I spent a few hours today solving the Laplace and Schrodinger equation on a variety of domains, and kept finding solutions to the separated equations that were orthogonal (polynomials) in $L^2$, e.g. the quantum harmonic oscillator
$$-i u_t = u_{xx}-x^2 u$$
which yield the eigenvalue problem for the separable solutions
$$X''(x)+(\lambda-x^2)X(x)=0~~~~~~~~\text{or equally}~~~~~~~w''(x)-2xw'(x)+(\lambda-1)w=0$$
where $X(x)=w(x)e^{-x^2/2}$. The solutions to this equation are the Hermite polynomials, which are orthogonal in $L^2$ on $[-1,1]$. The Schrodinger equation for the hydrogen atom
$$i u_t = -\frac{1}{2}\nabla^2 u -\frac{u}{r}$$
has separable solutions in terms of the Laguerre and Legendre polynomials, again orthogonal, and Chebuchev polynomials appear in other circumstances. I'm wondering what it is about these physical problems that produces solutions with these properties, and how if at all these properties impact the physical phenomena they describe. Does this have physical significance or a physical explanation related to the symmetry of the problem?
The second answer to this question is quite relevant, but it's quite "hand-wavy", and I don't fully understand his argument.
