Verify the relation from Sakurai pr. 5.16(a) using the ground state of the 3D SHO.  Verification of the potential derivative expectations for higher values of n do not complete in reasonable times.

ClearAll[hbar, m, omega, x0,display, psi, eta, x, y, z, psi3, braket, r, theta, phi, braOpKet, expDv]

$Assumptions=(And@@(#>0&)/@{hbar, m, omega});
display = {hbar -> ℏ, omega -> ω} ;
x0 = Sqrt[hbar/m/omega] ;
(*eta = 1/x0 ;*)
psi[n_] := (1/Sqrt[ 2^n n!]) (m omega/Pi/hbar)^(1/4) Exp[-#^2/2/x0^2] HermiteH[n,#/x0] & ;
(*x = #1 Sin[#2] Cos[#3] &;
y = #1 Sin[#2] Sin[#3] &;
z = #1 Cos[#2] & ;
psi3[n_] := psi[n,x[#1, #2,#3]] psi[n,y[#1, #2,#3]] psi[n,z[#1, #2,#3]] & ;
braket[f_, g_] := Integrate[ r^2 Sin[theta] f[r,theta, phi] g[r, theta, phi], {r, 0, Infinity}, {theta, 0, Pi}, {phi, 0, 2 Pi}] ;*)

psi3[n_] := (*( m omega/hbar )^(-3/4)*)psi[n][#1] psi[n][#2] psi[n][#3] & ;
braket[f_, g_] := Integrate[  f[x,y,z] g[x,y,z], {x, -Infinity, Infinity},{y, -Infinity, Infinity},{z, -Infinity, Infinity}] ;
braOpKet[f_, o_, g_] := braket[ f, o[#1,#2,#3] g[#1,#2,#3] & ] ;


(psi3[#][x,y,z] &/@ (Range[3]-1)) /. display // Column
braket[psi3[#], psi3[#]] & /@ (Range[3]-1) // Simplify

((((2 Pi hbar^2/m) (psi3[#][0,0,0])^2) &/@ (Range[3]-1)) // Simplify)/. display

expDv[n_] :=braOpKet[psi3[n], m omega^2 Sqrt[#1^2 + #2^2 + #3^2] &, psi3[n]] ;

(*too slow for n>1*)
((expDv[#] // Simplify)&/@ (Range[1]-1)) /. display

(*HermiteH[#,x]&/@ (Range[4]-1)*)