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, theta,phi]

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, theta,-phi]

psi[2,1,0,r] psic[2,1,0,r] r^2

          4           2
         r  Cos[theta]
Out[13]= --------------
                r
            32 E  Pi

Integrate[Integrate[Integrate[% r^2 Sin[theta],{phi,-Pi,Pi}],{theta,0,Pi}],{r,0,Infinity}]


a=4/5*{Sqrt[3/5],Sqrt[1/15],-Sqrt[1/3]}  
b=-3/5*{Sqrt[1/10],Sqrt[2/5],Sqrt[1/2]}   
c=a+b      
N[Norm[c]]
Out[11]= 1.
N[c^2]                                                                 

Out[12]= {0.184849, 0.0298993, 0.785252}

aa={Sqrt[3/5],Sqrt[1/15],-Sqrt[1/3]}  
Norm[aa]                                                               

Out[25]= 1

aa^2                                                                   

          3  1   1
Out[26]= {-, --, -}
          5  15  3


h={{2,I},{-I,2}}
{ew,vs}=Eigensystem[h]
Out[15]= {{3, 1}, {{I, 1}, {-I, 1}}}
v3=vs[[1]]/Norm[vs[[1]]]
v1=vs[[2]]/Norm[vs[[2]]]
1/Sqrt[2] v3+ 1/Sqrt[2] v1
Out[18]= {0, 1}
1/Sqrt[2] v3 Exp[-I 3 E0 t] + 1/Sqrt[2] v1  Exp[-I  E0 t] 
