shoR[n_,m_,r_,phi_]=r^Abs[m] LaguerreL[n, Abs[m], r^2] Exp[-r^2/2] Exp[I m phi]/Sqrt[ Pi Pochhammer[n+1, Abs[m]]]


Table[Integrate[Integrate[shoR[n,3,r,-phi] shoR[3,3,r,phi] r ,{phi,-Pi,Pi}],{r,0,Infinity}],{n,0,5}]
Out[2]= {0, 0, 0, 1, 0, 0}
Note: dx dy = r dr dphi = area element

Table[Integrate[Integrate[shoR[n,0,r,-phi] shoR[3,0,r,phi] r ,{phi,-Pi,Pi}],{r,0,Infinity}],{n,0,5}]
Out[3]= {0, 0, 0, 1, 0, 0}

Table[Integrate[Integrate[shoR[n,0,r,-phi] shoR[n,0,r,phi] r ,{phi,-Pi,Pi}],{r,0,Infinity}],{n,0,5}]    
Out[4]= {1, 1, 1, 1, 1, 1}

Table[Integrate[Integrate[shoR[3,m,r,-phi] shoR[3,m,r,phi] r ,{phi,-Pi,Pi}],{r,0,Infinity}],{m,0,5}]    
Out[5]= {1, 1, 1, 1, 1, 1}

shoX[n_,x_]=HermiteH[n,x] Exp[-x^2/2] /Sqrt[Sqrt[Pi] 2^n n!]


Table[Integrate[shoX[n,x] shoX[3,x] ,{x,-Infinity,+Infinity}],{n,0,5}]
Out[6]= {0, 0, 0, 1, 0, 0}

Table[Integrate[shoX[n,x] shoX[n,x] ,{x,-Infinity,+Infinity}],{n,0,5}]
Out[7]= {1, 1, 1, 1, 1, 1}

ContourPlot[sho[5,x] shoX[0,y],{x,-5,5},{y,-5,5},AspectRatio->Automatic]    
ContourPlot[Re[shoR[1,3,Sqrt[x^2+y^2], ArcTan[x, y]]],{x,-5,5},{y,-5,5},AspectRatio->Automatic] 
ContourPlot[Abs[shoR[1,3,Sqrt[x^2+y^2], ArcTan[x, y]]],{x,-5,5},{y,-5,5},AspectRatio->Automatic] 
Note: Abs[Exp[I phi]]=1

a=Table[Integrate[Integrate[shoR[k,(5-2 k),r,-phi] shoX[5,r Cos[phi]] shoX[0,r Sin[phi]] r ,{phi,-Pi,Pi}],{r,0,Infinity}],{k,0,2}]

                           5
                     -Sqrt[-]
              1            2   Sqrt[5]
Out[21]= {---------, --------, -------}
          4 Sqrt[2]     4         4


f[r_,phi_]=a[[1]] 2 Re[shoR[0,5,r,-phi]]+a[[2]] 2 Re[shoR[1,3,r,-phi]]+a[[3]] 2 Re[shoR[2,1,r,-phi]] 
ContourPlot[f[Sqrt[x^2+y^2], ArcTan[x, y]],{x,-5,5},{y,-5,5},AspectRatio->Automatic] 