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How can one formally prove the orthogonality of hydrogen atom wave functions?

I understand how the angular part is orthogonal, and I know that the radial part is orthogonal iff the quantum number $l$ is the same for both radial terms, i.e. we have $R_{n,\ell},R_{n',\ell}$ (see related post). And because this is given by the angular orthogonality, the result follows.

What I'm missing is the formal proof, and also some reference on the fact that when integrating the radial part, the two Laguerres have different arguments, i.e. the integral is not $$\int_0^\infty \rho^{\alpha}e^{-\rho}L_n^{(\alpha)}(\rho)L_m^{(\alpha)}(\rho)d\rho$$ as in Wikipedia, but each wave function provides a different $\rho$ depending on the energy level $n$ via $\rho=\frac{2r}{na_0}$. This seem to translate to a nontrivial change in the expression for the integral.

I was also looking at the Rodriguez formula, and it seems the difference in arguments translates to nontrivial change of these terms as well.

This is why I'm looking for a thorough and detailed proof of the orthogonality of the radial part, assuming $\ell=\ell'$ from the spherical harmonics, which i was unable to find just by Googling or through our lecture notes.

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    $\begingroup$ As far as I know, and precisely because they have different arguments, it is neigh impossible to prove orthogonality directly. Instead one relies on properties of solutions to Sturm-Liouville problems. $\endgroup$ – ZeroTheHero Oct 31 '17 at 14:14
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    $\begingroup$ You may find this article interesting: A Laguerre Polynomial Orthogonality and the Hydrogen Atom. $\endgroup$ – Michael Seifert Oct 31 '17 at 14:16
  • $\begingroup$ @MichaelSeifert nice reference. $\endgroup$ – ZeroTheHero Oct 31 '17 at 14:34
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    $\begingroup$ @ZeroTheHero do you mean that the solution to a hermitian operator forms a complete set and stuff like that? $\endgroup$ – Yoni Oct 31 '17 at 14:46
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    $\begingroup$ This is how I remember it and to quote Wiki: "Orthogonality follows from the fact that Schrödinger's equation is a Sturm–Liouville equation (in Schrödinger's formulation) or that observables are given by hermitian operators (in Heisenberg's formulation)". Seeking direct proof of orthogonality for complicated functions like the hydrogen atom wave functions seems therefore a rather futile waste of time, in my humble opinion. As solutions to a Sturm–Liouville equations, they are intrinsically orthogonal. $\endgroup$ – Gert Oct 31 '17 at 16:18
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You do have a nontrivial point in that the orthogonality relation for the Laguerre polynomials as they appear in the hydrogenic eigenfunctions, $$ \int_0^\infty L_{n-\ell-1}^{(2\ell+1)}(2r/n)L_{n'-\ell-1}^{(2\ell+1)}(2r/n') e^{-\frac{n+n'}{nn'}r} r^{2\ell+2} \mathrm dr = 0 \qquad \text{for }n\neq n', $$ is structurally very different to the standard orthogonality relation, $$ \int_0^\infty L_{n}^{(2\ell+1)}(r)L_{n'}^{(2\ell+1)}(r) e^{-r} r^{2\ell+1} \mathrm dr = 0 \qquad \text{for }n\neq n', $$ as provided in e.g. Wikipedia or the DLMF.

As pointed out in the comments, the orthogonality of hydrogenic wavefunctions follows directly from the general Sturm-Liouville theory (they're eigenfunctions of a hermitian Sturm-Liouville operator with different eigenvalues and that's all you should need) so that an elementary proof of the altered orthogonality relation seems a bit of a waste of time.

However, that obviously hasn't stopped other people from looking for it - one suitable example appears to be

A Laguerre Polynomial Orthogonality and the Hydrogen Atom. Charles F. Dunkl. arXiv:math-ph/0011021.

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It can be shown that the operator $H=\Delta + \frac{1}{r}$ is essentially self-adjoint on a domain containing the eigenfunctions $\psi_{nlm}$. This means in particular that it is symmetric for any $\psi_{nlm}$ and $\psi_{n'lm}$. From this, the normality of the different eigenspaces follows as usual, like for any symmetric or self-adjoint operator.

The arguments involved are a bit tedious but for the most part just use basic Hilbert space techniques. See for example chapter 9 of Brian C. Hall's quantum mechanics book, especially sections 9.8 and 9.9.

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