Proof of Orthogonality of Hydrogen Atom Wave Functions

Question

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.

Answer 1

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.

Answer 2

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.