Difference between revisions of "Azimutal equation"
m (Text replacement  "\$([^\$]+)\$" to "\\(\1\\)") 

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[[Laplacian in spherical coordinates]]. The [[Azimutal equation]] can be written as follows: 
[[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 
+  while \(m\) is integer parameter. 
−  Function 
+  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 
+  This leads to certain requirements on values of parameter \(L\). 
Using notations 
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 
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== 
==Solution== 

−  It is convenient to search for the 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 
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 
+  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 
+  Replacement \( s^2=1c^2\) gives 
−  +  \(\displaystyle (1c^2) F''(c )  2cF'(c )+ \left( L  \frac{m^2}{1c^2} \right) F(c )=0 \) 

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

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

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However, the restriction is necessary to use the [[Legendre function]] for the [[Hydrogen wave function]]. 
However, the restriction is necessary to use the [[Legendre function]] for the [[Hydrogen wave function]]. 

−  Then, 
+  Then, \(\ell\) is assumed to be non–negative integer number. 
==References== 
==References== 
Latest revision as of 18:45, 30 July 2019
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 \( s^2=1c^2\) gives
\(\displaystyle (1c^2) F''(c )  2cF'(c )+ \left( L  \frac{m^2}{1c^2} \right) F(c )=0 \)
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
http://hyperphysics.phyastr.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