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:

#include<math.h>
#include<stdio.h>
#define DO(x,y) for(x=0;x<y;x++)
#define DB double
#define N 12
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<N;n++) BC[m][n]=BC[m-1][n-1]+BC[m-1][n];
 DO(m,N+1){ DO(n,N+1) printf("%4d ",BC[m][n]);
           printf("\n");}
DB L[N+1][2*N+2]; DO(n,N+1) DO(m,2*N+2) {L[n][m]=0;}
DO(n,N+1){l=pow(-1,n); for(k=0;k<=n;k++){ L[n][2*k]=BC[n][k]*l; l=-l;}}
DO(n,N+1){ DO(k,2*N+1){ printf("%4.0lf ",L[n][k]);} printf("\n");}
DO(n,N+1){DO(m,n)
          DO(l,2*N) L[n][l]=L[n][l+1]*(l+1)/2./(m+1);
} // n differentiations and normalization
DO(n,N+1){ DO(k,n+1) printf("%10.4f",L[n][k]); printf("\n");}
} // end of program

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