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 .

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.