vanvleck 1993$ math
Mathematica 9.0 for Linux x86 (64-bit)
Copyright 1988-2013 Wolfram Research, Inc.
 -- Java graphics initialized -- 

vanvleck 1993$ math
Mathematica 9.0 for Linux x86 (64-bit)
Copyright 1988-2013 Wolfram Research, Inc.
 -- Java graphics initialized -- 

In[1]:= psi[n_,l_,m_,r_]=2/(n^2 Sqrt[Pochhammer[n-l,2l+1]]) (2r/n)^l LaguerreL[n-l-1,2l+1,2r/n] Exp[-r/n] SphericalHarmonicY[l, m, thet
a,phi]                                                                                                                                 

         1 + l  r l                                2 r
        2      (-)  LaguerreL[-1 - l + n, 1 + 2 l, ---] SphericalHarmonicY[l, m, theta, phi]
                n                                   n
Out[1]= ------------------------------------------------------------------------------------
                              r/n  2
                             E    n  Sqrt[Pochhammer[-l + n, 1 + 2 l]]

In[2]:= psic[n_,l_,m_,r_]=2/(n^2 Sqrt[Pochhammer[n-l,2l+1]]) (2r/n)^l LaguerreL[n-l-1,2l+1,2r/n] Exp[-r/n] SphericalHarmonicY[l, m, the
ta,-phi]                                                                                                                               

         1 + l  r l                                2 r
        2      (-)  LaguerreL[-1 - l + n, 1 + 2 l, ---] SphericalHarmonicY[l, m, theta, -phi]
                n                                   n
Out[2]= -------------------------------------------------------------------------------------
                               r/n  2
                              E    n  Sqrt[Pochhammer[-l + n, 1 + 2 l]]

In[3]:=                                                                                                                                
In[3]:= lm={{0,0},{1,1},{1,0},{1,-1},{2,2},{2,1},{2,0},{2,-1},{2,-2}}                                                                  

Out[3]= {{0, 0}, {1, 1}, {1, 0}, {1, -1}, {2, 2}, {2, 1}, {2, 0}, {2, -1}, {2, -2}}

In[4]:=                                                                                                                                
In[4]:= Table[Table[                                                                                                                   
        Integrate[Integrate[Integrate[psic[3,First[lm[[i]]],Last[lm[[i]]],r]  psi[3,First[lm[[j]]],Last[lm[[j]]],r] r^2 Sin[theta],{phi
,-Pi,Pi}],{theta,0,Pi}],{r,0,Infinity}],                                                                                               
        {i,1,9}],{j,1,9}]                                                                                                              

Out[4]= {{1, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, 
 
>    {0, 0, 0, 0, 1, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}, 
 
>    {0, 0, 0, 0, 0, 0, 0, 0, 1}}

In[5]:= MatrixForm[%]                                                                                                                  

Out[5]//MatrixForm= 1   0   0   0   0   0   0   0   0

                    0   1   0   0   0   0   0   0   0

                    0   0   1   0   0   0   0   0   0

                    0   0   0   1   0   0   0   0   0

                    0   0   0   0   1   0   0   0   0

                    0   0   0   0   0   1   0   0   0

                    0   0   0   0   0   0   1   0   0

                    0   0   0   0   0   0   0   1   0

                    0   0   0   0   0   0   0   0   1

In[6]:= Table[Table[                                                                                                                   
        Integrate[Integrate[Integrate[psic[3,First[lm[[i]]],Last[lm[[i]]],r] r Cos[theta] psi[3,First[lm[[j]]],Last[lm[[j]]],r] r^2 Sin
[theta],{phi,-Pi,Pi}],{theta,0,Pi}],{r,0,Infinity}],                                                                                   
        {i,1,9}],{j,1,9}]                                                                                                              

                                                                 9
Out[6]= {{0, 0, -3 Sqrt[6], 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, -(-), 0, 0, 0}, {-3 Sqrt[6], 0, 0, 0, 0, 0, -3 Sqrt[3], 0, 0}, 
                                                                 2
 
                             9                                          9
>    {0, 0, 0, 0, 0, 0, 0, -(-), 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, -(-), 0, 0, 0, 0, 0, 0, 0}, 
                             2                                          2
 
                                                       9
>    {0, 0, -3 Sqrt[3], 0, 0, 0, 0, 0, 0}, {0, 0, 0, -(-), 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0}}
                                                       2

In[7]:= MatrixForm[%]                                                                                                                  

Out[7]//MatrixForm= 

                    0            0            -3 Sqrt[6]   0            0            0            0            0            0

                                                                                       9
                                                                                     -(-)
                    0            0            0            0            0              2          0            0            0



                    -3 Sqrt[6]   0            0            0            0            0            -3 Sqrt[3]   0            0

                                                                                                                 9
                                                                                                               -(-)
                    0            0            0            0            0            0            0              2          0



                    0            0            0            0            0            0            0            0            0

                                   9
                                 -(-)
                    0              2          0            0            0            0            0            0            0



                    0            0            -3 Sqrt[3]   0            0            0            0            0            0

                                                             9
                                                           -(-)
                    0            0            0              2          0            0            0            0            0



                    0            0            0            0            0            0            0            0            0

In[8]:= Eigensystem[%%]                                                                                                                

                   9     9   9  9
Out[8]= {{-9, 9, -(-), -(-), -, -, 0, 0, 0}, {{Sqrt[2], 0, Sqrt[3], 0, 0, 0, 1, 0, 0}, {Sqrt[2], 0, -Sqrt[3], 0, 0, 0, 1, 0, 0}, 
                   2     2   2  2
 
>     {0, 0, 0, 1, 0, 0, 0, 1, 0}, {0, 1, 0, 0, 0, 1, 0, 0, 0}, {0, 0, 0, -1, 0, 0, 0, 1, 0}, {0, -1, 0, 0, 0, 1, 0, 0, 0}, 
 
                                         1
>     {0, 0, 0, 0, 0, 0, 0, 0, 1}, {-(-------), 0, 0, 0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0, 0, 0, 0}}}
                                      Sqrt[2]

In[9]:=                                                                                                                                
