View source for Radial equation for hydrogen atom

[[Radial equation for hydrogen atom]] refers to one of equations that appear at the [[separation of variables]] for the 
[[Stationary Schroedinger equation]] for the [[Hydrogen atom]] 
in spherical coordinates.

==Coordinates and the equation==
$x=\cos(\theta)\cos(\phi)$<br>
$y=\cos(\theta)\sin(\phi)$<br>
$z=\sin(\phi)$

The wave function appears as $\psi=R(r )\Theta(\theta)\Phi(\phi)$
in the [[spherical system of coordinates]] $r, \theta, \phi$;
then the Angular part of the equation gives

$\Phi(\phi)=\exp(\pm \mathrm i m \phi)$

$\Theta(\theta)=\displaystyle
P_{\ell,m}(\sin(\theta)) =
\mathrm{LegendreP}_{\ell,m}(\sin(\theta)) $,
<br>
where [[LegendreP]] is associated [[Legendre function]],

and for $R$, the separation gives the following equation

$\displaystyle
\frac{1}{r^2} \partial_r (r^2 R') 
- \frac{\ell(\ell\!+\!1)}{r^2} R 
+ \left( \frac{2\mu} {\hbar^2} \frac{e^2}{r} + \frac{2\mu} {\hbar^2} E \right) R = 0
$

Here 
$R=R(r )$,
$\mu$ is [[effective mass]] of electron,
$\hbar$ is [[Planck constant]],
$e$ is charge of electron,
$\ell$ is non–negative integer parameter, that comes from the equations for the angular part of the wave function.
Parameter $E$ appears as eigenvalue of the [[Hamiltonian]] and, therefore, is interpreted as [[energy]] of the stationary state.

==[[Bohr radius]]==
Define the [[Bohr radius]] $\displaystyle ~a\!=\!\frac{\hbar^2}{\mu\, e^2}$

Let $R(r )=F(x)$, where $x$ is new dimensionless variable that has nothing to do with variable $x$ used in the initial Cartesian system of coordinates. Then, for $F$, the equation becomes

$\displaystyle
\frac{1}{x^2} \partial_x (x^2 F') 
- \frac{\ell(\ell\!+\!1)}{x^2} F 
+ \left( \frac{1}{x} + \frac{\hbar^2}{\mu e^4}  E \right) F = 0
$

Parameter $\displaystyle -\varepsilon=\frac{\hbar^2}{\mu e^4}  E $ can be interpreted as dimension-less energy.

$\displaystyle
F''+ \frac{2}{x} F' 
- \frac{\ell(\ell\!+\!1)}{x^2} F 
+ \left( \frac{1}{x} - \varepsilon \right) F = 0
$

Seach for the solution $F(x)=\exp(-kx) f(x)$, that takes into account the asymptotic behaviour of the solution at $x\gg 1$, assuming, that $k^2=\varepsilon$. The substitution gives:

$\displaystyle
 f'' -2k f' +\frac{2}{x} f' +\frac{1\!-\!2k}{x}f - \frac{\ell(\ell\!+\!1)}{x^2} f= 0$

for $\ell \!=\!0$, this equation has, among other, solution $f(x)\!=\!\rm const$ at $k\!=\!1/2$.

The Mathematica code

<nowiki>
DSolve[f''[x] - 1/n f'[x] + 2/x f'[x] + (1 - 1/n)/x f[x] - (m (m + 1))/x^2 f[x] == 0, {f[x]}, x]
</nowiki>

among other solution, suggests for f[x] the following:

x^m LaguerreL[-1-m+n, 1+2m, x/n]

This expression suggests values $\displaystyle k=\frac{1}{2n}$; then, for positive integer $n$, the solution $f$ is polynomial,
easily expressed through the [[Associated Laguerre polynomial]]
<ref>http://mathworld.wolfram.com/AssociatedLaguerrePolynomial.html
</ref>

==Solution==
The physically-meaningful solution can be written as <ref>
https://www.physics.drexel.edu/~tim/open/hydrofin/hyd.pdf
TIMOTHY JONES. ELEMENTARY QUANTUM MECHANICAL MODEL OF THE HYDROGEN ATOM.
2009-02-11.
</ref>:

$\displaystyle
R(r )=\frac{N}{an} 
\left( \frac{r}{an}\right)^\ell 
\exp\!\left(- \frac{r}{an} \right) ~ 
L_{n-\ell-1}^{~2 \ell +1} \left( \frac{2r}{an} \right)$

where $n$ is positive integer parameter called also [[principal quantum number]],
$N$ is normalisation factor (that may depend on the quantum numbers) and

$L_{n-\ell-1}^{~2 \ell +1}(t)=\mathrm{LaguerreL}(n\!-\!\ell\!-\!1,2 \ell \!+\!1,t)$

is the [[LaguerreL]] polynomial.


==Wave function==

Often, the angular part of the wave function is denoted with 

$
\displaystyle
Y_n^m
(\theta,\phi) =
\sqrt{\frac{2n\!+\!1}{4\pi} \, \frac{(n\!-\!m)!}{(n\!+\!m)!} } ~
\exp(\mathrm i m \phi) ~
P_\ell^{\,n}(\sin \theta)
$

Then, the wave function can be written as follows:

$\displaystyle
\psi_{n,\ell,m}=
\sqrt{
\left( \frac{2}{na} \right)^3 \frac{(n\!-\!\ell\!-\!1)!}{2 n (n\!+\!1)!}
}
~
\exp\left(\frac{-r}{na}\right)  ~
L_{n-\ell-1}^{~ 2\ell+1}\!\left(\frac{2r}{na}\right) ~
Y_n^m (\theta,\phi) 
$

==[[Jim Branson]]==
Expression of the Radial wave function through the polynomials [[LaguerreL]] 
looks as a patch, due to the expressions in the subscript and superscript of the polynomial $L$. Namely for evaluation of the radial wave function, the polynomials can be represented in a little bit more direct way.

Jim Branson suggests another (and perhaps equivalent) representation of the solution, that seems to be more suitable namely for the Radial equation for Hydrogen atom <ref>
http://quantummechanics.ucsd.edu/ph130a/130_notes/node236.html
Jim Branson. Solution of Hydrogen Radial Equation. 2013-04-22 </ref>:

Let 
$\displaystyle
\rho=\sqrt{\frac{-8\mu E}{\hbar^2}\,}\,r$ , and let
$~\displaystyle \lambda = \frac{Z e^2}{\hbar} \sqrt{\frac{-\mu}{2 E}\,}$

Then, the radial equation can be written as follows:

$\displaystyle
\frac{\mathrm d^2 R}{\mathrm d \rho^2}
+ \frac{2}{\rho} \, \frac{\mathrm d R}{\mathrm d \rho}
- \frac{\ell(\ell\!+\!1)}{\rho^2}  R
+ \left(\frac{\lambda}{\rho}-\frac{1}{4}\right) R=0
$

It is convenient to denote dependence of $R$ on $\rho$ with some name; let it be first latter of the last name of Jim Branson, id est, $R(r )=B(\rho)$. The equation for $B$ can be written as follows:

$\displaystyle
B''(\rho) + \frac{2}{\rho} B'(\rho)
- \frac{\ell(\ell\!+\!1)}{\rho^2}  B(\rho)
+ \frac{\lambda}{\rho} B(\rho)-\frac{1}{4} B(\rho)=0
$

Then, the only positive integer values of $\lambda$ happen to be allowed. These values determine energy $E$ and, therefore, relation between $r$ and $\rho$. Through this $\rho$,  the solution can be written as follows:

$\displaystyle B_{n,\ell}=\rho^\ell \sum_{k=0}^\infty a_k \rho^k \exp(-\rho/2)$

where

$\displaystyle
a_{k+1}=\frac{k+\ell+1-n}{(k+1)(k+2\ell+2)} a_k
$

Here, $a$ are dimensionless coefficients of expansion; they should not be confused with the [[Bohr radius]] $a$. The last expression can be written in a little bit more compact form, replacing $k\!+\!1$ to $k$:

$\displaystyle
a_{k}=\frac{k+\ell - n}{k\, (k+2\ell+1)} a_{k-1}
$

Practically, the summation above stops, as the nominator of the fraction becomes zero;
the last term corresponds to $k=n-1-\ell$; and the only positive integer $n$ correspond to the normalisable eigenstates of the Hamiltonian. 
The Radial part of the Wave function can be expressed with 

$\displaystyle R_{n,\ell}(r )=B_{n,\ell}\left(\sqrt{\frac{-8\mu E}{\hbar^2}\,}r \right)
=F_{n,\ell}\left(\frac{2\mu Z e^2}{\hbar^2} r \right)
$

where 

$\displaystyle F_{n,\ell}(x)=B_{n,\ell} (x/n)$

and this $F$ satisfies the dimensionless radial Schroedinger equation

$\displaystyle
{F_{n,\ell}}''(x) + \frac{2}{x} {F_{n,\ell}}'(x)
- \frac{\ell(\ell\!+\!1)}{x^2}  F_{n,\ell}(x)
+ \frac{1}{x} F_{n,\ell}(\rho)-\frac{1}{4 n^2} F_{n,\ell}(x)=0$

where $-\frac{1}{4 n^2}$ plays role of the dimensionless energy.
Тhe dimensional energy 

$\displaystyle E=-
\frac{(Ze^2)^2 \mu}{2\hbar^2 n^2}
$

The radial wave functions are orthogonal;

$\displaystyle
\int_0^\infty \, F_{n,\ell} (x)\,  F_{m,\ell} (x)\,  x^2 \, \mathrm d x = \Nu_{n,\ell} \delta _{n,m}~ ~$
where $~\delta~$ is [[Kronecker symbol]].

==References==
<references/>

https://en.wikipedia.org/wiki/Bohr_radius

http://quantummechanics.ucsd.edu/ph130a/130_notes/node233.html The Radial Wavefunction Solutions. 
Defining the [[Bohr radius]], $\displaystyle a_0={\hbar\over\alpha mc} $

==Keywords==
[[Atomic physics]],
[[Laplacian in spherical coordinates]],
[[Quantum mechanics]]

[[Category:Atomic physics]]
[[Category:Laplacian in spherical coordinates]]
[[Category:Quantum mechanics]]
[[Category:Radial equation for hydrogen atom]]