{16}{6561\times 6}\int r^3(6-r)^2e^{-\frac23r}\,dr\int\!\!\!\int |Y_{1,0}|^2\,d\theta\,d\varphi\\
&=&-\frac{16}{6561\times 6}\int (r^5-12r^4+36r^3)e^{-\frac23r}\,dr\\
&=&-\frac{16}{6561\times 6}\left[\Gamma(6)\left(\frac32\right)^6-12\Gamma(5)\left(\frac32\right)^5+36\Gamma(4)\left(\frac32\right)^4\right]\\
&=&-\frac{16}{6561\times 6}\left(\frac58\times 3^7-3^7+\frac12\times 3^7 \right)\\
&=&-\frac19
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac1{18}-\frac19\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage

\subsection{3dxy}
\begin{eqnarray}
\psi_\mathrm{3d_{xy}}&=&\frac4{\sqrt{6561\times 8\pi}}r^2e^{-\frac{r}3}\sin^2 \theta\sin \varphi\cos \varphi\\
&=&\frac{4}{\sqrt{6561\times 30}}r^2e^{-\frac{r}3}\frac{i}{\sqrt 2}(Y_{2,+2}-Y_{2,-2})
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{3d_{xy}}|\psi_\mathrm{3d_{xy}}\right>&=&\frac{16}{6561\times 30}\int r^4e^{-\frac23r}\frac{|Y_{2,+2}-Y_{2,-2}|^2}2\,dV\\
&=&\frac{16}{6561\times 30}\int r^6e^{-\frac23r}\,dr\int\!\!\!\int\frac{|Y_{2,+2}-Y_{2,-2}|^2}2\sin \theta \,d\theta\,d\varphi\\
&=&\frac{16}{6561\times 30}\Gamma(7)\left(\frac32\right)^7\\
&=&\frac{16}{6561\times 30}\left(\frac58\times 3^9\right)\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3d_{xy}}\left|-\frac12\nabla^2\right|\psi_\mathrm{3d_{xy}}\right>\\
&=&-\frac4{6561\times 30}\int r^2e^{-\frac{r}3}(Y_{2,+2}-Y_{2,-2})^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)r^2e^{-\frac{r}3}(Y_{2,+2}-Y_{2,-2})\,dV\\
&=&-\frac8{6561\times 30}\int \left(\frac19r^6-2r^5\right)e^{-\frac23r}\,dr\int\!\!\!\int \frac{|Y_{2,+2}-Y_{2,-2}|^2}2\,d\theta d\varphi\\
&=&-\frac8{6561\times 30}\left[\frac19\Gamma(7)\left(\frac32\right)^7-2\Gamma(6)\left(\frac32\right)^6\right]\\
&=&-\frac8{6561\times 30}\left(\frac58\times 3^7-\frac54\times 3^7\right)\\
&=&\frac1{18}
\end{eqnarray*}

\newpage

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3d_{xy}}\left|-\frac1{r}\right|\psi_\mathrm{3d_{xy}}\right>\\
&=&-\frac{16}{6561\times 30}\int r^3e^{-\frac23r}\frac{|Y_{2,+2}-Y_{2,-2}|^2}2\,dV\\
&=&-\frac{16}{6561\times 30}\int r^5e^{-\frac23r}\,dr\int\!\!\!\int \frac{|Y_{2,+2}-Y_{2,-2}|^2}2\sin \theta\,d\theta d\varphi\\
&=&-\frac{16}{6561\times 30}\Gamma(6)\left(\frac32\right)^6\\
&=&-\frac19
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac1{18}-\frac19\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage

\subsection{3dyz}
\begin{eqnarray}
\psi_\mathrm{3d_{yz}}&=&\frac4{\sqrt{6561\times 8\pi}}r^2e^{-\frac{r}3}\sin \theta\cos \theta\sin \varphi\\
&=&\frac{4}{\sqrt{6561\times 30}}r^2e^{-\frac{r}3}\frac{i}{\sqrt 2}(-Y_{2,+1}-Y_{2,-1})
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{3d_{yz}}|\psi_\mathrm{3d_{yz}}\right>&=&\frac{16}{6561\times 30}\int r^4e^{-\frac23r}\frac{|-Y_{2,+1}-Y_{2,-1}|^2}2\,dV\\
&=&\frac{16}{6561\times 30}\int r^6e^{-\frac23r}\,dr\int\!\!\!\int\frac{|-Y_{2,+1}-Y_{2,-1}|^2}2\sin \theta \,d\theta\,d\varphi\\
&=&\frac{16}{6561\times 30}\Gamma(7)\left(\frac32\right)^7\\
&=&\frac{16}{6561\times 30}\left(\frac58\times 3^9\right)\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3d_{yz}}\left|-\frac12\nabla^2\right|\psi_\mathrm{3d_{yz}}\right>\\
&=&-\frac4{6561\times 30}\int r^2e^{-\frac{r}3}(-Y_{2,+1}-Y_{2,-1})^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)r^2e^{-\frac{r}3}(-Y_{2,+1}-Y_{2,-1})\,dV\\
&=&-\frac8{6561\times 30}\int \left(\frac19r^6-2r^5\right)e^{-\frac23r}\,dr\int\!\!\!\int \frac{|-Y_{2,+1}-Y_{2,-2}|^2}2\,d\theta d\varphi\\
&=&-\frac8{6561\times 30}\left[\frac19\Gamma(7)\left(\frac32\right)^7-2\Gamma(6)\left(\frac32\right)^6\right]\\
&=&-\frac8{6561\times 30}\left(\frac58\times 3^7-\frac54\times 3^7\right)\\
&=&\frac1{18}
\end{eqnarray*}

\newpage

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3d_{yz}}\left|-\frac1{r}\right|\psi_\mathrm{3d_{yz}}\right>\\
&=&-\frac{16}{6561\times 30}\int r^3e^{-\frac23r}\frac{|-Y_{2,+1}-Y_{2,-1}|^2}2\,dV\\
&=&-\frac{16}{6561\times 30}\int r^5e^{-\frac23r}\,dr\int\!\!\!\int \frac{|-Y_{2,+1}-Y_{2,-1}|^2}2\sin \theta\,d\theta d\varphi\\
&=&-\frac{16}{6561\times 30}\Gamma(6)\left(\frac32\right)^6\\
&=&-\frac19
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac1{18}-\frac19\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage

\subsection{3dzx}
\begin{eqnarray}
\psi_\mathrm{3d_{zx}}&=&\frac4{\sqrt{6561\times 8\pi}}r^2e^{-\frac{r}3}\sin \theta\cos \theta\cos \varphi\\
&=&\frac{4}{\sqrt{6561\times 30}}r^2e^{-\frac{r}3}\frac1{\sqrt 2}(-Y_{2,+1}+Y_{2,-1})
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{3d_{zx}}|\psi_\mathrm{3d_{zx}}\right>&=&\frac{16}{6561\times 30}\int r^4e^{-\frac23r}\frac{|-Y_{2,+1}+Y_{2,-1}|^2}2\,dV\\
&=&\frac{16}{6561\times 30}\int r^6e^{-\frac23r}\,dr\int\!\!\!\int\frac{|-Y_{2,+1}+Y_{2,-1}|^2}2\sin \theta \,d\theta\,d\varphi\\
&=&\frac{16}{6561\times 30}\Gamma(7)\left(\frac32\right)^7\\
&=&\frac{16}{6561\times 30}\left(\frac58\times 3^9\right)\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3d_{zx}}\left|-\frac12\nabla^2\right|\psi_\mathrm{3d_{zx}}\right>\\
&=&-\frac4{6561\times 30}\int r^2e^{-\frac{r}3}(-Y_{2,+1}+Y_{2,-1})^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)r^2e^{-\frac{r}3}(-Y_{2,+1}+Y_{2,-1})\,dV\\
&=&-\frac8{6561\times 30}\int \left(\frac19r^6-2r^5\right)e^{-\frac23r}\,dr\int\!\!\!\int \frac{|-Y_{2,+1}+Y_{2,-2}|^2}2\,d\theta d\varphi\\
&=&-\frac8{6561\times 30}\left[\frac19\Gamma(7)\left(\frac32\right)^7-2\Gamma(6)\left(\frac32\right)^6\right]\\
&=&-\frac8{6561\times 30}\left(\frac58\times 3^7-\frac54\times 3^7\right)\\
&=&\frac1{18}
\end{eqnarray*}

\newpage

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3d_{zx}}\left|-\frac1{r}\right|\psi_\mathrm{3d_{zx}}\right>\\
&=&-\frac{16}{6561\times 30}\int r^3e^{-\frac23r}\frac{|-Y_{2,+1}+Y_{2,-1}|^2}2\,dV\\
&=&-\frac{16}{6561\times 30}\int r^5e^{-\frac23r}\,dr\int\!\!\!\int \frac{|-Y_{2,+1}+Y_{2,-1}|^2}2\sin \theta\,d\theta d\varphi\\
&=&-\frac{16}{6561\times 30}\Gamma(6)\left(\frac32\right)^6\\
&=&-\frac19
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac1{18}-\frac19\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage



\subsection{3dx${}^2$-y${}^2$}
\begin{eqnarray}
\psi_\mathrm{3d_{x^2-y^2}}&=&\frac4{\sqrt{6561\times 8\pi}}r^2e^{-\frac{r}3}\sin^2 \theta(\cos^2 \varphi-\sin^2\varphi)\\
&=&\frac{4}{\sqrt{6561\times 30}}r^2e^{-\frac{r}3}\frac1{\sqrt 2}(Y_{2,+2}+Y_{2,-2})
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{3d_{x^2-y^2}}|\psi_\mathrm{3d_{x^2-y^2}}\right>&=&\frac{16}{6561\times 30}\int r^4e^{-\frac23r}\frac{|Y_{2,+2}+Y_{2,-2}|^2}2\,dV\\
&=&\frac{16}{6561\times 30}\int r^6e^{-\frac23r}\,dr\int\!\!\!\int\frac{|Y_{2,+2}+Y_{2,-2}|^2}2\sin \theta \,d\theta\,d\varphi\\
&=&\frac{16}{6561\times 30}\Gamma(7)\left(\frac32\right)^7\\
&=&\frac{16}{6561\times 30}\left(\frac58\times 3^9\right)\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3d_{xy}}\left|-\frac12\nabla^2\right|\psi_\mathrm{3d_{xy}}\right>\\
&=&-\frac4{6561\times 30}\int r^2e^{-\frac{r}3}(Y_{2,+2}+Y_{2,-2})^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)r^2e^{-\frac{r}3}(Y_{2,+2}+Y_{2,-2})\,dV\\
&=&-\frac8{6561\times 30}\int \left(\frac19r^6-2r^5\right)e^{-\frac23r}\,dr\int\!\!\!\int \frac{|Y_{2,+2}+Y_{2,-2}|^2}2\,d\theta d\varphi\\
&=&-\frac8{6561\times 30}\left[\frac19\Gamma(7)\left(\frac32\right)^7-2\Gamma(6)\left(\frac32\right)^6\right]\\
&=&-\frac8{6561\times 30}\left(\frac58\times 3^7-\frac54\times 3^7\right)\\
&=&\frac1{18}
\end{eqnarray*}

\newpage

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3d_{xy}}\left|-\frac1{r}\right|\psi_\mathrm{3d_{xy}}\right>\\
&=&-\frac{16}{6561\times 30}\int r^3e^{-\frac23r}\frac{|Y_{2,+2}+Y_{2,-2}|^2}2\,dV\\
&=&-\frac{16}{6561\times 30}\int r^5e^{-\frac23r}\,dr\int\!\!\!\int \frac{|Y_{2,+2}+Y_{2,-2}|^2}2\sin \theta\,d\theta d\varphi\\
&=&-\frac{16}{6561\times 30}\Gamma(6)\left(\frac32\right)^6\\
&=&-\frac19
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac1{18}-\frac19\\
&=&-\frac1{18}
\end{eqnarray*}

\newpage

\subsection{3d3x${}^2$-r${}^2$}
\begin{eqnarray}
\psi_\mathrm{3d_{3z^2-r^2}}&=&\frac1{\sqrt{6561\times 6\pi}}r^2e^{-\frac{r}3}(3\cos^2 \theta-1)\\
&=&\frac{4}{\sqrt{6561\times 30}}r^2e^{-\frac{r}3}Y_{2,0}
\end{eqnarray}

\subsubsection{normalization}
\begin{eqnarray*}
\left<\psi_\mathrm{3d_{3z^2-r^2}}|\psi_\mathrm{3d_{3z^2-r^2}}\right>&=&\frac{16}{6561\times 30}\int r^4e^{-\frac23r}|Y_{2,0}|^2\,dV\\
&=&\frac{16}{6561\times 30}\int r^6e^{-\frac23r}\,dr\int\!\!\!\int|Y_{2,0}|^2\sin \theta \,d\theta\,d\varphi\\
&=&\frac{16}{6561\times 30}\Gamma(7)\left(\frac32\right)^7\\
&=&\frac{16}{6561\times 30}\left(\frac58\times 3^9\right)\\
&=&1
\end{eqnarray*}

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3d_{3z^2-r^2}}\left|-\frac12\nabla^2\right|\psi_\mathrm{3d_{3z^2-r^2}}\right>\\
&=&-\frac4{6561\times 30}\int r^2e^{-\frac{r}3}Y_{2,0}^\ast\left(\frac{\partial^2}{\partial r^2}+\frac2{r}\frac{\partial}{\partial r}+\frac1{r^2}\Lambda\right)r^2e^{-\frac{r}3}Y_{2,0}\,dV\\
&=&-\frac8{6561\times 30}\int \left(\frac19r^6-2r^5\right)e^{-\frac23r}\,dr\int\!\!\!\int |Y_{2,0}|^2\,d\theta d\varphi\\
&=&-\frac8{6561\times 30}\left[\frac19\Gamma(7)\left(\frac32\right)^7-2\Gamma(6)\left(\frac32\right)^6\right]\\
&=&-\frac8{6561\times 30}\left(\frac58\times 3^7-\frac54\times 3^7\right)\\
&=&\frac1{18}
\end{eqnarray*}

\newpage

\subsubsection{$\left$}

\begin{eqnarray*}
\left&\equiv&\left<\psi_\mathrm{3d_{3z^2-r^2}}\left|-\frac1{r}\right|\psi_\mathrm{3d_{3z^2-r^2}}\right>\\
&=&-\frac{16}{6561\times 30}\int r^3e^{-\frac23r}|Y_{2,0}|^2\,dV\\
&=&-\frac{16}{6561\times 30}\int r^5e^{-\frac23r}\,dr\int\!\!\!\int |Y_{2,0}|^2\sin \theta\,d\theta d\varphi\\
&=&-\frac{16}{6561\times 30}\Gamma(6)\left(\frac32\right)^6\\
&=&-\frac19
\end{eqnarray*}

\subsubsection{$\left$}
\begin{eqnarray*}
\left&=&\left+\left\\
&=&\frac1{18}-\frac19\\
&=&-\frac1{18}
\end{eqnarray*}

\end{document}