Difference between revisions of "Legendre function"

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m (Text replacement - "\$([^\$]+)\$" to "\\(\1\\)")
 
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[[Legendre function]] is solution $F$ of equation
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[[Legendre function]] is solution \(F\) of equation
   
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$\displaystyle (1-x^2) F''(x) - 2xF'(x)+ \left( L - \frac{m^2}{1-x^2} \right) F(x)=0 $
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\(\displaystyle (1-x^2) F''(x) - 2xF'(x)+ \left( L - \frac{m^2}{1-x^2} \right) F(x)=0 \)
   
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where $m$ and $L$ are constant parameters. This equation comes from the [[Azimutal equation]], that appears at the
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where \(m\) and \(L\) are constant parameters. This equation comes from the [[Azimutal equation]], that appears at the
 
[[separation of variables]] for the [[Laplacian in spherical coordinates]].
  
[[separation of variables]] for the [[Laplacian in spherical coordinates]].
   
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For the application to [[quantum mechanics]] of the [[hydrogen atom]], it is assumed, that $m$ is integer.
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For the application to [[quantum mechanics]] of the [[hydrogen atom]], it is assumed, that \(m\) is integer.
   
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The solutions can be expressed in terms of polynomials, if $L=\ell(\ell+1)$ for some integer $\ell \ge |m|$; id est,
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The solutions can be expressed in terms of polynomials, if \(L=\ell(\ell+1)\) for some integer \(\ell \ge |m|\); id est,
   
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$\displaystyle (1\!-\!x^2) F''(x) - 2xF'(x)+ \left( \ell(\ell\!+\!1) - \frac{m^2}{1\!-\!x^2} \right) F(x)=0 $
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\(\displaystyle (1\!-\!x^2) F''(x) - 2xF'(x)+ \left( \ell(\ell\!+\!1) - \frac{m^2}{1\!-\!x^2} \right) F(x)=0 \)
   
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The solution $F(x)$ is denoted with
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The solution \(F(x)\) is denoted with
   
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$\mathrm{LegendreP} [\ell, m, x]=P_{\ell,m}(x)$
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\(\mathrm{LegendreP} [\ell, m, x]=P_{\ell,m}(x)\)
   
 
Identifier [[LegendreP]] is recognised by the [[Mathematica]] software.
  
Identifier [[LegendreP]] is recognised by the [[Mathematica]] software.
   
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==Case $m=0$ ==
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==Case \(m=0\) ==
   
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For $m=0$, the equation becomes
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For \(m=0\), the equation becomes
   
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$\displaystyle (1\!-\!x^2) F''(x) - 2xF'(x)+ \ell(\ell\!+\!1) F(x)=0 $
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\(\displaystyle (1\!-\!x^2) F''(x) - 2xF'(x)+ \ell(\ell\!+\!1) F(x)=0 \)
   
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For $\ell\!=\!0$, the solution is constant.
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For \(\ell\!=\!0\), the solution is constant.
   
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For $\ell\!=\!1$, the solution is linear function.
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For \(\ell\!=\!1\), the solution is linear function.
   
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For $\ell\!=\!1$, the solution $F(x)$ is quadratic function, proportional to $-1+3x^2$.
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For \(\ell\!=\!1\), the solution \(F(x)\) is quadratic function, proportional to \(-1+3x^2\).
   
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For integer non-negative $\ell$, the solution is called [[LegendreP]]$_\ell$, or "the [[Legendre polynomial]]"
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For integer non-negative \(\ell\), the solution is called [[LegendreP]]\(_\ell\), or "the [[Legendre polynomial]]"
 
<ref>https://reference.wolfram.com/language/ref/LegendreP.html
  
<ref>https://reference.wolfram.com/language/ref/LegendreP.html
 
</ref>.
  
</ref>.
   
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Another solution called [[LegendreQ]]$_\ell$ is singular, it can be expressed through logarithms
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Another solution called [[LegendreQ]]\(_\ell\) is singular, it can be expressed through logarithms
 
<ref>https://reference.wolfram.com/language/ref/LegendreQ.html</ref>.
  
<ref>https://reference.wolfram.com/language/ref/LegendreQ.html</ref>.
   
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The polynomial [[LegendreP]] can be generated with the Rodriguez formula
  
The polynomial [[LegendreP]] can be generated with the Rodriguez formula
   
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$\displaystyle
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\(\displaystyle
 
\mathrm{LegendreP}_n(z) = \frac{1}{2^n n!}
 
\mathrm{LegendreP}_n(z) = \frac{1}{2^n n!}
 
\frac{\mathrm d\, (z^2\!-\!1)^n }
  
\frac{\mathrm d\, (z^2\!-\!1)^n }
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{\mathrm d z\, ^n}$
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{\mathrm d z\, ^n}\)
   
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==Various $m$==
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==Various \(m\)==
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For more general case, when $m$ has no need to be zero, the solution can be expressed through the Legendre function, $F(x)=$ [[LegendreP]]$_{\ell,m}(x)$.
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For more general case, when \(m\) has no need to be zero, the solution can be expressed through the Legendre function, \(F(x)=\) [[LegendreP]]\(_{\ell,m}(x)\).
   
 
The [[Legendre function]] can be defined through the Legendre polynomial:
  
The [[Legendre function]] can be defined through the Legendre polynomial:
   
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$\displaystyle
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\(\displaystyle
 
\mathrm{LegendreP}_{\ell,m}(x) = (-1)^m \Big(1\!-\!x^2\Big)^{m/2}
  
\mathrm{LegendreP}_{\ell,m}(x) = (-1)^m \Big(1\!-\!x^2\Big)^{m/2}
 
\frac{\mathrm d\, \mathrm{LegendreP}_{\ell}(x)}
  
\frac{\mathrm d\, \mathrm{LegendreP}_{\ell}(x)}
 
{\mathrm d x\,^m}
  
{\mathrm d x\,^m}
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\)
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$
  
   
 
Usually, namely this solution is called [[Legendre function]].
  
Usually, namely this solution is called [[Legendre function]].
   
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If $m\!=\!0$, then the Legendre function
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If \(m\!=\!0\), then the Legendre function
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$\mathrm{LegendreP}_{\ell,m}=
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\(\mathrm{LegendreP}_{\ell,m}=
 
\mathrm{LegendreP}_{\ell,0}=
  
\mathrm{LegendreP}_{\ell,0}=
 
\mathrm{LegendreP}_{\ell}
  
\mathrm{LegendreP}_{\ell}
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\)
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$
  
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appears as polynomial of order $\ell$, defined in the previous section.
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appears as polynomial of order \(\ell\), defined in the previous section.
   
 
==References==
  
==References==

Latest revision as of 18:44, 30 July 2019

Legendre function is solution \(F\) of equation

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

where \(m\) and \(L\) are constant parameters. This equation comes from the Azimutal equation, that appears at the separation of variables for the Laplacian in spherical coordinates.

For the application to quantum mechanics of the hydrogen atom, it is assumed, that \(m\) is integer.

The solutions can be expressed in terms of polynomials, if \(L=\ell(\ell+1)\) for some integer \(\ell \ge |m|\); id est,

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

The solution \(F(x)\) is denoted with

\(\mathrm{LegendreP} [\ell, m, x]=P_{\ell,m}(x)\)

Identifier LegendreP is recognised by the Mathematica software.

Case \(m=0\)

For \(m=0\), the equation becomes

\(\displaystyle (1\!-\!x^2) F''(x) - 2xF'(x)+ \ell(\ell\!+\!1) F(x)=0 \)

For \(\ell\!=\!0\), the solution is constant.

For \(\ell\!=\!1\), the solution is linear function.

For \(\ell\!=\!1\), the solution \(F(x)\) is quadratic function, proportional to \(-1+3x^2\).

For integer non-negative \(\ell\), the solution is called LegendreP\(_\ell\), or "the Legendre polynomial" [1].

Another solution called LegendreQ\(_\ell\) is singular, it can be expressed through logarithms [2].

The only LegendreP is necessary for the direct application to the quantum mechanics with radial symmetry.

The polynomial LegendreP can be generated with the Rodriguez formula

\(\displaystyle \mathrm{LegendreP}_n(z) = \frac{1}{2^n n!} \frac{\mathrm d\, (z^2\!-\!1)^n } {\mathrm d z\, ^n}\)

Various \(m\)

For more general case, when \(m\) has no need to be zero, the solution can be expressed through the Legendre function, \(F(x)=\) LegendreP\(_{\ell,m}(x)\).

The Legendre function can be defined through the Legendre polynomial:

\(\displaystyle \mathrm{LegendreP}_{\ell,m}(x) = (-1)^m \Big(1\!-\!x^2\Big)^{m/2} \frac{\mathrm d\, \mathrm{LegendreP}_{\ell}(x)} {\mathrm d x\,^m} \)

Usually, namely this solution is called Legendre function.

If \(m\!=\!0\), then the Legendre function \(\mathrm{LegendreP}_{\ell,m}= \mathrm{LegendreP}_{\ell,0}= \mathrm{LegendreP}_{\ell} \) appears as polynomial of order \(\ell\), defined in the previous section.

References

  1. ↑ https://reference.wolfram.com/language/ref/LegendreP.html
  2. ↑ https://reference.wolfram.com/language/ref/LegendreQ.html

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

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

http://people.math.sfu.ca/~cbm/aands/page_333.htm Legendre function

Keywords

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

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