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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.

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    $\begingroup$ Not being glib with you, but The Unreasonable Effectiveness of Mathematics in the Natural Science. Your question is quite broad, perhaps limit it to say, just the wave functions of a 1 d potential well, it should be the same principle, no matter how complicated later systems are. $\endgroup$
    – user140606
    Commented Jan 12, 2017 at 0:30
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    $\begingroup$ You're right. I'll try to make it more specific. I just noticed basically the same result in solving a bunch of rather unrelated problems, and thought (as ACuriousMind has to some extent demonstrated) that there might be a very broad conceptual explanation for the pattern. $\endgroup$
    – JAustin
    Commented Jan 12, 2017 at 0:39
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    $\begingroup$ I should say that ACuriousMind knows a lot, lot more than I do.and that the spectral theorem is widely applicable. But I am going to guess that , as you can see from my reference to Wegner's quote, ACM's answer may not be explainable much deeper, without heading towards philosophy, if you are particularly interested in the connection between physical systems and how math explains them. I thought that was actually part of your question, the way you worded it. Ironically I came on line to delete my comment above, as I thought the answer sorted it all out for you:) best of luck with it anyway. $\endgroup$
    – user140606
    Commented Jan 12, 2017 at 1:41
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    $\begingroup$ @TaMeCeart Probably a typo but the quote is from Wigner. $\endgroup$ Commented Jan 12, 2017 at 1:52

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You have discovered the spectral theorem - (generalized) eigenvectors of self-adjoint operators like the Hamiltonian are orthogonal to each other and if the spectrum is discrete, they form an orthonormal basis.

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