Integral involving orthogonal functions (quantum harmonic oscillator eigenstates)

I have a question about the capabilities of the Integrate command for general parameters.

We know that there is a system of functions:

[CurlyPhi][n_, x_] = 
  HermiteH[n, x]/Sqrt[2^n*n!*Sqrt[[Pi]]]*Exp[-0.5 x^2];

and we know that for nonnegative integer n parameters, these functions are orthonormal in the usual sense of integration. However the following integral:

Integrate[[CurlyPhi][a, x]*[CurlyPhi][b, 
   x], {x, -[Infinity], [Infinity]}, 
 Assumptions -> {a [Element] Integers, a >= 0, b [Element] Integers,
    b >= 0} ]

supposedly does not converge (error Integrate::idiv). The correct result should be a Kronecker-delta function.

Have I made a coding error in the assumptions, or should some other command than Integrate be used?

φ[n_ /; NumericQ, x_] = 
  HermiteH[n, x]/Sqrt[2^n*n!*Sqrt[π]]*Exp[-0.5 x^2];
Unprotect[Integrate];
 Integrate[φ[n1_, x]*φ[n2_, 
      x], {x, -∞, ∞}] /; (n1 =!= n2 ) = 0;
 Integrate[φ[n1_, x]*φ[n1_, 
     x], {x, -∞, ∞}] = 1;
Protect[Integrate];

Integrate[φ[n1, x]*φ[n2, x], {x, -∞, ∞}]
Integrate[φ[n1, x]*φ[n1, x], {x, -∞, ∞}]