Integral involving orthogonal functions (quantum harmonic oscillator eigenstates)

Question

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?

Answer 1

What do mean by "not converge"? MMA simply gives the input back because a and b have no value. MMA can not evaluate the integral symbolically. For pos. integer a and b it will evaluate. If you need symbolic evaluation, you can teach MMA by e.g.:

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];

Then the following evaluate as you request:

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

By the way, "convergence" has quite another meaning, look it up.

Answer 2

The general solution to any Schrödinger eigenvalue problem with boundary value of the current operator over the boundary equal to zero and discrete discrete spectrum rests on the fact, that the general solution is not square integrable.

For the general algebraic eigenvalues, the solution, starting regularly at $x=0$, explodes exponentially on one of the boundary points. At the eigenvalue it changes its sign from $\pm \infty \to \mp \infty$.

The general solution of the oscillator problem is

    DSolve[(-1/2 D[#, x, x] + 1/2 x^2 # == (n + 1/2) # &)
      [E^(-x^2/2) g[x]], g[x], x]

    {{g[x] ->  C[1] HermiteH[n, x] + 
               C[2] Hypergeometric1F1[-(n/2), 1/2, x^2]}}

These functions ar special cases of $_1F_1$ and difficult to handle for integer parameters. But $C[2]\to 0, n\in N_+$ is the classical solution on $\mathbb R$ with $\partial^{n}\psi(\pm \infty)=0 \implies j(\pm \infty)=0$

The general solutions with no measure factor imposed, are parabolic cylinder functions

     DSolve[(-1/2 D[ #, x, x] + 1/2 x^2 # == (n + 1/2) # &)[g[x]], g[x], x]

      {{g[x] -> C[2] ParabolicCylinderD[-1 - n, I Sqrt[2] x] + 
                C[1] ParabolicCylinderD[n, Sqrt[2] x]}}

Both subspaces are identical with the $e^{\pm x^2/2} H[n,x]$, decaying on one side and exploding on the other side. All $L^2$-integrals will be considered non existent for general parameters.

Demo, Log scale on x range

 Manipulate[
    Plot[ParabolicCylinderD[n, Sqrt[2] Sinh[x]], {x, -3, 3}], 
  {{n, 3.998}, 3.5, 5.5}, ControlPlacement -> Top]

Near to an integer eigenvalue there is a minimum near zero. Finding the eigenvalue means shifting the turning point to $\infty$. As a mechanical problem it amounts to threading the needle at infinity with an elastic rod by control of the zero point of the elastic restoring force $-x \psi$, small at the origin, stiff for large x. This is an alternative aspect of the famous index theorems for positive differential operators on manifolds with boundaries, that rest on the diffusion processes of random paths.