LegendreP

LegendreP refers to the Legendre function and to the Legendre polynomial.

Legendre polynomial
The Legendre polynomial \(\mathrm{LegendreP}_{\ell}\) can be defined with the Rodrigues formula

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

In Mathematica, the polynomials above are denoted with identifier LegendreP

In C++, for integer values of subscript, evaluation of the coefficients of the polynomial can be performed with the code shown in the example below: int BC[N+1][N+1]; // Binomial coefficients; int main{ int m,n,k,l; DO(n,N+1){BC[n][0]=1; BC[n][1]=n; BC[n][n]=1;} // Pascal triangle for(m=2;m<N+1;m++) for(n=2;n
 * 2) include
 * 3) define DO(x,y) for(x=0;x<y;x++)
 * 4) define DB double
 * 5) define N 12

The output is 1.0000 0.0000   1.0000   -0.5000    0.0000    1.5000    0.0000   -1.5000    0.0000    2.5000    0.3750    0.0000   -3.7500    0.0000    4.3750    0.0000    1.8750    0.0000   -8.7500    0.0000    7.8750   -0.3125    0.0000    6.5625    0.0000  -19.6875    0.0000   14.4375    0.0000   -2.1875    0.0000   19.6875    0.0000  -43.3125    0.0000   26.8125    0.2734    0.0000   -9.8438    0.0000   54.1406    0.0000  -93.8438    0.0000   50.2734    0.0000    2.4609    0.0000  -36.0938    0.0000  140.7656    0.0000 -201.0938    0.0000   94.9609   -0.2461    0.0000   13.5352    0.0000 -117.3047    0.0000  351.9141    0.0000 -427.3242    0.0000  180.4258    0.0000   -2.7070    0.0000   58.6523    0.0000 -351.9141    0.0000  854.6484    0.0000 -902.1289    0.0000  344.4492    0.2256    0.0000  -17.5957    0.0000  219.9463    0.0000 -997.0898    0.0000 2029.7900    0.0000-1894.4707    0.0000  660.1943 It can be verified with Mathematica code N[Table[Expand[D[(x^2-1)^n/2^n/n!, {x, n}]], {n,0,12}], 7]

or N[Table[Expand[LegendreP[n, x]], {n,0,12}], 7]

that both do the same {1.000000, x, -0.5000000 + 1.500000 x^2, -1.500000 x + 2.500000 x^3, 0.3750000 - 3.750000 x^2 + 4.375000 x^4, 1.875000 x - 8.750000 x^3 + 7.875000 x^5, -0.3125000 + 6.562500 x^2 - 19.68750 x^4 + 14.43750 x^6, -2.187500 x + 19.68750 x^3 - 43.31250 x^5 + 26.81250 x^7, 0.2734375 - 9.843750 x^2 + 54.14063 x^4 - 93.84375 x^6 + 50.27344 x^8, 2.460938 x - 36.09375 x^3 + 140.7656 x^5 - 201.0938 x^7 +  94.96094 x^9, -0.2460938 + 13.53516 x^2 - 117.3047 x^4 + 351.9141 x^6 - 427.3242 x^8 + 180.4258 x^10, -2.707031 x + 58.65234 x^3 - 351.9141 x^5 + 854.6484 x^7 - 902.1289 x^9 + 344.4492 x^11, 0.2255859 - 17.59570 x^2 + 219.9463 x^4 - 997.0898 x^6 + 2029.790 x^8 - 1894.471 x^10 + 660.1943 x^12}

Legendre function
Then, the Legendre function \(\mathrm{LegendreP}_{\ell,m}\) appears as

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

In such a way, \(\mathrm{LegendreP}_{\ell,0}=\mathrm{LegendreP}_{\ell}\)

Legendre equation
Function \(P_{\ell, m}=\mathrm{LegendreP}_{\ell,m}\) satisfies the Legendre equation

\(\displaystyle (1\!-\!x^2) P_{\ell,m}^{~\prime\prime}(x) - 2 x P_{\ell,m}^{~\prime}(x) +\left( \ell (\ell+1) - \frac{m^2}{1\!-\!x^2} \right) P_{\ell,m}(x) =0 \)

Usage
The Legendre function is used to construct the Hydrogen wave function.

Argument of function \(\mathrm{LegendreP}_{\ell, m}\) has sense of \(\cos(\theta)\), if the azimuthal angle \(\theta\) ix measured from the "z" avis, and sense of \(\sin(\theta)\), if the angle is counted from the equatorial plane; \(\ell\) has sense of the orbital momentum od the corresponding wave function, and \(m\) has sense of projection of the orbital moment on the axis of symmetry of the wave function. Parameter \(m\) is called also magnetic quantum number.

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