Azimutal equation

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Azimutal equation appears as result of separation of variables for the Schroedinger equation with central symmetry, using the Laplacian in spherical coordinates. The Azimutal equation can be written as follows:

$\displaystyle \frac{1}{\sin(\theta)} \partial_\theta \Big( \sin(\theta) \Theta'(\theta) \Big) + \left( L - \frac{m^2}{s^2} \right) \Theta(\theta)=0 $

while $m$ is integer parameter. Function $\Theta$ is assumed to be regular at least in some vicinity of the real axis and periodic with period $2\pi$. This leads to certain requirements on values of parameter $L$.

Using notations $\Theta=\Theta(\theta)$, $\Theta'=\Theta'(\theta)$, $s=\sin(\theta)$, $c=\cos(\theta)$,

the azimutal equation can be written in shorter (but equivalent) form

$\displaystyle \frac{1}{s} \partial_\theta ( s \Theta' ) + \left( L - \frac{m^2}{s^2} \right) \Theta=0 $

Solution

It is convenient to search for the solution $\Theta$ in the following form:

$\Theta(\theta)=F(c )=F(\cos(\theta))$

The substitution into the Azimutal equation gives

$\displaystyle \frac{1}{s} \partial_\theta \Big( s F'(\cos(\theta) s \Big) + \left( L - \frac{m^2}{s^2} \right) F(\cos(\theta) ) =0 $

Using $\partial_\theta c=-s$, this equation van be rewritten as follows:

$\displaystyle s^2 F''(c ) - 2cF'(c )+ \left( L - \frac{m^2}{s^2} \right) F(c )=0 $

Replacement $ s^2=1-c^2$ gives

$\displaystyle (1-c^2) F''(c ) - 2cF'(c )+ \left( L - \frac{m^2}{1-c^2} \right) F(c )=0 $

For $L=\ell(\ell+1)$, the solution $F$ is called Legendre function. Similar notations are used in Wikipedia, $c$ appears as $x$ and $\ell$ appears as $\lambda$; in general, no restriction on values $\ell$ is assumed [1].

However, the restriction is necessary to use the Legendre function for the Hydrogen wave function. Then, $\ell$ is assumed to be non–negative integer number.

References

  1. https://en.wikipedia.org/wiki/Legendre_function

http://hyperphysics.phy-astr.gsu.edu/hbase/math/legend.html#c2

Keywords

Atomic physics, Azimutal equation‎, Hydrogen wave function, Laplacian in spherical coordinates, Laplacian, Legendre function, Legendre polynomial, Molecular physics, Quantum mechanics, Schroedinger equation