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I have read that the eigenfunctions of any hermitian operator can be used as a basis to express any function, but I have only ever really seen the eigenfunctions of the Hamiltonian used. Why is this? Could I express the hydrogen electron in terms of momentum eigenfunctions?

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  • $\begingroup$ That' s more of physicist speak for "I don't care that the mathematicians have spent a century on figuring out functional analysis for me and that it's actually much more complicated than I ever want to know, I just assume it's slap happy simple and move on.". The problem lies in "any function", which is a mighty large space and which is definitely not correct. If you replace it with "most functions I will ever have to concern myself with in physics", then you are getting somewhere, though. $\endgroup$ – CuriousOne Jul 11 '16 at 3:04
  • $\begingroup$ You use spherical harmonics when expressing the hydrogen atom solution, right? Spherical harmonics are eigenfunctions of $L_z$ and $L^2$. $\endgroup$ – valerio Jul 11 '16 at 6:23
  • $\begingroup$ Wavefunction in the coordinate representation can be on the physical level of rigorousness is the decomposition in terms of the eigenfunctions of the coordinate operator. Analogously its Fourier transform - the wavefunction in the momentum represenetation, is a decomposition in terms of the eigenfunctions of the momentum operator. Physical level of rigorousness because those "eigenfunctions" don't belong to the Hilbert space. $\endgroup$ – OON Jul 11 '16 at 7:51
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General wave functions can be expressed in terms of any set of eigenfunctions.

But for bound systems, the energy eigenfunctions have a couple of appealing properties that make them popular:

  1. The energy eigenfunctions are the solutions to the time-independent problem, so you can work on a steady-state system. This often makes the math a lot easier.

  2. The energy eigenvalues may be easily accessible in the lab. So, after making some necessary assumptions and approximation to get through the math you can check that your results agree well with reality. And spectroscopy is an incredibly fine-grained tool, so you can check that your detailed results agree with reality as you relax one approximation after another and approach an analytic or numeric solution to the full physics of the situation.

I don't have any particular thoughts about the use of energy eigenfunction over any other set for unbound systems, however.

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