\question
{Shoichi Takahata}
{02/03/12}
{physics2070@yahoo.co.jp}
{2-3}
%question
{
Plot the rough shape of $R_{nl}$ in ${\rm Eq}.\,(2.2)$ 
for 1s, 2s and 2p states 
as a function of r. 
Explain how these functions are orthogonal to each other.
}
%answer
{
We consider a hydrogen atom. 
From Eq.$\,(2.6)$ in the book, 
the form of of $R_{nl}(r)$ for an electron of a hydrogen atom $(Z=1)$
is expressed by 

\begin{equation}
R_{nl}(r)
=C_{nl}\,{\rm exp}\left(-\frac{r}{n{a}_{0}}\right) 
\left(\frac{r}{{a}_{0}}\right)^{l} {G}_{\rm nl}
\left(\frac{r}{{a}_{0}}\right) \label{eq:Rnl}\,,
\end{equation}

where $C_{nl}$ denotes the normalization constant. 
${G}_{nl}$ denotes the Laguerre polynomials 
as a function of $\rho=r/{a}_{0}$ 
(where ${\it {a}_{0}}$ is the Bohr radius, 
${a}_{0}={\hbar}^{2}/m{e}^{2}$) 
and is defined by

\begin{equation}
{G}_{nl}(\rho)
=L^{2l+1}_{n+l}\left( \frac{2\rho}{n} \right)
={\left.
\frac{d^{\,2l+1}}{{dx}^{2l+1}}
\left(\exp(x)
\frac{d^{\,n+l}}{{dx}^{n+l}}{x}^{n+l}\exp(-x)\right) 
\right|}_{x=\frac{2\rho}{n}} \,.
\end{equation}

Since $({\it n,l})=(1,0)\,,(2,0)\,,{\rm and}\,\,(2,1)$ 
correspond to 1s, 2s and 2p states, respectively, 
normalized $R_{nl}$ of Eq.\,(\ref{eq:Rnl}) for 1s, 2s and 2p states 
as a function of $r/{a}_{0}$ can be written as follows:

\begin{eqnarray}
R_{10}(r) &=& 
C_{10} \,
{\rm exp}\displaystyle{
\left(-\frac{r}{{a}_{0}}\right)} \label{eq:R10} \,,\\ 
R_{20}(r) &=& 
C_{20} 
\displaystyle{\left(1-\frac{r}{2{a}_{0}}\right)}
{\rm exp}\displaystyle{
\left(-\frac{r}{2{a}_{0}}\right)} \label{eq:R20} \,,\\ 
R_{21}(r) &=& 
C_{21} 
\displaystyle{\left(\frac{r}{{a}_{0}}\right)}
{\rm exp}\displaystyle{
\left(-\frac{r}{2{a}_{0}}\right)} \label{eq:R21} \,,
\end{eqnarray}

where the normalization constants are given by
$C_{10}=2{a_0}^{-3/2}$, 
$C_{20}={a_0}^{-3/2}/\sqrt{2}$, and
$C_{21}={a_0}^{-3/2}/2\sqrt{6}$\,,
which can be obtained by the normalization condition as:

\begin{equation}
\int_{0}^{\infty} dr{r}^{2} 
{{R}_{nl}(r)}^{\ast} {R}_{nl}(r)=1 \label{eq:normalization} \,.
\end{equation}

\begin{figure}[h]
\begin{center}
\includegraphics[width=10cm,clip]{Q2-3-1.eps}\\
\begin{minipage}{10cm}
\caption{The radial wave functions of a hydrogen atom
for 1s, 2s and 2p orbitals.
\label{fig:Q2-3-1}
}
\end{minipage}
\end{center}
\end{figure}

In Fig.\,2-3-1, 
we plot $R_{nl}/C_{nl}$ for 1s, 2s, and 2p orbital
as a function of $r/a_{0}$. 
The inner product of radial wave functions 
${I}_{n l {n}^{\prime} {l}^{\prime}}$
of Eqs.\,(\ref{eq:R10}), (\ref{eq:R20}) and (\ref{eq:R21}) 
is given by 

\begin{equation}
{I}_{n l {n}^{\prime} {l}^{\prime}}
=\int_{0}^{\infty} dr{r}^{2} 
{{R}_{nl}(r)}^{\ast} {R}_{{n}^{\prime}{l}^{\prime}}(r) 
\label{eq:I} \,.
\end{equation}

Using the formula

\begin{equation}
\int_{0}^{\infty} dr{r}^{m}{\rm exp}(-\alpha r) 
={m!}\,{{\alpha}^{-(m+1)}} \,,
\end{equation}

where $\alpha$ is a positive constant 
and {\it m} is non-negative integer, 
we can calculate Eq.\,(\ref{eq:I}) as follows:

\begin{eqnarray}
{I}_{1020}
&=&
\int_{0}^{\infty} dr{r}^{2} 
{{R}_{10}(r)}^{\ast} {R}_{20}(r)
\nonumber \\
&=&
\frac{\sqrt{2}}{{a_0}^{3}} \int_{0}^{\infty} dr{r}^{2} 
{\rm exp}\left(-\frac{3r}{2{a}_{0}}\right)
-\frac{\sqrt{2}}{{a_0}^{3}} \frac{1}{2a_0} \int_{0}^{\infty} dr{r}^{3} 
{\rm exp}\left(-\frac{3r}{2{a}_{0}}\right)
\nonumber \\
&=&
\frac{16\sqrt{2}}{27}-\frac{\sqrt{2}}{2} \frac{32}{27}
\nonumber \\
&=&
0 \,, \\
{I}_{1021}
&=&
\int_{0}^{\infty} dr{r}^{2} 
{{R}_{10}(r)}^{\ast} {R}_{21}(r)
\nonumber \\
&=&
\frac{1}{\sqrt{6} {a_0}^3} \frac{1}{a_0}
\int_{0}^{\infty} dr{r}^{3} 
{\rm exp}\left(-\frac{3r}{2{a}_{0}}\right)
\nonumber \\
&=&
\frac{32}{27\sqrt{6}}
\nonumber \\
&\neq&
0 \,, \\
{I}_{2021}
&=&
\int_{0}^{\infty} dr{r}^{2} 
{{R}_{20}(r)}^{\ast} {R}_{21}(r)
\nonumber \\
&=&
\frac{1}{4\sqrt{3} {a_0}^3} \frac{1}{a_0}
\int_{0}^{\infty} dr{r}^{3} 
{\rm exp}\left(-\frac{r}{a_0}\right)
-\frac{1}{4\sqrt{3} {a_0}^3} \frac{1}{2 {a_0}^2}
\int_{0}^{\infty} dr{r}^{4} 
{\rm exp}\left(-\frac{r}{a_0}\right)
\nonumber \\
&=&
\frac{6}{4\sqrt{3}}-\frac{12}{4\sqrt{3}}
\nonumber \\
&=&
-\frac{6}{4\sqrt{3}}
\nonumber \\
&\neq&
0 \,,
\end{eqnarray}

which show that 
radial wave functions for only 1s and 2s states 
are orthogonal to each other. 
For 1s and 2p (2s and 2p) orbitals, 
the orthogonal condition is satisfied by
the orthogonality of spherical harmonics $Y_{lm}$.
}