K+-  =  Shift Operatoren for total angular momentum l: Calculations

B. Schnizer

schnizer@itp.tu-graz.ac.at

2008-11-10

Clear[a, b, f1, f2, f3, sys, l, so, su, sp, sm]

The recurrence relaiton (70) is expressed as the following substitution rule:

\(r2\  = \ 
      x\ LegendreP[l, m, 
            x]\ \  -> \ 
\ \ \ \ \ \ \(\(\((\ \((l\  + \ 1\  - \ 
                    m)\)\ LegendreP[l + 1, m, \ 
                  x]\ \  + \ \((l\  + \ m)\)\ \ LegendreP[l - 1, m, 
                  x]\ )\)/\((2  l\  + \ 1)\)\)\(\ \)\)\)

The recurrence relaiton (71) is derived in two steps:

f1  =  D[LegendreP[l, m, x] , x]

f2  =  
      Cancel[(1  -  x^2) FullSimplify[f1  /.  r2]] // 
        Simplify

f3  =  r2[[2]]

These two equation are solved for  (Subscript[P, l + 1]^m)(x) and (Subscript[P, l - 1]^m)(x)while we keep in  mind that 
a = x LegendreP[l,m,x]  and  b = (1 - x^2) D[LegendreP[l,m,x] ,x] ,

sys  =  {f3  ==  a,  f2  ==  b}

so  =  
      Solve[sys,  {LegendreP[1 + l, m, x], LegendreP[-1 + l, m, x]}] // 
        Simplify

Now the original meaning of a and b is cast into the following substitution rule:

\(su\  = \ {\ 
        a\  -> \ x\ LegendreP[l, m, 
              x], \ 
	\ \ b\  -> \ \ \((1\  - \ x^2)\)\ D[
              LegendreP[l, m, x]\ , x]\ \ }\)

sop  =  
      Simplify[Simplify[ -((-a + b - a l)/(1 + l  -  
                      m))  /. su]  /. r2]

som  =  
      Simplify[Expand[
            Simplify[ -((-b - a l)/(l +  m)) /.  su]] /. 
          r2]

sp  =  Simplify[Simplify[ a - b + a l  /. su]  /. r2]

sm  =  
      Simplify[ Expand[Simplify[ b + a l /.  su]] /. r2]

This gives equations (68) and (69):

\(TraditionalForm\`\(K\_+\)\) P[l,m,x] =  (x (l + 1) - (1 - x^2) D[,x]) P[l,m,x]  = (l - m + 1) P[l+1,m,x], 
\(TraditionalForm\`\(K\_-\)\) P[l,m,x] = (x l + (1 - x^2) D[,x]) P[l,m,x] = (l + m)P[l-1,m,x]