Azimutal equation

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 .

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

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