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<title>GNU Scientific Library &ndash; Reference Manual: Normalized Hydrogenic Bound States</title>

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<a name="Normalized-Hydrogenic-Bound-States"></a>
<div class="header">
<p>
Next: <a href="Coulomb-Wave-Functions.html#Coulomb-Wave-Functions" accesskey="n" rel="next">Coulomb Wave Functions</a>, Up: <a href="Coulomb-Functions.html#Coulomb-Functions" accesskey="u" rel="up">Coulomb Functions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<hr>
<a name="Normalized-Hydrogenic-Bound-States-1"></a>
<h4 class="subsection">7.7.1 Normalized Hydrogenic Bound States</h4>

<dl>
<dt><a name="index-gsl_005fsf_005fhydrogenicR_005f1"></a>Function: <em>double</em> <strong>gsl_sf_hydrogenicR_1</strong> <em>(double <var>Z</var>, double <var>r</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhydrogenicR_005f1_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hydrogenicR_1_e</strong> <em>(double <var>Z</var>, double <var>r</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the lowest-order normalized hydrogenic bound
state radial wavefunction <em>R_1 := 2Z \sqrt{Z} \exp(-Z r)</em>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fsf_005fhydrogenicR"></a>Function: <em>double</em> <strong>gsl_sf_hydrogenicR</strong> <em>(int <var>n</var>, int <var>l</var>, double <var>Z</var>, double <var>r</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fhydrogenicR_005fe"></a>Function: <em>int</em> <strong>gsl_sf_hydrogenicR_e</strong> <em>(int <var>n</var>, int <var>l</var>, double <var>Z</var>, double <var>r</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><p>These routines compute the <var>n</var>-th normalized hydrogenic bound state
radial wavefunction,
</p>
<div class="example">
<pre class="example">R_n := 2 (Z^{3/2}/n^2) \sqrt{(n-l-1)!/(n+l)!} \exp(-Z r/n) (2Zr/n)^l
          L^{2l+1}_{n-l-1}(2Zr/n).  
</pre></div>

<p>where <em>L^a_b(x)</em> is the generalized Laguerre polynomial (see <a href="Laguerre-Functions.html#Laguerre-Functions">Laguerre Functions</a>).
The normalization is chosen such that the wavefunction <em>\psi</em> is
given by 
<em>\psi(n,l,r) = R_n Y_{lm}</em>.   
</p></dd></dl>




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