Group theory and Hydrogen-like wave functions

Question

I'm reading Pauling and Wilson's QM, in one place it is written that the $n^2$ wave functions $\psi_{n,l,m}(r,\theta,\phi)$ (for an H-like atom) with the same value of $n$ form a completed group. How? What is the law of composition for which they even form a group? I couldn't get any references online or offline. Any suggestions/references are welcome.

Answer 1

They are a complete set, or grouping (not group), of functions solving the Schroedinger equation for such an atom with energy characterized by that posited n. That is, they are a complete basis of all solutions for that energy.

When it comes to symmetry groups commuting with the hamiltonian, there is first SO(3), with quadratic Casimir ${\mathbf L}\cdot {\mathbf L}$ eigenvalues $\ell (\ell+1)$, $\ell= 0,1,2,...,n-1$, that is, n different irreducible representations of SO(3), each with dimension $2\ell+1$. One of the three generators of SO(3), namely $L_z$ has eigenvalues m, $|m|\leq \ell$, and connects states in a representation labelled by $\ell$; but cannot connect states for different $\ell$s—to do that you need generators of the larger symmetry of these atoms, SO(4)~ SO(3)×SO(3).

You might easily show the totality of states then is $\sum_{\ell=0}^{n-1} (2\ell +1)=n^2$. They comprise a complete expansion basis for all solutions with that energy.