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

Table[Integrate[psi[n,x]^2,{x,-Infinity,Infinity}],{n,0,4}]

Out[3]= {1, 1, 1, 1, 1}


Table[Table[Integrate[psi[m,x] psi[n,x],{x,-Infinity,Infinity}],{n,0,3}],{m,0,3}]

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

Table[Simplify[(-D[D[psi[n,x],x],x] + x^2 psi[n,x])/psi[n,x]],{n,0,4}]

Out[5]= {1, 3, 5, 7, 9}

Table[Integrate[psi[n,x] x psi[n,x],{x,-Infinity,Infinity}],{n,0,4}]

Out[6]= {0, 0, 0, 0, 0}

Table[Integrate[psi[n,x] x^2 psi[n,x],{x,-Infinity,Infinity}],{n,0,4}]

         1  3  5  7  9
Out[7]= {-, -, -, -, -}
         2  2  2  2  2

Table[Integrate[psi[n,x] (1/I) D[psi[n,x],x] ,{x,-Infinity,Infinity}],{n,0,4}]

Out[8]= {0, 0, 0, 0, 0}

Table[Integrate[psi[n,x] (-D[ D[psi[n,x],x],x]) ,{x,-Infinity,Infinity}],{n,0,4}]

          1  3  5  7  9
Out[10]= {-, -, -, -, -}
          2  2  2  2  2

Table[Integrate[psi[2,x] x psi[n,x],{x,-Infinity,Infinity}],{n,0,5}]

                        3
Out[11]= {0, 1, 0, Sqrt[-], 0, 0}
                        2

Table[Integrate[psi[2,x]  (1/I) D[psi[n,x],x],{x,-Infinity,Infinity}],{n,0,5}]

                           3
Out[12]= {0, I, 0, -I Sqrt[-], 0, 0}
                           2
---for these last two: x & p essentially are (a^dagger +- a) and so connect only to states
one above or one below the initial state, i.e., all the undisplayed integrals are zero.